{"version":1,"lectureId":"01M14V024XDHCN3G9SQFQ9ZS42","attempt":0,"publication":{"slug":"why-the-rainbow-sits-at-42-degrees","title":"Why the Rainbow Sits at 42 Degrees","subject":"physics","summary":"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.","metaDescription":"Derive the rainbow's 42 degree angle from one raindrop: refraction, one internal reflection, and the minimum of the deviation curve.","transcript":"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. 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. 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.","watch":{"version":1,"scenes":[{"title":"The Bow in the Sky","start":0,"end":96.92022916666667,"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=<function>, t_range=(0.411516846067488, 2.7300758075223053))","preview_violet":"a ParametricCurve [magenta] drawn in preview (function=<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":[{"start":0,"say":"A rainbow is not a coloured object hanging over a particular field. It is a viewing geometry that follows the observer.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[7.1635,"card is hidden from the screen — left the board."]]},{"start":7.7635,"say":"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.","live":null,"does":[[7.7635,"preview_heading is shown on the screen, written out."],[7.7635,"preview is shown on the screen, written out."],[10.503,"preview_horizon is shown on the screen, written out."],[11.501999999999999,"preview_red is shown on the screen, drawn."],[11.501999999999999,"preview_caption is shown on the screen, written out."],[11.802,"preview_violet is shown on the screen, drawn."],[11.838,"preview_head is shown on the screen, written out."],[15.659,"preview_answer is shown on the screen, written out."],[15.972000000000001,"preview_answer (the \"42\" part) is emphasized."],[20.3375,"preview is hidden from the screen — left the board."],[20.3375,"preview_horizon is hidden from the screen — preview left the board."],[20.3375,"preview_head is hidden from the screen — preview left the board."],[20.3375,"preview_red is hidden from the screen — preview left the board."],[20.3375,"preview_violet is hidden from the screen — preview left the board."],[20.3375,"preview_answer is hidden from the screen — left the board."],[20.3375,"preview_caption is hidden from the screen — left the board."],[20.3375,"preview_heading is hidden from the screen — left the board."],[20.3375,"preview_answer (the \"42\" part) is no longer emphasized."]]},{"start":20.9375,"say":"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.","live":null,"does":[[20.9375,"question is shown on the screen, written out."],[30.5855,"question moves to a new place on the board."]]},{"start":31.185499999999998,"say":"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.","live":["question"],"does":[[31.185499999999998,"sky is shown on the screen, written out."],[32.114,"sun_high is shown on the screen, written out."],[32.364,"sun_mid is shown on the screen, written out."],[33.995,"eye is shown on the screen, written out."],[36.921,"axis is shown on the screen, drawn."]]},{"start":41.665499999999994,"say":"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.","live":["question","sky","sun_high","sun_mid","eye","axis"],"does":[[42.16499999999999,"drop is shown on the screen, written out."],[42.81499999999999,"into_drop is shown on the screen, written out."],[45.16,"to_eye is shown on the screen, drawn."],[47.11099999999999,"bow_angle is shown on the screen, written out."],[50.08299999999999,"bow_angle is emphasized."],[57.48999999999999,"lower_shell is shown on the screen, drawn."],[59.99799999999999,"bow_angle is no longer emphasized."]]},{"start":60.598,"say":"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.","live":["question","sky","sun_high","sun_mid","eye","axis","drop","into_drop","to_eye","bow_angle","lower_shell"],"does":[[61.329,"eye_two is shown on the screen, written out."],[63.187000000000005,"axis_two is shown on the screen, drawn."],[65.219,"drop_two is shown on the screen, written out."],[66.6,"ray_two is shown on the screen, drawn."],[71.987,"eye_two is hidden from the screen."],[71.987,"axis_two is hidden from the screen."],[71.987,"drop_two is hidden from the screen."],[71.987,"ray_two is hidden from the screen."]]},{"start":72.587,"say":"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.","live":null,"does":[[75.44300000000001,"high_axis is shown on the screen, drawn."],[78.299,"below_bow is shown on the screen, written out."],[80.26100000000001,"high_axis is hidden from the screen."],[80.26100000000001,"below_bow is hidden from the screen."],[81.58500000000001,"drop is indicated — a transient flash."]]},{"start":83.706,"say":"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? 