# Why the Rainbow Sits at 42 Degrees > A rainbow is one of the few everyday sights whose explanation is completely within reach: it needs Snell's law, a circle, and a single derivative. We follow one ray of sunlight into one spherical raindrop, watch it refract, reflect once off the back and refract out again, and add up the three turns it makes to get the deviation angle. Then we slide the entry point across the face of the drop and watch that angle fall, flatten and rise: the minimum is where the outgoing rays pile up, and it is the reason a bow is bright at all. Setting the derivative to zero gives the angle exactly, and because water bends violet a little harder than red, every colour has its own minimum. That is the width of the bow, the order of its colours, and why the sky inside it is brighter than the sky outside. - Canonical watch page: [Why the Rainbow Sits at 42 Degrees](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-08-28T22:51:54.000Z - Updated: 2026-08-28T22:51:54.000Z - Duration: PT556S (9 minutes 16 seconds) - Chapters: 3 - Views: 1 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14V024XDHCN3G9SQFQ9ZS42/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14V024XDHCN3G9SQFQ9ZS42/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14V024XDHCN3G9SQFQ9ZS42/0/dark/poster.jpg) ## Description Derive the rainbow's 42 degree angle from one raindrop: refraction, one internal reflection, and the minimum of the deviation curve. ## Chapters - [00:00–01:36.92 · The Bow in the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=0) - [01:36.92–06:43.117 · One Drop, One Ray](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667) - [06:43.117–09:16 · Assembling the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334) ## Transcript ### [00:00 · The Bow in the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=0) A rainbow is not a coloured object hanging over a particular field. It is a viewing geometry that follows the observer. This is the destination. The visible bow is an arc centred on the point opposite the sun, and its outer red edge sits about forty two degrees from that direction. We now have to explain why. Here is the observation to explain. To see a rainbow, the sun must be behind you. That is a fact about geometry, and it is our first clue. Stand with the sun at your back and follow the shadow of your head away from the sun. That line is the anti-solar direction, the centre line for every rainbow you see. A drop out here catches the sunlight and sends its red light back to your eye. The angle between that returning ray and the centre line is about forty two degrees. Rotate that direction right around the centre line and the eligible drops form a thin conical shell, whose two edges are the red lines of this side view. Now step sideways. The cone moves with your eye, and a different drop joins the new line of sight. The rainbow is fixed by an angle, not by a place in the shower. Raise the sun and the anti-solar direction tilts down. The whole bow then follows it below the horizon. Lower the sun and the bow rises again. So the question has two parts. Why is the angle about forty two degrees, and why does the light bunch up there instead of spreading evenly across the sky? Both answers are inside one raindrop. ### [01:36.92 · One Drop, One Ray](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667) Refraction is the bending of light as it crosses into water. An incoming ray and the centre of a spherical drop determine one flat slice where we can follow that bend. Seen head on, the slice is a circle. The same circle, centre, and ray will stay with us all the way to the answer. At the surface, the normal is the radius through the point of contact. The incoming ray makes the incidence angle i with that normal. At that surface the light turns toward the normal as it enters water, so the angle inside, r, is smaller than i. Snell's law gives the size of that bend. Sine i equals n times sine r, and water has n about one point three three. The ray reaches the back wall. Most light escapes and is lost, but a small fraction reflects. Symmetry makes the incoming and reflected angles there equal to the same r. At the front surface the reflected ray meets one more radius. It leaves water, opens from r back to i, and becomes the ray that can reach an eye. Each pair of angle marks appeared only for the event it named. The exit pair has now carried the same law to the last surface, so it can leave too. Now measure how far the ray turns at each event. At the entrance it turns through i minus r. The reflection is the large turn. Equal angles r leave a straight angle minus two r between the old direction and the new one. Leaving the drop adds another i minus r. All three local turns now stand beside the path: i minus r, one hundred eighty minus two r, and i minus r. Add those three turns. The deviation is one hundred eighty degrees plus two i minus four r. The angle D compares the original forward direction with the direction that actually leaves the drop. A ray through the centre has i and r both zero, so the expression gives a turn of one hundred eighty degrees, straight back. Snell's law also makes r the inverse sine of sine i over n, leaving i as the only free angle. Set the entry low on the drop, at twenty degrees. The ray returns almost the way it arrived, with a deviation of about one hundred sixty degrees. Slide the entry point upward. The deviation falls through one hundred fifty, one hundred forty five, and one hundred forty. Then it slows. There, it has stopped coming down. Keep sliding toward the rim. Only after the minimum does the curve climb again, and it keeps climbing as the entry point approaches the edge. The flat bottom makes the minimum important. These two rays enter at forty eight and seventy degrees, far apart on the face of the drop. Yet their outgoing directions differ by less than a quarter of a degree. A broad band of entry points therefore sends light back in nearly one direction, so the light piles up instead of spreading thin. That concentration is the rainbow. To locate it exactly, use the flat curve condition: at the minimum, the derivative of D with respect to i is zero. Solving that condition says d r by d i must equal one half. The angle inside must change half as fast as the incidence angle outside. Differentiate Snell's law. Cosine i equals n cosine r times d r by d i. Substituting one half gives cosine i equal to n cosine r over two. Now square both sides. Four cosine squared i equals n squared cosine squared r. Replace cosine squared r with one minus sine squared r. Snell's law then replaces n squared sine squared r with sine squared i. Finally, sine squared i is one minus cosine squared i. Collect the cosine terms. Cosine i is the square root of n squared minus one, divided by three. For water, i is fifty nine point four degrees and r is forty point two. Putting both into the turn formula gives a minimum deviation of about one hundred thirty eight degrees. Return to the special ray at the minimum. Its path is still the same path through the same drop; only the chosen entry point has moved. Carry the incoming direction through the exit point. Forward to backward is a straight angle of one hundred eighty degrees. The ray has already turned through D, about one hundred thirty eight degrees. Subtract that deviation from the straight angle. The remaining angle between the outgoing ray and straight back is forty two degrees. That is the promised number. Snell's law fixes the path inside one circle, the three turns create a minimum at one hundred thirty eight degrees, and the view back toward the sun leaves forty two degrees. ### [06:43.117 · Assembling the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334) White sunlight contains many colours, and water bends each one by a slightly different amount. Red bends least; violet bends most. Follow the least-deviated red path through the drop, then the violet path. They enter close together, separate through the two refractions, and leave in different directions. For red, n is one point three three one and the viewing angle is forty two point three degrees. For violet, n is one point three four four and the viewing angle is forty point six degrees. A single drop therefore does not send an entire rainbow to one eye. Its red and violet leave along different lines, so one viewpoint receives at most one narrow part of that colour fan. Put two drops back in the shower. One is higher above the anti-solar line and one is lower, while the sunlight reaches both from behind the observer. The higher drop can send its red ray to the eye. Its violet ray passes over the observer's head and is missed. The lower drop can send violet to the same eye. Its red ray passes below the observer's feet. Red therefore arrives from the larger viewing angle and sits on the outside of the primary bow. Violet arrives from the smaller angle and sits inside. The anti-solar point is the centre of the geometry. The horizon lies above it whenever the sun is above the horizon. Rotate the primary viewing directions around that centre. Red traces the outer arc and violet traces the inner arc, with every point supplied by a different drop in the shower. The complete geometry is circular, but the ground hides the portion below the horizon. What remains visible is an arc, not a full ring. Look just inside the primary bow, then just outside it. The inside is brighter because a broad band of rays returns on that side of the minimum. The region just outside receives far fewer primary rays. A second internal reflection creates a secondary bow farther out. Its colour order reverses: violet is outside and red is inside. One drop supplies the path, the minimum deviation supplies the brightness, dispersion supplies the colours, and rotation around the anti-solar point supplies the arc. Together they make the rainbow in the sky. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14V024XDHCN3G9SQFQ9ZS42/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [The Bow in the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=0) Span: 00:00–01:36.92 (0s–96.92022916666667s). #### Objects - axis: an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in sky (start=(1.0, 1.0), end=(8.1, 1.0)) - axis\_two: a Line \[cyan\] drawn in sky (start=(1.0, 1.65), end=(7.8, 1.65), dashed=True) - below\_bow: a Point \[magenta\] labelled "upright("bow below horizon")" drawn in sky (location=(5.5, -0.72)) - bow\_angle: an Angle \[green\] labelled "42 degree" drawn in sky (vertex=(1.0, 1.0), sides=((8.1, 1.0), (4.492780679743753, 4.144913849886634)), radius=1.4) - card: a Title that says "Optics — Why the Rainbow Sits at 42 Degrees" - drop: a Circle \[blue\] drawn in sky (center=(4.492780679743753, 4.144913849886634), radius=0.25, filled=True) - drop\_two: a Circle \[cyan\] drawn in sky (center=(4.084051025731187, 4.426892016389262), radius=0.21, filled=True) - eye: a Point \[text\] labelled "upright("you")" drawn in sky (location=(1.0, 1.0)) - eye\_two: a Point \[cyan\] labelled "upright("new viewpoint")" drawn in sky (location=(1.0, 1.65)) - high\_axis: an Arrow \[magenta\] drawn in sky (start=(1.0, 1.0), end=(7.8, -0.9)) - into\_drop: an Arrow \[yellow\] drawn in sky (start=(2.5927806797437527, 4.144913849886634), end=(4.182780679743753, 4.144913849886634)) - lower\_shell: a Line \[red\] drawn in sky (start=(1.0, 1.0), end=(3.229434476432183, -1.0073918190765747), dashed=True) - preview: a Figure (x\_range=(-3.4, 3.4), y\_range=(-1.5, 2.1), aspect=(6.8, 3.6)) - preview\_answer: a Math \[text\] that says "$theta approx 42 degree$" - preview\_caption: a Tex \[text\] that says "The visible bow is a coloured arc around the anti-solar point." - preview\_head: a Point \[gray\] labelled "upright("anti-solar point")" drawn in preview (location=(0.0, -1.2)) - preview\_heading: a Heading that says "The Destination" - preview\_horizon: a Line \[gray\] labelled "upright("horizon")" drawn in preview (start=(-3.3, 0.0), end=(3.3, 0.0), dashed=True) - preview\_red: a ParametricCurve \[red\] drawn in preview (function=\, t\_range=(0.411516846067488, 2.7300758075223053)) - preview\_violet: a ParametricCurve \[magenta\] drawn in preview (function=\, t\_range=(0.4569091986615711, 2.684683454928222)) - question: a Tex \[text\] that says "Why does the bow keep the same angle from the shadow of your own head, whichever shower you happen to be looking at?" - ray\_two: an Arrow \[cyan\] drawn in sky (start=(4.084051025731187, 4.426892016389262), end=(1.0, 1.65)) - sky: a Figure (x\_range=(0.0, 8.5), y\_range=(-1.3, 5.2), aspect=(8.5, 6.5)) - sun\_high: an Arrow \[yellow\] labelled "upright("sunlight")" drawn in sky (start=(0.2, 4.8), end=(2.4, 4.8)) - sun\_mid: an Arrow \[yellow\] drawn in sky (start=(0.2, 4.2), end=(2.4, 4.2)) - to\_eye: an Arrow \[red\] drawn in sky (start=(4.302780679743752, 3.974913849886634), end=(1.1400000000000001, 1.12)) #### Beats ##### [00:00](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=0) Narration: A rainbow is not a coloured object hanging over a particular field. It is a viewing geometry that follows the observer. