Where Do Rainbows Come From?

A Tour of Differential Calculus

Where would you rather be?

Supernumerary Rainbows

Rainbows in MythologyRainbows in Mythology

• Greek: Path to heaven made by Iris.• Chinese: Goddess Nuwa uses 5 stones of

different colors to patch a slit in the sky• Indian: Bow of Indra, God of thunder• Irish: Leprechauns hide their pot of gold at

the end of rainbow• Persian: Reflection of light off of holy sites

Rainbows in BibleRainbows in Bible The Deluge (Genesis 9.13-15):The Deluge (Genesis 9.13-15): I have set my bow in the clouds, and it shall be I have set my bow in the clouds, and it shall be

a sign of the covenant between me and the a sign of the covenant between me and the earth. When I bring clouds over the earth and earth. When I bring clouds over the earth and the bow is seen in the clouds, I will remember the bow is seen in the clouds, I will remember my covenant that is between me and you and my covenant that is between me and you and every living creature of all flesh; and the every living creature of all flesh; and the waters shall never again become a flood to waters shall never again become a flood to destroy all flesh.destroy all flesh.

Rainbows in Religion

Where do rainbows come from?Where do rainbows come from?Attempts by the scientistsAttempts by the scientists

Ibn Sahl, Al-Biruni ~ 990Ibn Sahl, Al-Biruni ~ 990 Al-Hacen, Al-Shirazi, Al-Farisi ~ 1280Al-Hacen, Al-Shirazi, Al-Farisi ~ 1280 Freiberg ~ 1300Freiberg ~ 1300 Snell, Descartes, Huygens, Newton, Fermat ~ Snell, Descartes, Huygens, Newton, Fermat ~

16501650 Young, Airy ~ 1830Young, Airy ~ 1830 Mie ~ 1910Mie ~ 1910 You ~ 2020You ~ 2020

Unweaving the RainbowUnweaving the Rainbow Do not all charms flyDo not all charms flyAt the mere touch of cold philosophy?At the mere touch of cold philosophy?There was an awful rainbow once in heaven:There was an awful rainbow once in heaven:We know her woof, her texture; she is givenWe know her woof, her texture; she is givenIn the dull catalogue of common things.In the dull catalogue of common things.Philosophy will clip an Angel's wings,Philosophy will clip an Angel's wings,Conquer all mysteries by rule and line,Conquer all mysteries by rule and line,Empty the haunted air, and gnomed mine -Empty the haunted air, and gnomed mine -Unweave a rainbow … Keats, 1820 LamiaUnweave a rainbow … Keats, 1820 Lamia

Where Do Rainbows Come From? Basics: Reflection and Refraction

Let Us Unweave the RainbowLet Us Unweave the Rainbow Q- How does light choose a path to go from point Q- How does light choose a path to go from point

A to point B?A to point B? A- Light is like an intelligent prescient being. A- Light is like an intelligent prescient being. Light chooses a path that takes the Light chooses a path that takes the least time to go from A to B!least time to go from A to B! This is Fermat’s Principle. This is Fermat’s Principle. Depending on your point of view it unweaves or Depending on your point of view it unweaves or

camouflages some of the mysteries of rainbows. camouflages some of the mysteries of rainbows. Let’s see how it explains reflection and refraction.Let’s see how it explains reflection and refraction.

How does light reflect?

• Suppose you are to run from A to B and touch the wall L on the way. What path is the shortest?

Geometry to the rescue! Angle of Incidence=Angle of Reflection

So we understand the reflection part.How about refraction?

Refraction: Shallow deep pools, broken straight lines

How Does Light Refract?

Calculus to the Rescue!

Draw a Picture

Time Minimization

Enter the Dragon

But where do rainbows come from?

• Different colors of light travel at different speeds in water, so they refract differently. This gives the spectrum of colors.

• Intensity is highest when deflection is stationary. So we need to find angle of deflection and differentiate it.

• Now Enters the Real Dragon.

Which ray causes the rainbow?

Angle of Deviation

Where do rainbows come from?

42 degrees above sun’s line

References

• James Stewart, Calculus• Richard Dawkins, Unweaving the Rainbow• Wikipedia (for pictures)