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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.","live":[],"does":[[96.92022916666667,"heading_path is shown on the screen, written out."],[96.92022916666667,"refraction is shown on the screen, written out."],[101.28522916666667,"ray_in is shown on the screen, drawn."],[102.12122916666667,"centre is shown on the screen, written out."],[104.22322916666667,"plane is shown on the screen, written out."],[108.87822916666667,"drop is shown on the screen, drawn."]]},{"start":115.26022916666668,"say":"At the surface, the normal is the radius through the point of contact. The incoming ray makes the incidence angle i with that normal.","live":["refraction","plane","heading_path","drop","centre","ray_in"],"does":[[117.11822916666668,"normal_entry is shown on the screen, drawn."],[120.94922916666667,"incidence_angle is shown on the screen, written out."]]},{"start":123.80172916666668,"say":"At that surface the light turns toward the normal as it enters water, so the angle inside, r, is smaller than i.","live":["refraction","plane","heading_path","drop","centre","ray_in","normal_entry","incidence_angle"],"does":[[125.36922916666667,"chord_one is shown on the screen, drawn."],[128.51522916666667,"refracted_angle is shown on the screen, written out."]]},{"start":131.51872916666667,"say":"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.","live":["refraction","plane","heading_path","drop","centre","ray_in","normal_entry","incidence_angle","chord_one","refracted_angle"],"does":[[131.86722916666668,"snell is shown on the screen, written out."],[134.45622916666667,"snell (the \"sin i\" part) is emphasized."],[135.58222916666668,"snell (the \"n thin sin r\" part) is emphasized."],[135.58222916666668,"snell (the \"sin i\" part) is no longer emphasized."],[138.9842291666667,"snell (the \"1.33\" part) is emphasized."],[138.9842291666667,"snell (the \"n thin sin r\" part) is no longer emphasized."],[139.98272916666667,"snell (the \"1.33\" part) is no longer emphasized."]]},{"start":140.58272916666667,"say":"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.","live":["refraction","snell","plane","heading_path","drop","centre","ray_in","normal_entry","incidence_angle","chord_one","refracted_angle"],"does":[[140.58272916666667,"incidence_angle is hidden from the screen."],[140.58272916666667,"refracted_angle is hidden from the screen."],[140.58272916666667,"normal_entry is hidden from the screen."],[143.74022916666667,"waste is shown on the screen, drawn."],[146.34122916666666,"waste is hidden from the screen."],[147.66422916666667,"normal_back is shown on the screen, drawn."],[148.57022916666668,"reflection_in is shown on the screen, written out."],[149.30122916666667,"chord_two is shown on the screen, drawn."],[151.24022916666667,"reflection_out is shown on the screen, written out."]]},{"start":153.0132291666667,"say":"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.","live":["refraction","snell","plane","heading_path","drop","centre","ray_in","chord_one","normal_back","reflection_in","chord_two","reflection_out"],"does":[[153.0132291666667,"reflection_in is hidden from the screen."],[153.0132291666667,"reflection_out is hidden from the screen."],[153.0132291666667,"normal_back is hidden from the screen."],[155.98522916666667,"normal_leave is shown on the screen, drawn."],[157.45922916666666,"ray_out is shown on the screen, drawn."],[159.19022916666665,"exit_inside is shown on the screen, written out."],[159.96722916666667,"exit_outside is shown on the screen, written out."]]},{"start":163.66772916666667,"say":"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.","live":["refraction","snell","plane","heading_path","drop","centre","ray_in","chord_one","chord_two","normal_leave","exit_inside","ray_out","exit_outside"],"does":[[164.19022916666668,"exit_outside is indicated — a transient flash."],[167.83522916666666,"exit_inside is indicated — a transient flash."],[171.49222916666668,"exit_inside is hidden from the screen."],[171.49222916666668,"exit_outside is hidden from the screen."],[171.49222916666668,"normal_leave is hidden from the screen."],[172.43272916666666,"plane moves to a new place on the board."],[172.43272916666666,"heading_path is hidden from the screen — left the board."],[172.43272916666666,"refraction is hidden from the screen — left the board."],[172.43272916666666,"snell is hidden from the screen — left the board."]]},{"start":173.03272916666668,"say":"Now measure how far the ray turns at each event. At the entrance it turns through i minus r.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out"],"does":[[173.03272916666668,"heading_turns is shown on the screen, written out."],[173.03272916666668,"claim is shown on the screen, written out."],[173.68322916666668,"turn_one_forward is shown on the screen, drawn."],[177.02622916666667,"turn_one is shown on the screen, written out."],[178.67522916666667,"turn_work is shown on the screen, written out."],[178.67522916666667,"turn_work (the \"i - r\" part) is emphasized."],[179.59822916666667,"turn_work (the \"i - r\" part) is no longer emphasized."]]