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=1.5): card: enter:write-left-to-right. - [00:7.163](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=7.1635): card is hidden from the screen — left the board. ##### [00:7.763](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=7.7635) Narration: This is the destination. The visible bow is an arc centred on the point opposite the sun, and its outer red edge sits about forty two degrees from that direction. We now have to explain why. Board: Unchanged from the preceding beat in this scene. Actions: - [00:7.763](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=7.7635): preview\_heading is shown on the screen, written out. - [00:7.763](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=7.7635): preview is shown on the screen, written out. - [00:10.503](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=10.503): preview\_horizon is shown on the screen, written out. - [00:11.502](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=11.501999999999999): preview\_red is shown on the screen, drawn. - [00:11.502](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=11.501999999999999): preview\_caption is shown on the screen, written out. - [00:11.802](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=11.802): preview\_violet is shown on the screen, drawn. - [00:11.838](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=11.838): preview\_head is shown on the screen, written out. - [00:15.659](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=15.659): preview\_answer is shown on the screen, written out. - [00:15.972](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=15.972000000000001): preview\_answer (the "42" part) is emphasized. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview is hidden from the screen — left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_horizon is hidden from the screen — preview left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_head is hidden from the screen — preview left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_red is hidden from the screen — preview left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_violet is hidden from the screen — preview left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_answer is hidden from the screen — left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_caption is hidden from the screen — left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_heading is hidden from the screen — left the board. - [00:20.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.3375): preview\_answer (the "42" part) is no longer emphasized. ##### [00:20.938](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.9375) Narration: Here is the observation to explain. To see a rainbow, the sun must be behind you. That is a fact about geometry, and it is our first clue. Board: Unchanged from the preceding beat in this scene. Actions: - [00:20.938](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=20.9375): question is shown on the screen, written out. - [00:30.585](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=30.5855): question moves to a new place on the board. ##### [00:31.185](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=31.185499999999998) Narration: Stand with the sun at your back and follow the shadow of your head away from the sun. That line is the anti-solar direction, the centre line for every rainbow you see. Board: question — a Tex \[text\] that says "Why does the bow keep the same angle from the shadow of your own head, whichever shower you happen to be looking at?" Actions: - [00:31.185](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=31.185499999999998): sky is shown on the screen, written out. - [00:32.114](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=32.114): sun\_high is shown on the screen, written out. - [00:32.364](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=32.364): sun\_mid is shown on the screen, written out. - [00:33.995](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=33.995): eye is shown on the screen, written out. - [00:36.921](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=36.921): axis is shown on the screen, drawn. ##### [00:41.665](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=41.665499999999994) Narration: A drop out here catches the sunlight and sends its red light back to your eye. The angle between that returning ray and the centre line is about forty two degrees. Rotate that direction right around the centre line and the eligible drops form a thin conical shell, whose two edges are the red lines of this side view. Board: question — a Tex \[text\] that says "Why does the bow keep the same angle from the shadow of your own head, whichever shower you happen to be looking at?"; sky — a Figure (x\_range=(0.0, 8.5), y\_range=(-1.3, 5.2), aspect=(8.5, 6.5)); sun\_high — an Arrow \[yellow\] labelled "upright("sunlight")" drawn in sky (start=(0.2, 4.8), end=(2.4, 4.8)); sun\_mid — an Arrow \[yellow\] drawn in sky (start=(0.2, 4.2), end=(2.4, 4.2)); eye — a Point \[text\] labelled "upright("you")" drawn in sky (location=(1.0, 1.0)); axis — an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in sky (start=(1.0, 1.0), end=(8.1, 1.0)) Actions: - [00:42.165](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=42.16499999999999): drop is shown on the screen, written out. - [00:42.815](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=42.81499999999999): into\_drop is shown on the screen, written out. - [00:45.16](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=45.16): to\_eye is shown on the screen, drawn. - [00:47.111](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=47.11099999999999): bow\_angle is shown on the screen, written out. - [00:50.083](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=50.08299999999999): bow\_angle is emphasized. - [00:57.49](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=57.48999999999999): lower\_shell is shown on the screen, drawn. - [00:59.998](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=59.99799999999999): bow\_angle is no longer emphasized. ##### [01:0.598](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=60.598) Narration: Now step sideways. The cone moves with your eye, and a different drop joins the new line of sight. The rainbow is fixed by an angle, not by a place in the shower. Board: question — a Tex \[text\] that says "Why does the bow keep the same angle from the shadow of your own head, whichever shower you happen to be looking at?"; sky — a Figure (x\_range=(0.0, 8.5), y\_range=(-1.3, 5.2), aspect=(8.5, 6.5)); sun\_high — an Arrow \[yellow\] labelled "upright("sunlight")" drawn in sky (start=(0.2, 4.8), end=(2.4, 4.8)); sun\_mid — an Arrow \[yellow\] drawn in sky (start=(0.2, 4.2), end=(2.4, 4.2)); eye — a Point \[text\] labelled "upright("you")" drawn in sky (location=(1.0, 1.0)); axis — an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in sky (start=(1.0, 1.0), end=(8.1, 1.0)); drop — a Circle \[blue\] drawn in sky (center=(4.492780679743753, 4.144913849886634), radius=0.25, filled=True); into\_drop — an Arrow \[yellow\] drawn in sky (start=(2.5927806797437527, 4.144913849886634), end=(4.182780679743753, 4.144913849886634)); to\_eye — an Arrow \[red\] drawn in sky (start=(4.302780679743752, 3.974913849886634), end=(1.1400000000000001, 1.12)); bow\_angle — an Angle \[green\] labelled "42 degree" drawn in sky (vertex=(1.0, 1.0), sides=((8.1, 1.0), (4.492780679743753, 4.144913849886634)), radius=1.4); lower\_shell — a Line \[red\] drawn in sky (start=(1.0, 1.0), end=(3.229434476432183, -1.0073918190765747), dashed=True) Actions: - [01:1.329](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=61.329): eye\_two is shown on the screen, written out. - [01:3.187](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=63.187000000000005): axis\_two is shown on the screen, drawn. - [01:5.219](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=65.219): drop\_two is shown on the screen, written out. - [01:6.6](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=66.6): ray\_two is shown on the screen, drawn. - [01:11.987](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=71.987): eye\_two is hidden from the screen. - [01:11.987](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=71.987): axis\_two is hidden from the screen. - [01:11.987](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=71.987): drop\_two is hidden from the screen. - [01:11.987](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=71.987): ray\_two is hidden from the screen. ##### [01:12.587](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=72.587) Narration: Raise the sun and the anti-solar direction tilts down. The whole bow then follows it below the horizon. Lower the sun and the bow rises again. Board: Unchanged from the preceding beat in this scene. Actions: - [01:15.443](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=75.44300000000001): high\_axis is shown on the screen, drawn. - [01:18.299](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=78.299): below\_bow is shown on the screen, written out. - [01:20.261](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=80.26100000000001): high\_axis is hidden from the screen. - [01:20.261](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=80.26100000000001): below\_bow is hidden from the screen. - [01:21.585](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=81.58500000000001): drop is indicated — a transient flash. ##### [01:23.706](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=83.706) Narration: So the question has two parts. Why is the angle about forty two degrees, and why does the light bunch up there instead of spreading evenly across the sky? Both answers are inside one raindrop. Board: Unchanged from the preceding beat in this scene. Actions: - [01:24.867](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=84.867): bow\_angle is emphasized. - [01:34.608](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=94.608): drop is indicated — a transient flash. - [01:35.629](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.6285625): bow\_angle is no longer emphasized. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): question is hidden from the screen — left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): sky is hidden from the screen — left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): sun\_high is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): sun\_mid is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): eye is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): axis is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): drop is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): into\_drop is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): to\_eye is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): bow\_angle is hidden from the screen — sky left the board. - [01:35.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=95.8785625): lower\_shell is hidden from the screen — sky left the board. ### Scene 2: [One Drop, One Ray](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667) Span: 01:36.92–06:43.117 (96.92022916666667s–403.1172083333334s). #### Objects - answer\_angles: a Math \[text\] that says "$i approx 59.4 degree, quad r approx 40.2 degree$" - answer\_deviation: a Math \[text\] that says "$D\_(min) approx 138 degree$" - answer\_stack: an Arithmetic \[text\] that says "$180 degree D\_(min) approx 138 degree theta approx 42 degree$" (operator='-', operands=('180 degree', 'D\_(min) approx 138 degree'), result='theta approx 42 degree') - calc: a Derivation \[text\] that says "$frac(dif D, dif i) &= 2 - 4 frac(dif r, dif i) \\ 0 &= 2 - 4 frac(dif r, dif i) \\ frac(dif r, dif i) &= frac(1, 2) \\ cos i &= n cos r thin frac(dif r, dif i) \\ cos i &= frac(n cos r, 2) \\ 4 cos^2 i &= n^2 cos^2 r \\ &= n^2 (1 - sin^2 r) \\ &=…$" - carried\_forward: a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(2.0053766742983252, -0.9999855455822798), dashed=True) - centre: a Point \[gray\] drawn in plane - centre\_ray: a Line \[gray\] drawn in plane (start=(-3.0, 0.0), dashed=True) - chord\_one: an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…) - chord\_two: an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…) - claim: a Tex \[text\] that says "The deviation $D$ is the ray's total change of direction." - curve: a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)) - drawn\_deviation: an Angle \[green\] drawn in plane (vertex=(0.1991726221617528, -0.9799644210792613), sides=((2.2991726221617528, -0.9799644210792613), (-1.656407267024752…, radius=0.7) - drawn\_return: an Angle \[magenta\] drawn in plane (vertex=(0.1991726221617528, -0.9799644210792613), sides=((-1.6564072670247523, -2.655322076878941), (-1.900827377838247…, radius=1.0) - drop: a Circle \[blue\] drawn in plane - exit\_inside: an Angle \[cyan\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((0.9387697563226889, 0.34454512710795854), (0.0, 0.0)), radius=0.5) - exit\_outside: an Angle \[cyan\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((0.008065011447487822, -1.4999783183734197), (-1.9010666026324…, radius=0.31) - flat: a TangentLine \[green\] drawn in graph (target='curve', x=59.4104730269434, length=17.0) - graph: an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)) - heading\_answer: a Heading that says "Turn the Deviation Around" - heading\_calc: a Heading that says "The Flat Spot, Exactly" - heading\_path: a Heading that says "One Drop, One Ray" - heading\_sweep: a Heading that says "Where the Rays Pile Up" - heading\_turns: a Heading that says "Three Turns, One Deviation" - high\_in: an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)) - high\_out: an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)) - i\_deg: a VariableNumber (initial\_value=50.0, format\_spec='.1f') - incidence\_angle: an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((-3.0, 0.766044443118978), (-0.9963207950141361, 