What Makes a Rainbow: Geometry, Optics, and Perception
Why rainbows are circles, why they're always 42°, and what you're actually seeing
Rainbows feel like magic — arcs of perfect color appearing after rain, always in the same sequence, always the same shape. But the physics is even more beautiful than the magic. Rainbows are geometry made visible, proof that light is a wave, and evidence that reality is stranger than it appears.
The Simple Explanation (That’s Incomplete)
Sunlight hits raindrops. White light separates into colors (dispersion). Light refracts entering the drop, reflects off the back, refracts again leaving. You see a rainbow.
This is true, but it raises questions:
- Why is the rainbow always curved?
- Why does red appear on top, violet on bottom?
- Why is the rainbow always the same angular size?
- Why can’t you reach the end of a rainbow?
Let’s go deeper.
Light as a Wave
White light from the Sun is a mix of wavelengths:
- Violet: ~400 nm
- Blue: ~450 nm
- Green: ~520 nm
- Yellow: ~580 nm
- Orange: ~600 nm
- Red: ~650 nm
When light enters water, it slows down. But different wavelengths slow down by different amounts.
Dispersion: The speed of light in a medium depends on wavelength. Shorter wavelengths (violet) slow down more than longer wavelengths (red).
This speed difference causes refraction angles to vary by color — that’s how a prism creates a spectrum.
The Path Through a Raindrop
Follow a ray of sunlight hitting a spherical raindrop:Real raindrops aren’t perfectly spherical (they’re slightly flattened), but close enough for rainbows.
The key: because of the specific geometry of a sphere and the refractive index of water, maximum intensity happens at a specific angle.
For red light: 42.3° from the anti-solar point. For violet light: 40.6° from the anti-solar point.
This is why rainbows are always the same size — the angle is set by physics, not by the raindrops' distance or size.
The Anti-Solar Point
The anti-solar point is directly opposite the Sun from your perspective.
If the Sun is behind you at 20° above the horizon, the anti-solar point is 20° below the horizon in front of you.
The rainbow forms a cone around this point, with half-angle ~42°.
You see an arc because the ground blocks the lower half of the cone. From an airplane, you can see a full circle rainbow.
"Every rainbow you see is centered on a line from the Sun through your eyes. Two people standing next to each other see different rainbows."
This is why you can’t reach the end of a rainbow — it moves with you. The rainbow isn’t “out there” at a fixed location. It’s a set of angles relative to your viewpoint.
Why Red Is on Top
The geometry determines color order:
- Red refracts less, exits at a larger angle (42.3°)
- Violet refracts more, exits at a smaller angle (40.6°)
From your perspective, looking at the rainbow:
- Red comes from drops higher in the sky (larger angle from anti-solar point)
- Violet comes from drops lower in the sky (smaller angle)
So red appears on top, violet on bottom. Always.
Secondary rainbows (fainter, outside the primary) reverse this order because light reflects twice inside the drop.
Why Rainbows Are Bright Where They Are
Here’s the subtle part: light exits the raindrop at all angles, not just 42°.
But at 42°, something special happens: caustic formation.
Many rays converge to the same exit angle, creating intensity concentration. This is a caustic — like the bright patterns at the bottom of a swimming pool.
At angles less than 42°, almost no light exits (the rays miss each other). This creates Alexander’s dark band — the noticeably darker region between primary and secondary rainbows.
Supernumerary Rainbows
Sometimes you see faint colored bands inside the main rainbow arc — pink, green, purple fringes.
These are supernumerary bows, and they’re proof that light is a wave.
Supernumerary bows are caused by interference — light waves from slightly different paths through the drop overlap and create constructive/destructive interference patterns.
Geometric optics (light as rays) can’t explain these. You need wave optics.
Thomas Young used supernumerary bows as evidence for the wave nature of light in the early 1800s.
Double Rainbows
A secondary rainbow forms when light reflects twice inside the raindrop before exiting.
- Primary rainbow: One internal reflection, 42° half-angle, red on top
- Secondary rainbow: Two internal reflections, 51° half-angle, violet on top (reversed!)
The secondary is fainter because some light escapes at each reflection (not all light bounces perfectly).
Between them: Alexander’s dark band, noticeably dimmer.