},{"start":180.19822916666666,"say":"The reflection is the large turn. Equal angles r leave a straight angle minus two r between the old direction and the new one.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","claim","heading_turns","turn_one_forward","turn_one"],"does":[[180.48222916666668,"turn_two_forward is shown on the screen, drawn."],[181.30722916666667,"turn_two is shown on the screen, written out."],[184.18622916666666,"turn_work is shown on the screen, written out."],[184.87122916666667,"turn_work (the \"180 degree - 2 r\" part) is emphasized."],[187.99422916666668,"turn_work (the \"180 degree - 2 r\" part) is no longer emphasized."]]},{"start":188.59422916666665,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","claim","heading_turns","turn_one_forward","turn_one","turn_two_forward","turn_two"],"does":[[188.94222916666666,"turn_three_forward is shown on the screen, drawn."],[190.06822916666664,"turn_three is shown on the screen, written out."],[190.39322916666666,"turn_work (the \"i - r\" part) is emphasized."],[190.69522916666665,"turn_work is shown on the screen, written out."],[195.53622916666666,"turn_work (the \"i - r\" part) is emphasized."],[195.53622916666666,"turn_work (the \"i - r\" part) is no longer emphasized."],[196.90622916666666,"turn_work (the \"i - r\" part) is no longer emphasized."],[196.90622916666666,"turn_work (the \"180 degree - 2 r\" part) is emphasized."],[198.71722916666664,"turn_work (the \"180 degree - 2 r\" part) is no longer emphasized."],[198.71722916666664,"turn_work (the \"i - r\" part) is emphasized."],[199.89572916666668,"turn_work (the \"i - r\" part) is no longer emphasized."]]},{"start":200.49572916666665,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","claim","heading_turns","turn_one_forward","turn_one","turn_two_forward","turn_two","turn_three_forward","turn_three"],"does":[[202.90522916666666,"turn_work is shown on the screen, written out."],[203.90322916666668,"turn_work (the \"180 degree\" part) is emphasized."],[205.08822916666665,"turn_work (the \"180 degree\" part) is no longer emphasized."],[205.08822916666665,"turn_work (the \"2 i\" part) is emphasized."],[206.00522916666665,"turn_work (the \"2 i\" part) is no longer emphasized."],[206.00522916666665,"turn_work (the \"4 r\" part) is emphasized."],[207.54922916666663,"turn_work (the \"4 r\" part) is no longer emphasized."],[208.74522916666666,"carried_forward is shown on the screen, drawn."],[210.05722916666667,"total_angle is shown on the screen, written out."]]},{"start":213.25772916666665,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","claim","heading_turns","turn_one_forward","turn_one","turn_two_forward","turn_two","turn_three_forward","turn_three","carried_forward","total_angle"],"does":[[214.33722916666665,"centre_ray is shown on the screen, written out."],[221.28022916666666,"centre_ray is hidden from the screen."],[223.25422916666668,"turn_work is shown on the screen, written out."],[223.25422916666668,"turn_work (the \"sin^(-1)(frac(sin i, n))\" part) is emphasized."],[228.26972916666665,"plane moves to a new place on the board."],[228.26972916666665,"claim is hidden from the screen — left the board."],[228.26972916666665,"heading_turns is hidden from the screen — left the board."],[228.26972916666665,"turn_work is hidden from the screen — left the board."],[228.26972916666665,"turn_work (the \"sin^(-1)(frac(sin i, n))\" part) is no longer emphasized."]]},{"start":228.86972916666667,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","turn_one_forward","turn_one","turn_two_forward","turn_two","turn_three_forward","turn_three","carried_forward","total_angle"],"does":[[228.86972916666667,"heading_sweep is shown on the screen, written out."],[228.86972916666667,"label_drop is shown on the screen, written out."],[228.86972916666667,"label_plot_text is shown on the screen, written out."],[231.05222916666668,"ray_in is redrawn as the numbers it depends on change."],[231.05222916666668,"chord_one is redrawn as the numbers it depends on change."],[231.05222916666668,"chord_two is redrawn as the numbers it depends on change."],[231.05222916666668,"ray_out is redrawn as the numbers it depends on change."],[231.05222916666668,"curve is redrawn as the numbers it depends on change."],[231.05222916666668,"tracker is redrawn as the numbers it depends on change."],[231.05222916666668,"graph is shown on the screen, written out."],[231.05222916666668,"turn_one is hidden from the screen."],[231.05222916666668,"turn_two is hidden from the screen."],[231.05222916666668,"turn_three is hidden from the screen."],[231.05222916666668,"turn_one_forward is hidden from the screen."],[231.05222916666668,"turn_two_forward is hidden from the screen."],[231.05222916666668,"turn_three_forward is hidden from the screen."],[231.05222916666668,"carried_forward is hidden from the screen."],[231.05222916666668,"total_angle is hidden from the screen."],[231.05222916666668,"i_deg ticks to 20.0."],[235.11522916666667,"tracker is shown on the screen, written out."],[235.11522916666667,"curve is shown on the screen, written out."],[236.3112291666667,"start_readout is shown on the screen, written out."],[236.6022291666667,"start_readout (the \"160\" part) is emphasized."],[237.90222916666667,"start_readout (the \"160\" part) is no longer emphasized."]]