1.18736888683…, radius=0.3) - label\_drop: a Tex \[text\] that says "One drop" - label\_plot\_text: a Tex \[text\] that says "Deviation as the entry point rises" - low\_in: an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)) - low\_out: an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)) - minimum\_marker: a PlotPoint \[green\] labelled "(59.4, 138)" drawn in graph (target='curve', x=59.4104730269434) - minimum\_readout: a Math \[text\] that says "$D\_(min) approx 138 degree$" - normal\_back: a Line \[gray\] drawn in plane (end=(1.361216146667899, 0.4995904343065399), dashed=True) - normal\_entry: a Line \[gray\] drawn in plane (end=(-0.9963207950141361, 1.1873688868344159), dashed=True) - normal\_leave: a Line \[gray\] drawn in plane (end=(0.008065011447487822, -1.4999783183734197), dashed=True) - plane: a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)) - ray\_in: an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…) - ray\_out: an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…) - reflection\_in: an Angle \[cyan\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((-0.6427876096865394, 0.766044443118978), (0.0, 0.0)), radius=0.31) - reflection\_out: an Angle \[cyan\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((0.0, 0.0), (0.005376674298325215, -0.9999855455822798)), radius=0.46) - refracted\_angle: an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.9387697563226889, 0.34454512710795854), (0.0, 0.0)), radius=0.43) - refraction: a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly." - snell: a Math \[text\] that says "$sin i = n thin sin r, quad n approx 1.33$" - start\_readout: a Math \[text\] that says "$D approx 160 degree$" - straight\_back: a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(-1.9008273778382474, -0.9799644210792613), dashed=True) - straight\_forward: a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(2.2991726221617528, -0.9799644210792613), dashed=True) - total\_angle: an Angle \[green\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((2.0053766742983252, -0.9999855455822798), (-1.901066602632478…, radius=0.7) - tracker: a PlotPoint \[red\] labelled "50.0" drawn in graph (target='curve', x=\) - turn\_one: an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32) - turn\_one\_forward: a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True) - turn\_three: an Angle \[magenta\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((-0.6946681372199475, -2.0083835500999587), (-1.90106660263247…, radius=0.32) - turn\_three\_forward: a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(-0.6946681372199475, -2.0083835500999587), dashed=True) - turn\_two: an Angle \[magenta\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((2.12493778082961, 0.028420640099693917), (0.00537667429832521…, radius=0.43) - turn\_two\_forward: a Line \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(2.12493778082961, 0.028420640099693917), dashed=True) - turn\_work: a Derivation \[text\] that says "$D\_1 &= i - r \\ D\_2 &= 180 degree - 2 r \\ D\_3 &= i - r \\ D &= 180 degree + 2 i - 4 r \\ &= 180 degree + 2 i - 4 sin^(-1)(frac(sin i, n))$" - waste: an Arrow \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(1.8877041759282258, 0.09164553750134685)) #### Beats ##### [01:36.92](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667) Narration: Refraction is the bending of light as it crosses into water. An incoming ray and the centre of a spherical drop determine one flat slice where we can follow that bend. Seen head on, the slice is a circle. The same circle, centre, and ray will stay with us all the way to the answer. Board: Empty. Actions: - [01:36.92](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667): heading\_path is shown on the screen, written out. - [01:36.92](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=96.92022916666667): refraction is shown on the screen, written out. - [01:41.285](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=101.28522916666667): ray\_in is shown on the screen, drawn. - [01:42.121](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=102.12122916666667): centre is shown on the screen, written out. - [01:44.223](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=104.22322916666667): plane is shown on the screen, written out. - [01:48.878](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=108.87822916666667): drop is shown on the screen, drawn. ##### [01:55.26](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=115.26022916666668) Narration: At the surface, the normal is the radius through the point of contact. The incoming ray makes the incidence angle i with that normal. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…) Actions: - [01:57.118](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=117.11822916666668): normal\_entry is shown on the screen, drawn. - [02:0.949](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=120.94922916666667): incidence\_angle is shown on the screen, written out. ##### [02:3.802](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=123.80172916666668) Narration: At that surface the light turns toward the normal as it enters water, so the angle inside, r, is smaller than i. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); normal\_entry — a Line \[gray\] drawn in plane (end=(-0.9963207950141361, 1.1873688868344159), dashed=True); incidence\_angle — an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((-3.0, 0.766044443118978), (-0.9963207950141361, 1.18736888683…, radius=0.3) Actions: - [02:5.369](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=125.36922916666667): chord\_one is shown on the screen, drawn. - [02:8.515](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=128.51522916666667): refracted\_angle is shown on the screen, written out. ##### [02:11.519](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=131.51872916666667) Narration: Snell's law gives the size of that bend. Sine i equals n times sine r, and water has n about one point three three. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); normal\_entry — a Line \[gray\] drawn in plane (end=(-0.9963207950141361, 1.1873688868344159), dashed=True); incidence\_angle — an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((-3.0, 0.766044443118978), (-0.9963207950141361, 1.18736888683…, radius=0.3); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); refracted\_angle — an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.9387697563226889, 0.34454512710795854), (0.0, 0.0)), radius=0.43) Actions: - [02:11.867](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=131.86722916666668): snell is shown on the screen, written out. - [02:14.456](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=134.45622916666667): snell (the "sin i" part) is emphasized. - [02:15.582](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=135.58222916666668): snell (the "n thin sin r" part) is emphasized. - [02:15.582](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=135.58222916666668): snell (the "sin i" part) is no longer emphasized. - [02:18.984](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=138.9842291666667): snell (the "1.33" part) is emphasized. - [02:18.984](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=138.9842291666667): snell (the "n thin sin r" part) is no longer emphasized. - [02:19.983](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=139.98272916666667): snell (the "1.33" part) is no longer emphasized. ##### [02:20.583](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=140.58272916666667) Narration: The ray reaches the back wall. Most light escapes and is lost, but a small fraction reflects. Symmetry makes the incoming and reflected angles there equal to the same r. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; snell — a Math \[text\] that says "$sin i = n thin sin r, quad n approx 1.33$"; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); normal\_entry — a Line \[gray\] drawn in plane (end=(-0.9963207950141361, 1.1873688868344159), dashed=True); incidence\_angle — an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((-3.0, 0.766044443118978), (-0.9963207950141361, 1.18736888683…, radius=0.3); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); refracted\_angle — an Angle \[cyan\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.9387697563226889, 0.34454512710795854), (0.0, 0.0)), radius=0.43) Actions: - [02:20.583](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=140.58272916666667): incidence\_angle is hidden from the screen. - [02:20.583](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=140.58272916666667): refracted\_angle is hidden from the screen. - [02:20.583](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=140.58272916666667): normal\_entry is hidden from the screen. - [02:23.74](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=143.74022916666667): waste is shown on the screen, drawn. - [02:26.341](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=146.34122916666666): waste is hidden from the screen. - [02:27.664](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=147.66422916666667): normal\_back is shown on the screen, drawn. - [02:28.57](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=148.57022916666668): reflection\_in is shown on the screen, written out. - [02:29.301](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=149.30122916666667): chord\_two is shown on the screen, drawn. - [02:31.24](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=151.24022916666667): reflection\_out is shown on the screen, written out. ##### [02:33.013](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=153.0132291666667) Narration: At the front surface the reflected ray meets one more radius. It leaves water, opens from r back to i, and becomes the ray that can reach an eye. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; snell — a Math \[text\] that says "$sin i = n thin sin r, quad n approx 1.33$"; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); normal\_back — a Line \[gray\] drawn in plane (end=(1.361216146667899, 0.4995904343065399), dashed=True); reflection\_in — an Angle \[cyan\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((-0.6427876096865394, 0.766044443118978), (0.0, 0.0)), radius=0.31); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); reflection\_out — an Angle \[cyan\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((0.0, 0.0), (0.005376674298325215, -0.9999855455822798)), radius=0.46) Actions: - [02:33.013](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=153.0132291666667): reflection\_in is hidden from the screen. - [02:33.013](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=153.0132291666667): reflection\_out is hidden from the screen. - [02:33.013](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=153.0132291666667): normal\_back is hidden from the screen. - [02:35.985](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=155.98522916666667): normal\_leave is shown on the screen, drawn. - [02:37.459](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=157.45922916666666): ray\_out is shown on the screen, drawn. - [02:39.19](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=159.19022916666665): exit\_inside is shown on the screen, written out. - [02:39.967](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=159.96722916666667): exit\_outside is shown on the screen, written out. ##### [02:43.668](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=163.66772916666667) Narration: Each pair of angle marks appeared only for the event it named. The exit pair has now carried the same law to the last surface, so it can leave too. Board: refraction — a Panel that says "Light entering water bends toward the normal. The refractive index tells us how strongly."; snell — a Math \[text\] that says "$sin i = n thin sin r, quad n approx 1.33$"; plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); heading\_path — a Heading that says "One Drop, One Ray"; drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); normal\_leave — a Line \[gray\] drawn in plane (end=(0.008065011447487822, -1.4999783183734197), dashed=True); exit\_inside — an Angle \[cyan\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((0.9387697563226889, 0.34454512710795854), (0.0, 0.0)), radius=0.5); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); exit\_outside — an Angle \[cyan\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((0.008065011447487822, -1.4999783183734197), (-1.9010666026324…, radius=0.31) Actions: - [02:44.19](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=164.19022916666668): exit\_outside is indicated — a transient flash. - [02:47.835](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=167.83522916666666): exit\_inside is indicated — a transient flash. - [02:51.492](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=171.49222916666668): exit\_inside is hidden from the screen. - [02:51.492](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=171.49222916666668): exit\_outside is hidden from the screen. - [02:51.492](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=171.49222916666668): normal\_leave is hidden from the screen. - [02:52.433](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=172.43272916666666): plane moves to a new place on the board. - [02:52.433](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=172.43272916666666): heading\_path is hidden from the screen — left the board. - [02:52.433](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=172.43272916666666): refraction is hidden from the screen — left the board. - [02:52.433](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=172.43272916666666): snell is hidden from the screen — left the board. ##### [02:53.033](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=173.03272916666668) Narration: Now measure how far the ray turns at each event. At the entrance it turns through i minus r. