Where Do Rainbows Come From?
A Tour of Differential Calculus
Where would you rather be?
Supernumerary Rainbows
Rainbows 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 Bible
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The Deluge (Genesis 9.13-15):
I have set my bow in the clouds, and it shall be
a sign of the covenant between me and the
earth. When I bring clouds over the earth and
the bow is seen in the clouds, I will remember
my covenant that is between me and you and
every living creature of all flesh; and the
waters shall never again become a flood to
destroy all flesh.
Rainbows in Religion
Where do rainbows come from?
Attempts by the scientists
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Ibn Sahl, Al-Biruni ~ 990
Al-Hacen, Al-Shirazi, Al-Farisi ~ 1280
Freiberg ~ 1300
Snell, Descartes, Huygens, Newton, Fermat ~ 1650
Young, Airy ~ 1830
Mie ~ 1910
You ~ 2020
Unweaving the Rainbow
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel's wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnomed mine Unweave a rainbow … Keats, 1820 Lamia
Where Do Rainbows Come From?
Basics: Reflection and Refraction
Let Us Unweave the Rainbow
 Q- How does light choose a path to go from point
A to point B?
 A- Light is like an intelligent prescient being.
Light chooses a path that takes the
least time to go from A to B!
 This is Fermat’s Principle.
 Depending on your point of view it unweaves or
camouflages some of the mysteries of rainbows.
 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)