},{"start":238.50222916666667,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","label_drop","label_plot_text","start_readout","graph","heading_sweep","curve","tracker"],"does":[[238.85022916666665,"ray_in is redrawn as the numbers it depends on change."],[238.85022916666665,"chord_one is redrawn as the numbers it depends on change."],[238.85022916666665,"chord_two is redrawn as the numbers it depends on change."],[238.85022916666665,"ray_out is redrawn as the numbers it depends on change."],[238.85022916666665,"curve is redrawn as the numbers it depends on change."],[238.85022916666665,"tracker is redrawn as the numbers it depends on change."],[238.85022916666665,"start_readout is hidden from the screen."],[238.85022916666665,"i_deg ticks to 59.4104730269434."],[248.42822916666665,"minimum_marker is shown on the screen, written out."],[248.42822916666665,"minimum_readout is shown on the screen, written out."],[249.12422916666665,"minimum_readout (the \"138\" part) is emphasized."],[249.87922916666668,"minimum_readout (the \"138\" part) is no longer emphasized."]]},{"start":250.47922916666667,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","label_drop","label_plot_text","minimum_readout","graph","heading_sweep","curve","tracker","minimum_marker"],"does":[[250.82722916666668,"ray_in is redrawn as the numbers it depends on change."],[250.82722916666668,"chord_one is redrawn as the numbers it depends on change."],[250.82722916666668,"chord_two is redrawn as the numbers it depends on change."],[250.82722916666668,"ray_out is redrawn as the numbers it depends on change."],[250.82722916666668,"curve is redrawn as the numbers it depends on change."],[250.82722916666668,"tracker is redrawn as the numbers it depends on change."],[250.82722916666668,"i_deg ticks to 82.0."],[253.42822916666668,"minimum_marker is indicated — a transient flash."]]},{"start":259.1832291666667,"say":"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.","live":null,"does":[[259.1832291666667,"ray_in is hidden from the screen."],[259.1832291666667,"chord_one is hidden from the screen."],[259.1832291666667,"chord_two is hidden from the screen."],[259.1832291666667,"ray_out is hidden from the screen."],[259.7052291666667,"flat is shown on the screen, written out."],[263.7802291666667,"low_in is shown on the screen, drawn."],[264.5002291666667,"high_in is shown on the screen, drawn."]]},{"start":268.4327291666667,"say":"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.","live":["plane","drop","centre","label_drop","label_plot_text","minimum_readout","graph","heading_sweep","curve","tracker","minimum_marker","low_in","high_in","flat"],"does":[[269.1642291666667,"low_out is shown on the screen, drawn."],[271.4162291666667,"high_out is shown on the screen, drawn."],[273.2042291666667,"low_in is indicated — a transient flash."],[273.51822916666674,"high_in is indicated — a transient flash."],[280.1007291666667,"tracker is hidden from the screen."]]},{"start":280.7007291666667,"say":"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.","live":["plane","drop","centre","label_drop","label_plot_text","minimum_readout","graph","heading_sweep","curve","minimum_marker","low_in","high_in","flat","low_out","high_out"],"does":[[280.7007291666667,"graph moves to a new place on the board."],[280.7007291666667,"heading_sweep is hidden from the screen — left the board."],[280.7007291666667,"label_drop is hidden from the screen — left the board."],[280.7007291666667,"label_plot_text is hidden from the screen — left the board."],[280.7007291666667,"minimum_readout is hidden from the screen — left the board."],[280.7007291666667,"plane is hidden from the screen — left the board."],[280.7007291666667,"drop is hidden from the screen — plane left the board."],[280.7007291666667,"centre is hidden from the screen — plane left the board."],[280.7007291666667,"low_in is hidden from the screen — plane left the board."],[280.7007291666667,"high_in is hidden from the screen — plane left the board."],[280.7007291666667,"low_out is hidden from the screen — plane left the board."],[280.7007291666667,"high_out is hidden from the screen — plane left the board."],[280.7007291666667,"heading_calc is shown on the screen, written out."],[280.7007291666667,"calc is shown on the screen, written out."],[290.7092291666667,"calc is shown on the screen, written out."]]