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…) Actions: - [02:53.033](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=173.03272916666668): heading\_turns is shown on the screen, written out. - [02:53.033](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=173.03272916666668): claim is shown on the screen, written out. - [02:53.683](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=173.68322916666668): turn\_one\_forward is shown on the screen, drawn. - [02:57.026](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=177.02622916666667): turn\_one is shown on the screen, written out. - [02:58.675](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=178.67522916666667): turn\_work is shown on the screen, written out. - [02:58.675](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=178.67522916666667): turn\_work (the "i - r" part) is emphasized. - [02:59.598](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=179.59822916666667): turn\_work (the "i - r" part) is no longer emphasized. ##### [03:0.198](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=180.19822916666666) Narration: The reflection is the large turn. Equal angles r leave a straight angle minus two r between the old direction and the new one. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); claim — a Tex \[text\] that says "The deviation $D$ is the ray's total change of direction."; heading\_turns — a Heading that says "Three Turns, One Deviation"; turn\_one\_forward — a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True); turn\_one — an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32) Actions: - [03:0.482](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=180.48222916666668): turn\_two\_forward is shown on the screen, drawn. - [03:1.307](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=181.30722916666667): turn\_two is shown on the screen, written out. - [03:4.186](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=184.18622916666666): turn\_work is shown on the screen, written out. - [03:4.871](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=184.87122916666667): turn\_work (the "180 degree - 2 r" part) is emphasized. - [03:7.994](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=187.99422916666668): turn\_work (the "180 degree - 2 r" part) is no longer emphasized. ##### [03:8.594](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=188.59422916666665) Narration: Leaving the drop adds another i minus r. All three local turns now stand beside the path: i minus r, one hundred eighty minus two r, and i minus r. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); claim — a Tex \[text\] that says "The deviation $D$ is the ray's total change of direction."; heading\_turns — a Heading that says "Three Turns, One Deviation"; turn\_one\_forward — a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True); turn\_one — an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32); turn\_two\_forward — a Line \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(2.12493778082961, 0.028420640099693917), dashed=True); turn\_two — an Angle \[magenta\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((2.12493778082961, 0.028420640099693917), (0.00537667429832521…, radius=0.43) Actions: - [03:8.942](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=188.94222916666666): turn\_three\_forward is shown on the screen, drawn. - [03:10.068](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=190.06822916666664): turn\_three is shown on the screen, written out. - [03:10.393](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=190.39322916666666): turn\_work (the "i - r" part) is emphasized. - [03:10.695](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=190.69522916666665): turn\_work is shown on the screen, written out. - [03:15.536](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=195.53622916666666): turn\_work (the "i - r" part) is emphasized. - [03:15.536](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=195.53622916666666): turn\_work (the "i - r" part) is no longer emphasized. - [03:16.906](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=196.90622916666666): turn\_work (the "i - r" part) is no longer emphasized. - [03:16.906](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=196.90622916666666): turn\_work (the "180 degree - 2 r" part) is emphasized. - [03:18.717](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=198.71722916666664): turn\_work (the "180 degree - 2 r" part) is no longer emphasized. - [03:18.717](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=198.71722916666664): turn\_work (the "i - r" part) is emphasized. - [03:19.896](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=199.89572916666668): turn\_work (the "i - r" part) is no longer emphasized. ##### [03:20.496](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=200.49572916666665) Narration: Add those three turns. The deviation is one hundred eighty degrees plus two i minus four r. The angle D compares the original forward direction with the direction that actually leaves the drop. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); claim — a Tex \[text\] that says "The deviation $D$ is the ray's total change of direction."; heading\_turns — a Heading that says "Three Turns, One Deviation"; turn\_one\_forward — a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True); turn\_one — an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32); turn\_two\_forward — a Line \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(2.12493778082961, 0.028420640099693917), dashed=True); turn\_two — an Angle \[magenta\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((2.12493778082961, 0.028420640099693917), (0.00537667429832521…, radius=0.43); turn\_three\_forward — a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(-0.6946681372199475, -2.0083835500999587), dashed=True); turn\_three — an Angle \[magenta\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((-0.6946681372199475, -2.0083835500999587), (-1.90106660263247…, radius=0.32) Actions: - [03:22.905](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=202.90522916666666): turn\_work is shown on the screen, written out. - [03:23.903](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=203.90322916666668): turn\_work (the "180 degree" part) is emphasized. - [03:25.088](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=205.08822916666665): turn\_work (the "180 degree" part) is no longer emphasized. - [03:25.088](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=205.08822916666665): turn\_work (the "2 i" part) is emphasized. - [03:26.005](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=206.00522916666665): turn\_work (the "2 i" part) is no longer emphasized. - [03:26.005](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=206.00522916666665): turn\_work (the "4 r" part) is emphasized. - [03:27.549](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=207.54922916666663): turn\_work (the "4 r" part) is no longer emphasized. - [03:28.745](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=208.74522916666666): carried\_forward is shown on the screen, drawn. - [03:30.057](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=210.05722916666667): total\_angle is shown on the screen, written out. ##### [03:33.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=213.25772916666665) Narration: A ray through the centre has i and r both zero, so the expression gives a turn of one hundred eighty degrees, straight back. Snell's law also makes r the inverse sine of sine i over n, leaving i as the only free angle. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); claim — a Tex \[text\] that says "The deviation $D$ is the ray's total change of direction."; heading\_turns — a Heading that says "Three Turns, One Deviation"; turn\_one\_forward — a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True); turn\_one — an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32); turn\_two\_forward — a Line \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(2.12493778082961, 0.028420640099693917), dashed=True); turn\_two — an Angle \[magenta\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((2.12493778082961, 0.028420640099693917), (0.00537667429832521…, radius=0.43); turn\_three\_forward — a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(-0.6946681372199475, -2.0083835500999587), dashed=True); turn\_three — an Angle \[magenta\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((-0.6946681372199475, -2.0083835500999587), (-1.90106660263247…, radius=0.32); carried\_forward — a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(2.0053766742983252, -0.9999855455822798), dashed=True); total\_angle — an Angle \[green\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((2.0053766742983252, -0.9999855455822798), (-1.901066602632478…, radius=0.7) Actions: - [03:34.337](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=214.33722916666665): centre\_ray is shown on the screen, written out. - [03:41.28](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=221.28022916666666): centre\_ray is hidden from the screen. - [03:43.254](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=223.25422916666668): turn\_work is shown on the screen, written out. - [03:43.254](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=223.25422916666668): turn\_work (the "sin^(-1)(frac(sin i, n))" part) is emphasized. - [03:48.27](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.26972916666665): plane moves to a new place on the board. - [03:48.27](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.26972916666665): claim is hidden from the screen — left the board. - [03:48.27](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.26972916666665): heading\_turns is hidden from the screen — left the board. - [03:48.27](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.26972916666665): turn\_work is hidden from the screen — left the board. - [03:48.27](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.26972916666665): turn\_work (the "sin^(-1)(frac(sin i, n))" part) is no longer emphasized. ##### [03:48.87](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.86972916666667) Narration: Set the entry low on the drop, at twenty degrees. The ray returns almost the way it arrived, with a deviation of about one hundred sixty degrees. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); turn\_one\_forward — a Line \[gray\] drawn in plane (start=(-0.6427876096865394, 0.766044443118978), end=(0.25721239031346066, 0.766044443118978), dashed=True); turn\_one — an Angle \[magenta\] drawn in plane (vertex=(-0.6427876096865394, 0.766044443118978), sides=((0.25721239031346066, 0.766044443118978), (0.9387697563226889,…, radius=0.32); turn\_two\_forward — a Line \[gray\] drawn in plane (start=(0.9387697563226889, 0.34454512710795854), end=(2.12493778082961, 0.028420640099693917), dashed=True); turn\_two — an Angle \[magenta\] drawn in plane (vertex=(0.9387697563226889, 0.34454512710795854), sides=((2.12493778082961, 0.028420640099693917), (0.00537667429832521…, radius=0.43); turn\_three\_forward — a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(-0.6946681372199475, -2.0083835500999587), dashed=True); turn\_three — an Angle \[magenta\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((-0.6946681372199475, -2.0083835500999587), (-1.90106660263247…, radius=0.32); carried\_forward — a Line \[gray\] drawn in plane (start=(0.005376674298325215, -0.9999855455822798), end=(2.0053766742983252, -0.9999855455822798), dashed=True); total\_angle — an Angle \[green\] drawn in plane (vertex=(0.005376674298325215, -0.9999855455822798), sides=((2.0053766742983252, -0.9999855455822798), (-1.901066602632478…, radius=0.7) Actions: - [03:48.87](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.86972916666667): heading\_sweep is shown on the screen, written out. - [03:48.87](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.86972916666667): label\_drop is shown on the screen, written out. - [03:48.87](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=228.86972916666667): label\_plot\_text is shown on the screen, written out. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): ray\_in is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): chord\_one is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): chord\_two is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): ray\_out is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): curve is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): tracker is redrawn as the numbers it depends on change. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): graph is shown on the screen, written out. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_one is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_two is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_three is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_one\_forward is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_two\_forward is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): turn\_three\_forward is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): carried\_forward is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): total\_angle is hidden from the screen. - [03:51.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=231.05222916666668): i\_deg ticks to 20.0. - [03:55.115](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=235.11522916666667): tracker is shown on the screen, written out. - [03:55.115](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=235.11522916666667): curve is shown on the screen, written out. - [03:56.311](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=236.3112291666667): start\_readout is shown on the screen, written out. - [03:56.602](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=236.6022291666667): start\_readout (the "160" part) is emphasized. - [03:57.902](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=237.90222916666667): start\_readout (the "160" part) is no longer emphasized. ##### [03:58.502](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.50222916666667) Narration: Slide the entry point upward. The deviation falls through one hundred fifty, one hundred forty five, and one hundred forty. Then it slows. There, it has stopped coming down. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); label\_drop — a Tex \[text\] that says "One drop"; label\_plot\_text — a Tex \[text\] that says "Deviation as the entry point rises"; start\_readout — a Math \[text\] that says "$D approx 160 degree$"; graph — an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)); heading\_sweep — a Heading that says "Where the Rays Pile Up"; curve — a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)); tracker — a PlotPoint \[red\] labelled "50.0" drawn in graph (target='curve', x=\) Actions: - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): ray\_in is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): chord\_one is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): chord\_two is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): ray\_out is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): curve is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): tracker is redrawn as the numbers it depends on change. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): start\_readout is hidden from the screen. - [03:58.85](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=238.85022916666665): i\_deg ticks to 59.4104730269434. - [04:8.428](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=248.42822916666665): minimum\_marker is shown on the screen, written out. - [04:8.428](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=248.42822916666665): minimum\_readout is shown on the screen, written out. - [04:9.124](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=249.12422916666665): minimum\_readout (the "138" part) is emphasized. - [04:9.879](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=249.87922916666668): minimum\_readout (the "138" part) is no longer emphasized. ##### [04:10.479](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.47922916666667) Narration: Keep sliding toward the rim. Only after the minimum does the curve climb again, and it keeps climbing as the entry point approaches the edge. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); label\_drop — a Tex \[text\] that says "One drop"; label\_plot\_text — a Tex \[text\] that says "Deviation as the entry point rises"; minimum\_readout — a Math \[text\] that says "$D\_(min) approx 138 degree$"; graph — an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)); heading\_sweep — a Heading that says "Where the Rays Pile Up"; curve — a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)); tracker — a PlotPoint \[red\] labelled "50.0" drawn in graph (target='curve', x=\); minimum\_marker — a PlotPoint \[green\] labelled "(59.4, 138)" drawn in graph (target='curve', x=59.4104730269434) Actions: - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): ray\_in is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): chord\_one is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): chord\_two is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): ray\_out is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): curve is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): tracker is redrawn as the numbers it depends on change. - [04:10.827](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=250.82722916666668): i\_deg ticks to 82.0. - [04:13.428](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=253.42822916666668): minimum\_marker is indicated — a transient flash. ##### [04:19.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.1832291666667) Narration: The flat bottom makes the minimum important. These two rays enter at forty eight and seventy degrees, far apart on the face of the drop. Board: Unchanged from the preceding beat in this scene. Actions: - [04:19.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.1832291666667): ray\_in is hidden from the screen. - [04:19.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.1832291666667): chord\_one is hidden from the screen. - [04:19.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.1832291666667): chord\_two is hidden from the screen. - [04:19.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.1832291666667): ray\_out is hidden from the screen. - [04:19.705](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=259.7052291666667): flat is shown on the screen, written out. - [04:23.78](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=263.7802291666667): low\_in is shown on the screen, drawn. - [04:24.5](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=264.5002291666667): high\_in is shown on the screen, drawn. ##### [04:28.433](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=268.4327291666667) Narration: Yet their outgoing directions differ by less than a quarter of a degree. A broad band of entry points therefore sends light back in nearly one direction, so the light piles up instead of spreading thin. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; label\_drop — a Tex \[text\] that says "One drop"; label\_plot\_text — a Tex \[text\] that says "Deviation as the entry point rises"; minimum\_readout — a Math \[text\] that says "$D\_(min) approx 138 degree$"; graph — an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)); heading\_sweep — a Heading that says "Where the Rays Pile Up"; curve — a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)); tracker — a PlotPoint \[red\] labelled "50.0" drawn in graph (target='curve', x=\); minimum\_marker — a PlotPoint \[green\] labelled "(59.4, 138)" drawn in graph (target='curve', x=59.4104730269434); low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); flat — a TangentLine \[green\] drawn in graph (target='curve', x=59.4104730269434, length=17.0) Actions: - [04:29.164](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=269.1642291666667): low\_out is shown on the screen, drawn. - [04:31.416](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=271.4162291666667): high\_out is shown on the screen, drawn. - [04:33.204](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=273.2042291666667): low\_in is indicated — a transient flash. - [04:33.518](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=273.51822916666674): high\_in is indicated — a transient flash. - [04:40.101](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.1007291666667): tracker is hidden from the screen. ##### [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667) Narration: That concentration is the rainbow. To locate it exactly, use the flat curve condition: at the minimum, the derivative of D with respect to i is zero. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; label\_drop — a Tex \[text\] that says "One drop"; label\_plot\_text — a Tex \[text\] that says "Deviation as the entry point rises"; minimum\_readout — a Math \[text\] that says "$D\_(min) approx 138 degree$"; graph — an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)); heading\_sweep — a Heading that says "Where the Rays Pile Up"; curve — a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)); minimum\_marker — a PlotPoint \[green\] labelled "(59.4, 138)" drawn in graph (target='curve', x=59.4104730269434); low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); flat — a TangentLine \[green\] drawn in graph (target='curve', x=59.4104730269434, length=17.0); low\_out — an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)); high\_out — an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)) Actions: - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): graph moves to a new place on the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): heading\_sweep is hidden from the screen — left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): label\_drop is hidden from the screen — left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): label\_plot\_text is hidden from the screen — left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): minimum\_readout is hidden from the screen — left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): plane is hidden from the screen — left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): drop is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): centre is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): low\_in is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): high\_in is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): low\_out is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): high\_out is hidden from the screen — plane left the board. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): heading\_calc is shown on the screen, written out. - [04:40.701](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=280.7007291666667): calc is shown on the screen, written out. - [04:50.709](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=290.7092291666667): calc is shown on the screen, written out. ##### [04:52.296](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=292.29572916666666) Narration: Solving that condition says d r by d i must equal one half. The angle inside must change half as fast as the incidence angle outside. Board: graph — an Axes (x\_range=(15.0, 85.0), y\_range=(136.0, 164.0), aspect=(5.7, 3.65)); curve — a FunctionPlot \[yellow\] drawn in graph (function=\, x\_range=(19.9, \)); minimum\_marker — a PlotPoint \[green\] labelled "(59.4, 138)" drawn in graph (target='curve', x=59.4104730269434); flat — a TangentLine \[green\] drawn in graph (target='curve', x=59.4104730269434, length=17.0); heading\_calc — a Heading that says "The Flat Spot, Exactly" Actions: - [04:56.382](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=296.3822291666667): calc is shown on the screen, written out. - [04:56.382](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=296.3822291666667): calc (the "frac(1, 2)" part) is emphasized. - [05:1.955](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=301.9552291666667): calc (the "frac(1, 2)" part) is no longer emphasized. ##### [05:2.555](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=302.55522916666666) Narration: Differentiate Snell's law. Cosine i equals n cosine r times d r by d i. Substituting one half gives cosine i equal to n cosine r over two. Board: Unchanged from the preceding beat in this scene. Actions: - [05:5.179](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=305.1792291666667): calc is shown on the screen, written out. - [05:10.473](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=310.47322916666667): calc is shown on the screen, written out. ##### [05:14.382](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=314.3817291666667) Narration: Now square both sides. Four cosine squared i equals n squared cosine squared r. Board: Unchanged from the preceding beat in this scene. Actions: - [05:14.974](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=314.97422916666676): calc is shown on the screen, written out. - [05:16.831](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=316.83122916666673): calc (the "4 cos^2 i" part) is emphasized. - [05:19.107](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=319.1072291666667): calc (the "4 cos^2 i" part) is no longer emphasized. - [05:19.107](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=319.1072291666667): calc (the "n^2 cos^2 r" part) is emphasized. - [05:21.464](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=321.46372916666667): calc (the "n^2 cos^2 r" part) is no longer emphasized. ##### [05:22.064](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=322.0637291666667) Narration: Replace cosine squared r with one minus sine squared r. Snell's law then replaces n squared sine squared r with sine squared i. Finally, sine squared i is one minus cosine squared i. Board: Unchanged from the preceding beat in this scene. Actions: - [05:24.327](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=324.3272291666667): calc is shown on the screen, written out. - [05:26.917](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=326.91722916666674): calc is shown on the screen, written out. - [05:32.013](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=332.01322916666675): calc is shown on the screen, written out. ##### [05:36.84](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=336.8397291666667) Narration: Collect the cosine terms. Cosine i is the square root of n squared minus one, divided by three. Board: Unchanged from the preceding beat in this scene. Actions: - [05:40.972](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=340.9722291666667): calc is shown on the screen, written out. - [05:40.972](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=340.9722291666667): calc (the "sqrt(frac(n^2 - 1, 3))" part) is emphasized. - [05:44.653](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=344.6527291666667): calc (the "sqrt(frac(n^2 - 1, 3))" part) is no longer emphasized. ##### [05:45.253](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=345.2527291666667) Narration: For water, i is fifty nine point four degrees and r is forty point two. Putting both into the turn formula gives a minimum deviation of about one hundred thirty eight degrees. Board: Unchanged from the preceding beat in this scene. Actions: - [05:47.11](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=347.1102291666667): answer\_angles is shown on the screen, written out. - [05:47.992](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=347.99222916666673): answer\_angles (the "59.4" part) is emphasized. - [05:49.629](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=349.62922916666673): answer\_angles (the "40.2" part) is emphasized. - [05:49.629](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=349.62922916666673): answer\_angles (the "59.4" part) is no longer emphasized. - [05:53.832](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=353.8322291666667): answer\_deviation is shown on the screen, written out. - [05:55.713](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=355.71322916666674): answer\_deviation (the "138" part) is emphasized. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): answer\_deviation moves to a new place on the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): answer\_angles is hidden from the screen — left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): calc is hidden from the screen — left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): graph is hidden from the screen — left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): curve is hidden from the screen — graph left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): minimum\_marker is hidden from the screen — graph left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): flat is hidden from the screen — graph left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): heading\_calc is hidden from the screen — left the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): plane is shown on the screen, faded in — cast on this board again. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): drop is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): centre is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): low\_in is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): high\_in is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): low\_out is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): high\_out is shown on the screen, faded in — plane came back to the board. - [05:57.258](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.2582291666667): answer\_deviation (the "138" part) is no longer emphasized. ##### [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667) Narration: Return to the special ray at the minimum. Its path is still the same path through the same drop; only the chosen entry point has moved. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); low\_out — an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)); high\_out — an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)); answer\_deviation — a Math \[text\] that says "$D\_(min) approx 138 degree$" Actions: - [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667): heading\_answer is shown on the screen, written out. - [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667): ray\_in is shown on the screen, written out. - [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667): chord\_one is shown on the screen, written out. - [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667): chord\_two is shown on the screen, written out. - [05:57.858](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=357.8582291666667): ray\_out is shown on the screen, written out. - [05:59.007](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=359.0072291666667): i\_deg ticks to 59.4104730269434. - [05:59.358](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=359.3582291666667): ray\_in is redrawn as the numbers it depends on change. - [05:59.358](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=359.3582291666667): chord\_one is redrawn as the numbers it depends on change. - [05:59.358](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=359.3582291666667): chord\_two is redrawn as the numbers it depends on change. - [05:59.358](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=359.3582291666667): ray\_out is redrawn as the numbers it depends on change. ##### [06:7.537](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=367.5367291666667) Narration: Carry the incoming direction through the exit point. Forward to backward is a straight angle of one hundred eighty degrees. The ray has already turned through D, about one hundred thirty eight degrees. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); low\_out — an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)); high\_out — an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)); answer\_deviation — a Math \[text\] that says "$D\_(min) approx 138 degree$"; heading\_answer — a Heading that says "Turn the Deviation Around" Actions: - [06:11.403](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=371.40322916666673): straight\_forward is shown on the screen, drawn. - [06:11.994](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=371.99422916666674): straight\_back is shown on the screen, drawn. - [06:14.212](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=374.2122291666667): answer\_stack is shown on the screen, written out. - [06:14.525](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=374.5252291666667): answer\_stack (the "180" part) is emphasized. - [06:16.697](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=376.6972291666667): answer\_stack (the "180" part) is no longer emphasized. - [06:17.022](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=377.02222916666676): drawn\_deviation is shown on the screen, written out. - [06:19.123](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=379.12322916666676): answer\_stack is shown on the screen, written out. - [06:19.402](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=379.40222916666676): answer\_stack (the "138" part) is emphasized. - [06:20.505](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=380.5047291666667): answer\_stack (the "138" part) is no longer emphasized. ##### [06:21.105](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=381.1047291666667) Narration: Subtract that deviation from the straight angle. The remaining angle between the outgoing ray and straight back is forty two degrees. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); low\_out — an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)); high\_out — an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)); answer\_deviation — a Math \[text\] that says "$D\_(min) approx 138 degree$"; heading\_answer — a Heading that says "Turn the Deviation Around"; straight\_forward — a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(2.2991726221617528, -0.9799644210792613), dashed=True); straight\_back — a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(-1.9008273778382474, -0.9799644210792613), dashed=True); drawn\_deviation — an Angle \[green\] drawn in plane (vertex=(0.1991726221617528, -0.9799644210792613), sides=((2.2991726221617528, -0.9799644210792613), (-1.656407267024752…, radius=0.7) Actions: - [06:21.406](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=381.4062291666667): answer\_stack is shown on the screen, drawn. - [06:22.206](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=382.20622916666673): answer\_stack is shown on the screen, drawn. - [06:24.727](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=384.72722916666675): drawn\_return is shown on the screen, written out. - [06:27.954](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=387.9542291666667): answer\_stack is shown on the screen, written out. - [06:28.267](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=388.2672291666667): answer\_stack (the "42" part) is emphasized. - [06:29.278](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=389.2777291666667): answer\_stack (the "42" part) is no longer emphasized. ##### [06:29.878](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=389.8777291666667) Narration: That is the promised number. Snell's law fixes the path inside one circle, the three turns create a minimum at one hundred thirty eight degrees, and the view back toward the sun leaves forty two degrees. Board: plane — a Figure (x\_range=(-3.3, 2.5), y\_range=(-3.0, 1.25), aspect=(5.8, 4.25)); drop — a Circle \[blue\] drawn in plane; centre — a Point \[gray\] drawn in plane; ray\_in — an Arrow \[yellow\] drawn in plane (start=(-3.0, sin((3.141592653589793 - (i\_deg / 57.29577951308232)))), end=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…); chord\_one — an Arrow \[yellow\] drawn in plane (start=(cos((3.141592653589793 - (i\_deg / 57.29577951308232))), sin((3…, end=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…); chord\_two — an Arrow \[yellow\] drawn in plane (start=(cos(((2.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333))…, end=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…); ray\_out — an Arrow \[red\] drawn in plane (start=(cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333)…, end=((cos((((4.0 \* arcsin((sin((i\_deg / 57.29577951308232)) / 1.333…); low\_in — an Arrow \[green\] drawn in plane (start=(-3.0, 0.7431448254773942), end=(-0.6691306063588582, 0.7431448254773942)); high\_in — an Arrow \[cyan\] drawn in plane (start=(-3.0, 0.9396926207859084), end=(-0.3420201433256687, 0.9396926207859084)); low\_out — an Arrow \[green\] drawn in plane (start=(-0.04306535999054507, -0.999072257030934), end=(-1.8555539220456703, -2.4948614174473827)); high\_out — an Arrow \[cyan\] drawn in plane (start=(0.3305247746393443, -0.943797315820304), end=(-1.9909749525578184, -2.843965470648581)); answer\_deviation — a Math \[text\] that says "$D\_(min) approx 138 degree$"; heading\_answer — a Heading that says "Turn the Deviation Around"; straight\_forward — a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(2.2991726221617528, -0.9799644210792613), dashed=True); straight\_back — a Line \[gray\] drawn in plane (start=(0.1991726221617528, -0.9799644210792613), end=(-1.9008273778382474, -0.9799644210792613), dashed=True); drawn\_deviation — an Angle \[green\] drawn in plane (vertex=(0.1991726221617528, -0.9799644210792613), sides=((2.2991726221617528, -0.9799644210792613), (-1.656407267024752…, radius=0.7); drawn\_return — an Angle \[magenta\] drawn in plane (vertex=(0.1991726221617528, -0.9799644210792613), sides=((-1.6564072670247523, -2.655322076878941), (-1.900827377838247…, radius=1.0) Actions: - [06:30.632](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=390.6322291666667): A box is drawn around answer\_stack. - [06:40.524](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=400.5242291666667): drawn\_return is indicated — a transient flash. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): answer\_deviation is hidden from the screen — left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): answer\_stack is hidden from the screen — left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): heading\_answer is hidden from the screen — left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): plane is hidden from the screen — left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): drop is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): centre is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): ray\_in is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): chord\_one is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): chord\_two is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): ray\_out is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): low\_in is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): high\_in is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): low\_out is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): high\_out is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): straight\_forward is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): straight\_back is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): drawn\_deviation is hidden from the screen — plane left the board. - [06:42.076](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=402.0755416666667): drawn\_return is hidden from the screen — plane left the board. ### Scene 3: [Assembling the Sky](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334) Span: 06:43.117–09:15.517 (403.1172083333334s–555.5166250000001s). #### Objects - anti\_line: an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in side (start=(1.0, 1.25), end=(8.0, 1.25)) - bowl: a Figure (x\_range=(-4.5, 4.5), y\_range=(-1.7, 3.5), aspect=(9.0, 5.2)) - caption\_drop: a Tex \[text\] that says "Path separation is exaggerated; the equations keep the true values." - caption\_primary: a Tex \[text\] that says "Colour spacing is exaggerated so the order stays visible." - caption\_secondary: a Tex \[text\] that says "Secondary bow: violet outside, red inside." - caption\_sky: a Tex \[text\] that says "Ray directions are widened