},{"start":292.29572916666666,"say":"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.","live":["graph","curve","minimum_marker","flat","heading_calc"],"does":[[296.3822291666667,"calc is shown on the screen, written out."],[296.3822291666667,"calc (the \"frac(1, 2)\" part) is emphasized."],[301.9552291666667,"calc (the \"frac(1, 2)\" part) is no longer emphasized."]]},{"start":302.55522916666666,"say":"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.","live":null,"does":[[305.1792291666667,"calc is shown on the screen, written out."],[310.47322916666667,"calc is shown on the screen, written out."]]},{"start":314.3817291666667,"say":"Now square both sides. Four cosine squared i equals n squared cosine squared r.","live":null,"does":[[314.97422916666676,"calc is shown on the screen, written out."],[316.83122916666673,"calc (the \"4 cos^2 i\" part) is emphasized."],[319.1072291666667,"calc (the \"4 cos^2 i\" part) is no longer emphasized."],[319.1072291666667,"calc (the \"n^2 cos^2 r\" part) is emphasized."],[321.46372916666667,"calc (the \"n^2 cos^2 r\" part) is no longer emphasized."]]},{"start":322.0637291666667,"say":"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.","live":null,"does":[[324.3272291666667,"calc is shown on the screen, written out."],[326.91722916666674,"calc is shown on the screen, written out."],[332.01322916666675,"calc is shown on the screen, written out."]]},{"start":336.8397291666667,"say":"Collect the cosine terms. Cosine i is the square root of n squared minus one, divided by three.","live":null,"does":[[340.9722291666667,"calc is shown on the screen, written out."],[340.9722291666667,"calc (the \"sqrt(frac(n^2 - 1, 3))\" part) is emphasized."],[344.6527291666667,"calc (the \"sqrt(frac(n^2 - 1, 3))\" part) is no longer emphasized."]]},{"start":345.2527291666667,"say":"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.","live":null,"does":[[347.1102291666667,"answer_angles is shown on the screen, written out."],[347.99222916666673,"answer_angles (the \"59.4\" part) is emphasized."],[349.62922916666673,"answer_angles (the \"40.2\" part) is emphasized."],[349.62922916666673,"answer_angles (the \"59.4\" part) is no longer emphasized."],[353.8322291666667,"answer_deviation is shown on the screen, written out."],[355.71322916666674,"answer_deviation (the \"138\" part) is emphasized."],[357.2582291666667,"answer_deviation moves to a new place on the board."],[357.2582291666667,"answer_angles is hidden from the screen — left the board."],[357.2582291666667,"calc is hidden from the screen — left the board."],[357.2582291666667,"graph is hidden from the screen — left the board."],[357.2582291666667,"curve is hidden from the screen — graph left the board."],[357.2582291666667,"minimum_marker is hidden from the screen — graph left the board."],[357.2582291666667,"flat is hidden from the screen — graph left the board."],[357.2582291666667,"heading_calc is hidden from the screen — left the board."],[357.2582291666667,"plane is shown on the screen, faded in — cast on this board again."],[357.2582291666667,"drop is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"centre is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"low_in is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"high_in is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"low_out is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"high_out is shown on the screen, faded in — plane came back to the board."],[357.2582291666667,"answer_deviation (the \"138\" part) is no longer emphasized."]]},{"start":357.8582291666667,"say":"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.","live":["plane","drop","centre","low_in","high_in","low_out","high_out","answer_deviation"],"does":[[357.8582291666667,"heading_answer is shown on the screen, written out."],[357.8582291666667,"ray_in is shown on the screen, written out."],[357.8582291666667,"chord_one is shown on the screen, written out."],[357.8582291666667,"chord_two is shown on the screen, written out."],[357.8582291666667,"ray_out is shown on the screen, written out."],[359.0072291666667,"i_deg ticks to 59.4104730269434."],[359.3582291666667,"ray_in is redrawn as the numbers it depends on change."],[359.3582291666667,"chord_one is redrawn as the numbers it depends on change."],[359.3582291666667,"chord_two is redrawn as the numbers it depends on change."],[359.3582291666667,"ray_out is redrawn as the numbers it depends on change."]]},{"start":367.5367291666667,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","low_in","high_in","low_out","high_out","answer_deviation","heading_answer"],"does":[[371.40322916666673,"straight_forward is shown on the screen, drawn."],[371.99422916666674,"straight_back is shown on the screen, drawn."],[374.2122291666667,"answer_stack is shown on the screen, written out."],[374.5252291666667,"answer_stack (the \"180\" part) is emphasized."],[376.6972291666667,"answer_stack (the \"180\" part) is no longer emphasized."],[377.02222916666676,"drawn_deviation is shown on the screen, written out."],[379.12322916666676,"answer_stack is shown on the screen, written out."],[379.40222916666676,"answer_stack (the \"138\" part) is emphasized."],[380.5047291666667,"answer_stack (the \"138\" part) is no longer emphasized."]]},{"start":381.1047291666667,"say":"Subtract that deviation from the straight angle. The remaining angle between the outgoing ray and straight back is forty two degrees.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","low_in","high_in","low_out","high_out","answer_deviation","heading_answer","straight_forward","straight_back","drawn_deviation"],"does":[[381.4062291666667,"answer_stack is shown on the screen, drawn."],[382.20622916666673,"answer_stack is shown on the screen, drawn."],[384.72722916666675,"drawn_return is shown on the screen, written out."],[387.9542291666667,"answer_stack is shown on the screen, written out."],[388.2672291666667,"answer_stack (the \"42\" part) is emphasized."],[389.2777291666667,"answer_stack (the \"42\" part) is no longer emphasized."]]},{"start":389.8777291666667,"say":"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.","live":["plane","drop","centre","ray_in","chord_one","chord_two","ray_out","low_in","high_in","low_out","high_out","answer_deviation","heading_answer","straight_forward","straight_back","drawn_deviation","drawn_return"],"does":[[390.6322291666667,"A box is drawn around answer_stack."],[400.5242291666667,"drawn_return is indicated — a transient flash."],[402.0755416666667,"answer_deviation is hidden from the screen — left the board."],[402.0755416666667,"answer_stack is hidden from the screen — left the board."],[402.0755416666667,"heading_answer is hidden from the screen — left the board."],[402.0755416666667,"plane is hidden from the screen — left the board."],[402.0755416666667,"drop is hidden from the screen — plane left the board."],[402.0755416666667,"centre is hidden from the screen — plane left the board."],[402.0755416666667,"ray_in is hidden from the screen — plane left the board."],[402.0755416666667,"chord_one is hidden from the screen — plane left the board."],[402.0755416666667,"chord_two is hidden from the screen — plane left the board."],[402.0755416666667,"ray_out is hidden from the screen — plane left the board."],[402.0755416666667,"low_in is hidden from the screen — plane left the board."],[402.0755416666667,"high_in is hidden from the screen — plane left the board."],[402.0755416666667,"low_out is hidden from the screen — plane left the board."],[402.0755416666667,"high_out is hidden from the screen — plane left the board."],[402.0755416666667,"straight_forward is hidden from the screen — plane left the board."],[402.0755416666667,"straight_back is hidden from the screen — plane left the board."],[402.0755416666667,"drawn_deviation is hidden from the screen — plane left the board."],[402.0755416666667,"drawn_return is hidden from the screen — plane left the board."]]}]},{"title":"Assembling the Sky","start":403.1172083333334,"end":555.5166250000001,"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=<function>, t_range=(0.3663488425130368, 2.7752438110767566))","primary_violet":"a ParametricCurve [magenta] drawn in bowl (function=<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=<function>, t_range=(0.3086759880022728, 2.83291666558752))","secondary_violet":"a ParametricCurve [magenta] drawn in bowl (function=<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":[{"start":403.1172083333334,"say":"White sunlight contains many colours, and water bends each one by a slightly different amount. Red bends least; violet bends most.","live":[],"does":[[403.1172083333334,"heading_drop is shown on the screen, written out."],[403.1172083333334,"caption_drop is shown on the screen, written out."],[403.1172083333334,"drop_plane is shown on the screen, written out."],[405.6602083333334,"drop is shown on the screen, drawn."]]},{"start":412.79620833333337,"say":"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.","live":["caption_drop","drop_plane","heading_drop","drop"],"does":[[414.7352083333334,"red_in is shown on the screen, drawn."],[414.9442083333334,"red_one is shown on the screen, drawn."],[415.2572083333334,"red_two is shown on the screen, drawn."],[415.60620833333337,"red_out is shown on the screen, drawn."],[416.74320833333337,"violet_in is shown on the screen, drawn."],[417.1152083333334,"violet_one is shown on the screen, drawn."],[420.98120833333337,"violet_two is shown on the screen, drawn."],[422.5252083333334,"violet_out is shown on the screen, drawn."]]