here to make their order visible." - drop: a Circle \[blue\] drawn in drop\_plane - drop\_high: a Circle \[blue\] drawn in side (center=(5.35, 4.3), radius=0.23, filled=True) - drop\_low: a Circle \[blue\] drawn in side (center=(5.7, 3.1), radius=0.23, filled=True) - drop\_plane: a Figure (x\_range=(-3.1, 2.0), y\_range=(-3.0, 1.3), aspect=(5.1, 4.3)) - eye: a Point \[text\] labelled "upright("you")" drawn in side (location=(1.0, 1.25)) - feed\_high: an Arrow \[yellow\] drawn in side (start=(3.6499999999999995, 4.3), end=(5.06, 4.3)) - feed\_low: an Arrow \[yellow\] drawn in side (start=(4.0, 3.1), end=(5.41, 3.1)) - head: a Point \[gray\] labelled "upright("anti-solar point")" drawn in bowl (location=(0.0, -1.2)) - heading\_bow: a Heading that says "The Shape of the Bow" - heading\_drop: a Heading that says "One Drop, Two Colours" - heading\_sky: a Heading that says "Why Red Is on the Outside" - horizon: a Line \[gray\] labelled "upright("horizon")" drawn in bowl (start=(-4.35, 0.0), end=(4.35, 0.0), dashed=True) - inside\_mark: a Point \[yellow\] drawn in bowl (location=(0.0, 1.55)) - outside\_mark: a Point \[yellow\] drawn in bowl (location=(0.0, 2.5)) - primary\_red: a ParametricCurve \[red\] drawn in bowl (function=\, t\_range=(0.3663488425130368, 2.7752438110767566)) - primary\_violet: a ParametricCurve \[magenta\] drawn in bowl (function=\, t\_range=(0.40437323624907384, 2.7372194173407194)) - red\_in: an Arrow \[red\] drawn in drop\_plane (start=(-2.95, 0.73), end=(-0.68, 0.73)) - red\_miss: an Arrow \[red\] drawn in side (start=(5.7, 3.1), end=(0.25, 0.2)) - red\_one: an Arrow \[red\] drawn in drop\_plane (start=(-0.68, 0.73), end=(0.94, 0.34)) - red\_out: an Arrow \[red\] drawn in drop\_plane (start=(0.0, -1.0), end=(-1.85, -2.55)) - red\_seen: an Arrow \[red\] drawn in side (start=(5.35, 4.3), end=(1.0, 1.25)) - red\_two: an Arrow \[red\] drawn in drop\_plane (start=(0.94, 0.34), end=(0.0, -1.0)) - red\_value: a Math \[text\] that says "$upright("red"): quad n = 1.331, quad theta = 42.3 degree$" - secondary\_red: a ParametricCurve \[red\] drawn in bowl (function=\, t\_range=(0.3086759880022728, 2.83291666558752)) - secondary\_violet: a ParametricCurve \[magenta\] drawn in bowl (function=\, t\_range=(0.28624596875053626, 2.855346684839257)) - side: a Figure (x\_range=(0.0, 8.3), y\_range=(-0.3, 5.4), aspect=(8.3, 5.7)) - sun\_one: an Arrow \[yellow\] labelled "upright("sunlight")" drawn in side (start=(0.2, 5.05), end=(2.2, 5.05)) - sun\_two: an Arrow \[yellow\] drawn in side (start=(0.2, 4.55), end=(2.2, 4.55)) - violet\_in: an Arrow \[magenta\] drawn in drop\_plane (start=(-2.95, 0.94), end=(-0.34, 0.94)) - violet\_miss: an Arrow \[magenta\] drawn in side (start=(5.35, 4.3), end=(0.25, 2.15)) - violet\_one: an Arrow \[magenta\] drawn in drop\_plane (start=(-0.34, 0.94), end=(0.87, 0.49)) - violet\_out: an Arrow \[magenta\] drawn in drop\_plane (start=(0.3, -0.95), end=(-0.92, -2.78)) - violet\_seen: an Arrow \[magenta\] drawn in side (start=(5.7, 3.1), end=(1.0, 1.25)) - violet\_two: an Arrow \[magenta\] drawn in drop\_plane (start=(0.87, 0.49), end=(0.3, -0.95)) - violet\_value: a Math \[text\] that says "$upright("violet"): quad n = 1.344, quad theta = 40.6 degree$" #### Beats ##### [06:43.117](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334) Narration: White sunlight contains many colours, and water bends each one by a slightly different amount. Red bends least; violet bends most. Board: Empty. Actions: - [06:43.117](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334): heading\_drop is shown on the screen, written out. - [06:43.117](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334): caption\_drop is shown on the screen, written out. - [06:43.117](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=403.1172083333334): drop\_plane is shown on the screen, written out. - [06:45.66](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=405.6602083333334): drop is shown on the screen, drawn. ##### [06:52.796](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=412.79620833333337) Narration: Follow the least-deviated red path through the drop, then the violet path. They enter close together, separate through the two refractions, and leave in different directions. Board: caption\_drop — a Tex \[text\] that says "Path separation is exaggerated; the equations keep the true values."; drop\_plane — a Figure (x\_range=(-3.1, 2.0), y\_range=(-3.0, 1.3), aspect=(5.1, 4.3)); heading\_drop — a Heading that says "One Drop, Two Colours"; drop — a Circle \[blue\] drawn in drop\_plane Actions: - [06:54.735](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=414.7352083333334): red\_in is shown on the screen, drawn. - [06:54.944](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=414.9442083333334): red\_one is shown on the screen, drawn. - [06:55.257](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=415.2572083333334): red\_two is shown on the screen, drawn. - [06:55.606](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=415.60620833333337): red\_out is shown on the screen, drawn. - [06:56.743](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=416.74320833333337): violet\_in is shown on the screen, drawn. - [06:57.115](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=417.1152083333334): violet\_one is shown on the screen, drawn. - [07:0.981](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=420.98120833333337): violet\_two is shown on the screen, drawn. - [07:2.525](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=422.5252083333334): violet\_out is shown on the screen, drawn. ##### [07:4.577](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=424.5767083333334) Narration: For red, n is one point three three one and the viewing angle is forty two point three degrees. For violet, n is one point three four four and the viewing angle is forty point six degrees. Board: caption\_drop — a Tex \[text\] that says "Path separation is exaggerated; the equations keep the true values."; drop\_plane — a Figure (x\_range=(-3.1, 2.0), y\_range=(-3.0, 1.3), aspect=(5.1, 4.3)); heading\_drop — a Heading that says "One Drop, Two Colours"; drop — a Circle \[blue\] drawn in drop\_plane; red\_in — an Arrow \[red\] drawn in drop\_plane (start=(-2.95, 0.73), end=(-0.68, 0.73)); red\_one — an Arrow \[red\] drawn in drop\_plane (start=(-0.68, 0.73), end=(0.94, 0.34)); red\_two — an Arrow \[red\] drawn in drop\_plane (start=(0.94, 0.34), end=(0.0, -1.0)); red\_out — an Arrow \[red\] drawn in drop\_plane (start=(0.0, -1.0), end=(-1.85, -2.55)); violet\_in — an Arrow \[magenta\] drawn in drop\_plane (start=(-2.95, 0.94), end=(-0.34, 0.94)); violet\_one — an Arrow \[magenta\] drawn in drop\_plane (start=(-0.34, 0.94), end=(0.87, 0.49)); violet\_two — an Arrow \[magenta\] drawn in drop\_plane (start=(0.87, 0.49), end=(0.3, -0.95)); violet\_out — an Arrow \[magenta\] drawn in drop\_plane (start=(0.3, -0.95), end=(-0.92, -2.78)) Actions: - [07:5.215](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=425.2152083333334): red\_value is shown on the screen, written out. - [07:6.364](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=426.3642083333334): red\_value (the "1.331" part) is emphasized. - [07:9.371](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=429.3712083333334): red\_value (the "1.331" part) is no longer emphasized. - [07:9.371](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=429.3712083333334): red\_value (the "42.3" part) is emphasized. - [07:11.949](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=431.9492083333334): violet\_value is shown on the screen, written out. - [07:11.949](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=431.9492083333334): red\_value (the "42.3" part) is no longer emphasized. - [07:13.249](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=433.2492083333334): violet\_value (the "1.344" part) is emphasized. - [07:16.175](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=436.1752083333334): violet\_value (the "1.344" part) is no longer emphasized. - [07:16.175](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=436.1752083333334): violet\_value (the "40.6" part) is emphasized. - [07:17.963](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=437.9632083333334): violet\_value (the "40.6" part) is no longer emphasized. ##### [07:18.563](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=438.56320833333336) Narration: A single drop therefore does not send an entire rainbow to one eye. Its red and violet leave along different lines, so one viewpoint receives at most one narrow part of that colour fan. Board: caption\_drop — a Tex \[text\] that says "Path separation is exaggerated; the equations keep the true values."; red\_value — a Math \[text\] that says "$upright("red"): quad n = 1.331, quad theta = 42.3 degree$"; violet\_value — a Math \[text\] that says "$upright("violet"): quad n = 1.344, quad theta = 40.6 degree$"; drop\_plane — a Figure (x\_range=(-3.1, 2.0), y\_range=(-3.0, 1.3), aspect=(5.1, 4.3)); heading\_drop — a Heading that says "One Drop, Two Colours"; drop — a Circle \[blue\] drawn in drop\_plane; red\_in — an Arrow \[red\] drawn in drop\_plane (start=(-2.95, 0.73), end=(-0.68, 0.73)); red\_one — an Arrow \[red\] drawn in drop\_plane (start=(-0.68, 0.73), end=(0.94, 0.34)); red\_two — an Arrow \[red\] drawn in drop\_plane (start=(0.94, 0.34), end=(0.0, -1.0)); red\_out — an Arrow \[red\] drawn in drop\_plane (start=(0.0, -1.0), end=(-1.85, -2.55)); violet\_in — an Arrow \[magenta\] drawn in drop\_plane (start=(-2.95, 0.94), end=(-0.34, 0.94)); violet\_one — an Arrow \[magenta\] drawn in drop\_plane (start=(-0.34, 0.94), end=(0.87, 0.49)); violet\_two — an Arrow \[magenta\] drawn in drop\_plane (start=(0.87, 0.49), end=(0.3, -0.95)); violet\_out — an Arrow \[magenta\] drawn in drop\_plane (start=(0.3, -0.95), end=(-0.92, -2.78)) Actions: - [07:23.404](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=443.4042083333334): red\_out is indicated — a transient flash. - [07:23.706](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=443.7062083333334): violet\_out is indicated — a transient flash. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): caption\_drop is hidden from the screen — left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): drop\_plane is hidden from the screen — left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): drop is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): red\_in is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): red\_one is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): red\_two is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): red\_out is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): violet\_in is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): violet\_one is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): violet\_two is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): violet\_out is hidden from the screen — drop\_plane left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): heading\_drop is hidden from the screen — left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): red\_value is hidden from the screen — left the board. - [07:29.581](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=449.5807083333334): violet\_value is hidden from the screen — left the board. ##### [07:30.181](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=450.18070833333337) Narration: Put two drops back in the shower. One is higher above the anti-solar line and one is lower, while the sunlight reaches both from behind the observer. Board: Empty. Actions: - [07:30.181](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=450.18070833333337): heading\_sky is shown on the screen, written out. - [07:30.181](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=450.18070833333337): caption\_sky is shown on the screen, written out. - [07:30.181](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=450.18070833333337): side is shown on the screen, written out. - [07:33.64](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=453.64020833333336): drop\_high is shown on the screen, written out. - [07:34.453](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=454.4532083333334): anti\_line is shown on the screen, drawn. - [07:36.183](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=456.18320833333337): drop\_low is shown on the screen, written out. - [07:37.181](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=457.1812083333334): sun\_one is shown on the screen, written out. - [07:37.959](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=457.9592083333334): sun\_two is shown on the screen, written out. - [07:37.959](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=457.9592083333334): feed\_high is shown on the screen, drawn. - [07:38.259](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=458.2592083333334): feed\_low is shown on the screen, drawn. - [07:38.865](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=458.8652083333334): eye is shown on the screen, written out. ##### [07:40.463](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=460.4632083333334) Narration: The higher drop can send its red ray to the eye. Its violet ray passes over the observer's head and is missed. Board: side — a Figure (x\_range=(0.0, 8.3), y\_range=(-0.3, 5.4), aspect=(8.3, 5.7)); caption\_sky — a Tex \[text\] that says "Ray directions are widened here to make their order visible."; heading\_sky — a Heading that says "Why Red Is on the Outside"; eye — a Point \[text\] labelled "upright("you")" drawn in side (location=(1.0, 1.25)); anti\_line — an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in side (start=(1.0, 1.25), end=(8.0, 1.25)); sun\_one — an Arrow \[yellow\] labelled "upright("sunlight")" drawn in side (start=(0.2, 5.05), end=(2.2, 5.05)); sun\_two — an Arrow \[yellow\] drawn in side (start=(0.2, 4.55), end=(2.2, 4.55)); drop\_high — a Circle \[blue\] drawn in side (center=(5.35, 4.3), radius=0.23, filled=True); drop\_low — a Circle \[blue\] drawn in side (center=(5.7, 3.1), radius=0.23, filled=True); feed\_high — an Arrow \[yellow\] drawn in side (start=(3.6499999999999995, 4.3), end=(5.06, 4.3)); feed\_low — an Arrow \[yellow\] drawn in side (start=(4.0, 3.1), end=(5.41, 3.1)) Actions: - [07:42.344](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=462.34420833333337): red\_seen is shown on the screen, drawn. - [07:44.225](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=464.2252083333334): violet\_miss is shown on the screen, drawn. - [07:46.14](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=466.1402083333334): eye is indicated — a transient flash. ##### [07:48.052](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=468.0522083333334) Narration: The lower drop can send violet to the same eye. Its red ray passes below the observer's feet. Board: side — a Figure (x\_range=(0.0, 8.3), y\_range=(-0.3, 5.4), aspect=(8.3, 5.7)); caption\_sky — a Tex \[text\] that says "Ray directions are widened here to make their order visible."; heading\_sky — a Heading that says "Why Red Is on the Outside"; eye — a Point \[text\] labelled "upright("you")" drawn in side (location=(1.0, 1.25)); anti\_line — an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in side (start=(1.0, 1.25), end=(8.0, 1.25)); sun\_one — an Arrow \[yellow\] labelled "upright("sunlight")" drawn in side (start=(0.2, 5.05), end=(2.2, 5.05)); sun\_two — an Arrow \[yellow\] drawn in side (start=(0.2, 4.55), end=(2.2, 4.55)); drop\_high — a Circle \[blue\] drawn in side (center=(5.35, 4.3), radius=0.23, filled=True); drop\_low — a Circle \[blue\] drawn in side (center=(5.7, 3.1), radius=0.23, filled=True); feed\_high — an Arrow \[yellow\] drawn in side (start=(3.6499999999999995, 4.3), end=(5.06, 4.3)); feed\_low — an Arrow \[yellow\] drawn in side (start=(4.0, 3.1), end=(5.41, 3.1)); red\_seen — an Arrow \[red\] drawn in side (start=(5.35, 4.3), end=(1.0, 1.25)); violet\_miss — an Arrow \[magenta\] drawn in side (start=(5.35, 4.3), end=(0.25, 2.15)) Actions: - [07:49.712](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=469.71220833333336): violet\_seen is shown on the screen, drawn. - [07:51.907](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=471.9072083333334): red\_miss is shown on the screen, drawn. - [07:53.648](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=473.6482083333334): eye is indicated — a transient flash. ##### [07:55.026](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=475.0262083333334) Narration: Red therefore arrives from the larger viewing angle and sits on the outside of the primary bow. Violet arrives from the smaller angle and sits inside. Board: side — a Figure (x\_range=(0.0, 8.3), y\_range=(-0.3, 5.4), aspect=(8.3, 5.7)); caption\_sky — a Tex \[text\] that says "Ray directions are widened here to make their order visible."; heading\_sky — a Heading that says "Why Red Is on the Outside"; eye — a Point \[text\] labelled "upright("you")" drawn in side (location=(1.0, 1.25)); anti\_line — an Arrow \[gray\] labelled "upright("anti-solar direction")" drawn in side (start=(1.0, 1.25), end=(8.0, 1.25)); sun\_one — an Arrow \[yellow\] labelled "upright("sunlight")" drawn in side (start=(0.2, 5.05), end=(2.2, 5.05)); sun\_two — an Arrow \[yellow\] drawn in side (start=(0.2, 4.55), end=(2.2, 4.55)); drop\_high — a Circle \[blue\] drawn in side (center=(5.35, 4.3), radius=0.23, filled=True); drop\_low — a Circle \[blue\] drawn in side (center=(5.7, 3.1), radius=0.23, filled=True); feed\_high — an Arrow \[yellow\] drawn in side (start=(3.6499999999999995, 4.3), end=(5.06, 4.3)); feed\_low — an Arrow \[yellow\] drawn in side (start=(4.0, 3.1), end=(5.41, 3.1)); red\_seen — an Arrow \[red\] drawn in side (start=(5.35, 4.3), end=(1.0, 1.25)); violet\_miss — an Arrow \[magenta\] drawn in side (start=(5.35, 4.3), end=(0.25, 2.15)); violet\_seen — an Arrow \[magenta\] drawn in side (start=(5.7, 3.1), end=(1.0, 1.25)); red\_miss — an Arrow \[red\] drawn in side (start=(5.7, 3.1), end=(0.25, 0.2)) Actions: - [07:58.509](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=478.5092083333334): red\_seen is indicated — a transient flash. - [08:3.42](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=483.4202083333334): violet\_seen is indicated — a transient flash. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): caption\_sky is hidden from the screen — left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): heading\_sky is hidden from the screen — left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): side is hidden from the screen — left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): eye is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): anti\_line is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): sun\_one is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): sun\_two is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): drop\_high is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): drop\_low is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): feed\_high is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): feed\_low is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): red\_seen is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): violet\_miss is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): violet\_seen is hidden from the screen — side left the board. - [08:4.383](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.3832083333334): red\_miss is hidden from the screen — side left the board. ##### [08:4.983](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.9832083333334) Narration: The anti-solar point is the centre of the geometry. The horizon lies above it whenever the sun is above the horizon. Board: Empty. Actions: - [08:4.983](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.9832083333334): heading\_bow is shown on the screen, written out. - [08:4.983](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=484.9832083333334): bowl is shown on the screen, written out. - [08:6.748](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=486.7482083333334): head is shown on the screen, written out. - [08:8.838](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=488.8382083333334): horizon is shown on the screen, drawn. ##### [08:13.037](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=493.0367083333334) Narration: Rotate the primary viewing directions around that centre. Red traces the outer arc and violet traces the inner arc, with every point supplied by a different drop in the shower. Board: bowl — a Figure (x\_range=(-4.5, 4.5), y\_range=(-1.7, 3.5), aspect=(9.0, 5.2)); heading\_bow — a Heading that says "The Shape of the Bow"; head — a Point \[gray\] labelled "upright("anti-solar point")" drawn in bowl (location=(0.0, -1.2)); horizon — a Line \[gray\] labelled "upright("horizon")" drawn in bowl (start=(-4.35, 0.0), end=(4.35, 0.0), dashed=True) Actions: - [08:13.443](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=493.4432083333334): caption\_primary is shown on the screen, written out. - [08:17.645](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=497.6452083333334): primary\_red is shown on the screen, drawn. - [08:19.561](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=499.5612083333334): primary\_violet is shown on the screen, drawn. ##### [08:25.107](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=505.1072083333334) Narration: The complete geometry is circular, but the ground hides the portion below the horizon. What remains visible is an arc, not a full ring. Board: bowl — a Figure (x\_range=(-4.5, 4.5), y\_range=(-1.7, 3.5), aspect=(9.0, 5.2)); caption\_primary — a Tex \[text\] that says "Colour spacing is exaggerated so the order stays visible."; heading\_bow — a Heading that says "The Shape of the Bow"; head — a Point \[gray\] labelled "upright("anti-solar point")" drawn in bowl (location=(0.0, -1.2)); horizon — a Line \[gray\] labelled "upright("horizon")" drawn in bowl (start=(-4.35, 0.0), end=(4.35, 0.0), dashed=True); primary\_red — a ParametricCurve \[red\] drawn in bowl (function=\, t\_range=(0.3663488425130368, 2.7752438110767566)); primary\_violet — a ParametricCurve \[magenta\] drawn in bowl (function=\, t\_range=(0.40437323624907384, 2.7372194173407194)) Actions: - [08:29.646](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=509.6462083333334): horizon is indicated — a transient flash. - [08:32.503](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=512.5032083333334): primary\_red is indicated — a transient flash. - [08:32.503](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=512.5032083333334): primary\_violet is indicated — a transient flash. ##### [08:34.903](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=514.9027083333334) Narration: Look just inside the primary bow, then just outside it. The inside is brighter because a broad band of rays returns on that side of the minimum. The region just outside receives far fewer primary rays. Board: Unchanged from the preceding beat in this scene. Actions: - [08:35.843](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=515.8432083333335): inside\_mark is shown on the screen, grown. - [08:37.843](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=517.8432083333335): inside\_mark is hidden from the screen. - [08:38.2](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=518.2002083333334): outside\_mark is shown on the screen, grown. - [08:40.2](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=520.2002083333334): outside\_mark is hidden from the screen. ##### [08:49.598](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=529.5977083333335) Narration: A second internal reflection creates a secondary bow farther out. Its colour order reverses: violet is outside and red is inside. Board: Unchanged from the preceding beat in this scene. Actions: - [08:50.143](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=530.1432083333334): caption\_secondary is shown on the screen, written out. - [08:56.436](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=536.4362083333334): secondary\_violet is shown on the screen, drawn. - [08:57.933](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=537.9332083333334): secondary\_red is shown on the screen, drawn. ##### [09:0.275](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=540.2752083333335) Narration: One drop supplies the path, the minimum deviation supplies the brightness, dispersion supplies the colours, and rotation around the anti-solar point supplies the arc. Together they make the rainbow in the sky. Board: bowl — a Figure (x\_range=(-4.5, 4.5), y\_range=(-1.7, 3.5), aspect=(9.0, 5.2)); caption\_primary — a Tex \[text\] that says "Colour spacing is exaggerated so the order stays visible."; caption\_secondary — a Tex \[text\] that says "Secondary bow: violet outside, red inside."; heading\_bow — a Heading that says "The Shape of the Bow"; head — a Point \[gray\] labelled "upright("anti-solar point")" drawn in bowl (location=(0.0, -1.2)); horizon — a Line \[gray\] labelled "upright("horizon")" drawn in bowl (start=(-4.35, 0.0), end=(4.35, 0.0), dashed=True); primary\_red — a ParametricCurve \[red\] drawn in bowl (function=\, t\_range=(0.3663488425130368, 2.7752438110767566)); primary\_violet — a ParametricCurve \[magenta\] drawn in bowl (function=\, t\_range=(0.40437323624907384, 2.7372194173407194)); secondary\_violet — a ParametricCurve \[magenta\] drawn in bowl (function=\, t\_range=(0.28624596875053626, 2.855346684839257)); secondary\_red — a ParametricCurve \[red\] drawn in bowl (function=\, t\_range=(0.3086759880022728, 2.83291666558752)) Actions: - [09:3.119](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=543.1192083333334): primary\_red is indicated — a transient flash. - [09:6.997](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=546.9972083333334): primary\_violet is indicated — a transient flash. - [09:8.878](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=548.8782083333334): head is indicated — a transient flash. - [09:10.445](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=550.4452083333334): secondary\_violet is indicated — a transient flash. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): bowl is hidden from the screen — left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): head is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): horizon is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): primary\_red is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): primary\_violet is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): secondary\_violet is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): secondary\_red is hidden from the screen — bowl left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): caption\_primary is hidden from the screen — left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): caption\_secondary is hidden from the screen — left the board. - [09:14.475](https://academa.ai/lectures/why-the-rainbow-sits-at-42-degrees?t=554.4749583333335): heading\_bow is hidden from the screen — left the board.