},{"start":424.5767083333334,"say":"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.","live":["caption_drop","drop_plane","heading_drop","drop","red_in","red_one","red_two","red_out","violet_in","violet_one","violet_two","violet_out"],"does":[[425.2152083333334,"red_value is shown on the screen, written out."],[426.3642083333334,"red_value (the \"1.331\" part) is emphasized."],[429.3712083333334,"red_value (the \"1.331\" part) is no longer emphasized."],[429.3712083333334,"red_value (the \"42.3\" part) is emphasized."],[431.9492083333334,"violet_value is shown on the screen, written out."],[431.9492083333334,"red_value (the \"42.3\" part) is no longer emphasized."],[433.2492083333334,"violet_value (the \"1.344\" part) is emphasized."],[436.1752083333334,"violet_value (the \"1.344\" part) is no longer emphasized."],[436.1752083333334,"violet_value (the \"40.6\" part) is emphasized."],[437.9632083333334,"violet_value (the \"40.6\" part) is no longer emphasized."]]},{"start":438.56320833333336,"say":"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.","live":["caption_drop","red_value","violet_value","drop_plane","heading_drop","drop","red_in","red_one","red_two","red_out","violet_in","violet_one","violet_two","violet_out"],"does":[[443.4042083333334,"red_out is indicated — a transient flash."],[443.7062083333334,"violet_out is indicated — a transient flash."],[449.5807083333334,"caption_drop is hidden from the screen — left the board."],[449.5807083333334,"drop_plane is hidden from the screen — left the board."],[449.5807083333334,"drop is hidden from the screen — drop_plane left the board."],[449.5807083333334,"red_in is hidden from the screen — drop_plane left the board."],[449.5807083333334,"red_one is hidden from the screen — drop_plane left the board."],[449.5807083333334,"red_two is hidden from the screen — drop_plane left the board."],[449.5807083333334,"red_out is hidden from the screen — drop_plane left the board."],[449.5807083333334,"violet_in is hidden from the screen — drop_plane left the board."],[449.5807083333334,"violet_one is hidden from the screen — drop_plane left the board."],[449.5807083333334,"violet_two is hidden from the screen — drop_plane left the board."],[449.5807083333334,"violet_out is hidden from the screen — drop_plane left the board."],[449.5807083333334,"heading_drop is hidden from the screen — left the board."],[449.5807083333334,"red_value is hidden from the screen — left the board."],[449.5807083333334,"violet_value is hidden from the screen — left the board."]]},{"start":450.18070833333337,"say":"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.","live":[],"does":[[450.18070833333337,"heading_sky is shown on the screen, written out."],[450.18070833333337,"caption_sky is shown on the screen, written out."],[450.18070833333337,"side is shown on the screen, written out."],[453.64020833333336,"drop_high is shown on the screen, written out."],[454.4532083333334,"anti_line is shown on the screen, drawn."],[456.18320833333337,"drop_low is shown on the screen, written out."],[457.1812083333334,"sun_one is shown on the screen, written out."],[457.9592083333334,"sun_two is shown on the screen, written out."],[457.9592083333334,"feed_high is shown on the screen, drawn."],[458.2592083333334,"feed_low is shown on the screen, drawn."],[458.8652083333334,"eye is shown on the screen, written out."]]},{"start":460.4632083333334,"say":"The higher drop can send its red ray to the eye. Its violet ray passes over the observer's head and is missed.","live":["side","caption_sky","heading_sky","eye","anti_line","sun_one","sun_two","drop_high","drop_low","feed_high","feed_low"],"does":[[462.34420833333337,"red_seen is shown on the screen, drawn."],[464.2252083333334,"violet_miss is shown on the screen, drawn."],[466.1402083333334,"eye is indicated — a transient flash."]]},{"start":468.0522083333334,"say":"The lower drop can send violet to the same eye. Its red ray passes below the observer's feet.","live":["side","caption_sky","heading_sky","eye","anti_line","sun_one","sun_two","drop_high","drop_low","feed_high","feed_low","red_seen","violet_miss"],"does":[[469.71220833333336,"violet_seen is shown on the screen, drawn."],[471.9072083333334,"red_miss is shown on the screen, drawn."],[473.6482083333334,"eye is indicated — a transient flash."]]},{"start":475.0262083333334,"say":"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.","live":["side","caption_sky","heading_sky","eye","anti_line","sun_one","sun_two","drop_high","drop_low","feed_high","feed_low","red_seen","violet_miss","violet_seen","red_miss"],"does":[[478.5092083333334,"red_seen is indicated — a transient flash."],[483.4202083333334,"violet_seen is indicated — a transient flash."],[484.3832083333334,"caption_sky is hidden from the screen — left the board."],[484.3832083333334,"heading_sky is hidden from the screen — left the board."],[484.3832083333334,"side is hidden from the screen — left the board."],[484.3832083333334,"eye is hidden from the screen — side left the board."],[484.3832083333334,"anti_line is hidden from the screen — side left the board."],[484.3832083333334,"sun_one is hidden from the screen — side left the board."],[484.3832083333334,"sun_two is hidden from the screen — side left the board."],[484.3832083333334,"drop_high is hidden from the screen — side left the board."],[484.3832083333334,"drop_low is hidden from the screen — side left the board."],[484.3832083333334,"feed_high is hidden from the screen — side left the board."],[484.3832083333334,"feed_low is hidden from the screen — side left the board."],[484.3832083333334,"red_seen is hidden from the screen — side left the board."],[484.3832083333334,"violet_miss is hidden from the screen — side left the board."],[484.3832083333334,"violet_seen is hidden from the screen — side left the board."],[484.3832083333334,"red_miss is hidden from the screen — side left the board."]]},{"start":484.9832083333334,"say":"The anti-solar point is the centre of the geometry. The horizon lies above it whenever the sun is above the horizon.","live":[],"does":[[484.9832083333334,"heading_bow is shown on the screen, written out."],[484.9832083333334,"bowl is shown on the screen, written out."],[486.7482083333334,"head is shown on the screen, written out."],[488.8382083333334,"horizon is shown on the screen, drawn."]]},{"start":493.0367083333334,"say":"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.","live":["bowl","heading_bow","head","horizon"],"does":[[493.4432083333334,"caption_primary is shown on the screen, written out."],[497.6452083333334,"primary_red is shown on the screen, drawn."],[499.5612083333334,"primary_violet is shown on the screen, drawn."]]},{"start":505.1072083333334,"say":"The complete geometry is circular, but the ground hides the portion below the horizon. What remains visible is an arc, not a full ring.","live":["bowl","caption_primary","heading_bow","head","horizon","primary_red","primary_violet"],"does":[[509.6462083333334,"horizon is indicated — a transient flash."],[512.5032083333334,"primary_red is indicated — a transient flash."],[512.5032083333334,"primary_violet is indicated — a transient flash."]]},{"start":514.9027083333334,"say":"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.","live":null,"does":[[515.8432083333335,"inside_mark is shown on the screen, grown."],[517.8432083333335,"inside_mark is hidden from the screen."],[518.2002083333334,"outside_mark is shown on the screen, grown."],[520.2002083333334,"outside_mark is hidden from the screen."]]},{"start":529.5977083333335,"say":"A second internal reflection creates a secondary bow farther out. Its colour order reverses: violet is outside and red is inside.","live":null,"does":[[530.1432083333334,"caption_secondary is shown on the screen, written out."],[536.4362083333334,"secondary_violet is shown on the screen, drawn."],[537.9332083333334,"secondary_red is shown on the screen, drawn."]]},{"start":540.2752083333335,"say":"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.","live":["bowl","caption_primary","caption_secondary","heading_bow","head","horizon","primary_red","primary_violet","secondary_violet","secondary_red"],"does":[[543.1192083333334,"primary_red is indicated — a transient flash."],[546.9972083333334,"primary_violet is indicated — a transient flash."],[548.8782083333334,"head is indicated — a transient flash."],[550.4452083333334,"secondary_violet is indicated — a transient flash."],[554.4749583333335,"bowl is hidden from the screen — left the board."],[554.4749583333335,"head is hidden from the screen — bowl left the board."],[554.4749583333335,"horizon is hidden from the screen — bowl left the board."],[554.4749583333335,"primary_red is hidden from the screen — bowl left the board."],[554.4749583333335,"primary_violet is hidden from the screen — bowl left the board."],[554.4749583333335,"secondary_violet is hidden from the screen — bowl left the board."],[554.4749583333335,"secondary_red is hidden from the screen — bowl left the board."],[554.4749583333335,"caption_primary is hidden from the screen — left the board."],[554.4749583333335,"caption_secondary is hidden from the screen — left the board."],[554.4749583333335,"heading_bow is hidden from the screen — left the board."]]}]}]},"durationSeconds":556,"chapters":[{"title":"The Bow in the Sky","startSeconds":0,"narration":"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."},{"title":"One Drop, One Ray","startSeconds":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. 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."},{"title":"Assembling the Sky","startSeconds":403.1172083333334,"narration":"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."}]}}