Triple and quadruple rainbows exist (three and four internal reflections) but are extremely rare and faint. They appear on the same side of the sky as the Sun, not opposite.
Circular Rainbows
From high altitude (airplane, mountain, helicopter), you can see a full circle rainbow.
The ground normally blocks the lower half. But if you’re above the rain, the full 42° cone is visible.
This proves rainbows are circles, not arcs — we just usually only see the top half.
Some lucky pilots and skydivers have photographed full-circle rainbows. They’re spectacular.
Moonbows
Rainbows can form from moonlight, not just sunlight.
Moonbows are much fainter (the Moon is ~400,000 times dimmer than the Sun) and often appear white to the human eye because our color vision doesn’t work well in low light.
But long-exposure photography reveals the colors — red, yellow, green, blue, violet, just like solar rainbows.
Same physics, different light source.
Polarization
Rainbow light is polarized — the electric field oscillates in a preferred direction.
Light that reflects once inside the drop (primary rainbow) is partially polarized. Secondary rainbows have different polarization.
If you look at a rainbow through polarizing sunglasses and rotate them, the rainbow brightness changes. This is evidence of polarization.
Bees and some other animals can see polarization naturally. They might perceive rainbows differently than we do.
Why You Can’t Reach the Endpoint
People ask: “What’s at the end of the rainbow?”
There is no endpoint. The rainbow is not a physical object at a location — it’s an optical phenomenon defined by angles.
As you move, the angles change, and you see light from different raindrops. The rainbow moves with you.
Two people standing apart see different rainbows (light from different sets of drops at the right angles for each observer).
This is why you can’t reach it. The rainbow doesn’t exist “out there” — it exists in the relationship between sunlight, water, and your eyes.
Rainbows on Other Worlds
On Mars: possible, but rare (very little rain)
On Titan (Saturn’s moon): methane rainbows! Titan has methane rain and a thick atmosphere. Methane has different refractive index, so the rainbow angles would differ.
On a planet with different atmospheric composition or different liquid droplets, rainbows would appear at different angles, possibly with different color sequences.
Physics is universal, but parameters vary.
The Math: Descartes and Snell
René Descartes worked out the geometry in 1637. The key is Snell’s law:
n₁ sin(θ₁) = n₂ sin(θ₂)Where n is refractive index, θ is angle from normal.
By tracing rays through a sphere and calculating where intensity peaks (via calculus), you get the rainbow angle: ~42°.
Descartes understood the geometry but didn’t know about wave interference or supernumerary bows. That had to wait for Young and wave optics.
Building Intuition
To understand rainbows:
Making Your Own Rainbow
You can create rainbows with:
- A garden hose (fine spray, Sun behind you)
- A glass of water and sunlight (refraction through curved surface)
- A CD or DVD (diffraction grating, not quite a rainbow but similar)
- Soap bubbles (interference colors, related phenomenon)
The physics is the same: dispersion, refraction, specific angles.
Why This Matters
Rainbows teach us:
Light is a wave: Supernumerary bows prove interference Geometry constrains physics: The 42° angle is inevitable given spheres and water’s refractive index Perception is relative: Your rainbow is different from your neighbor’s rainbow Beauty emerges from law: The elegance of rainbow colors comes from Maxwell’s equations and geometry
- Dispersion separates white light into colors
- Geometry determines the specific 42° angle
- Caustics create intensity peaks
- Interference produces supernumerary bows
- The rainbow moves with you — it's not "out there"
My Takeaway
Rainbows are geometry made visible.
Every rainbow is an demonstration of:
- Snell’s law (refraction)
- Wavelength-dependent dispersion
- Spherical geometry
- Caustic formation
- Wave interference (if you look closely)
The next time you see a rainbow, remember: you're seeing light sorted by wavelength, refracted through millions of spherical droplets, all at exactly 42° from the anti-solar point.
You're seeing physics.
And the fact that this physics produces something so aesthetically perfect — the smooth gradient of colors, the precise arc, the ethereal quality — that’s not design. That’s emergence.
Beauty from law. Order from chaos. Structure from simplicity.
That’s what rainbows are.
Resources: “Rainbows, Halos, and Glories” by Robert Greenler. Also try making your own rainbow with a garden hose on a sunny day — understanding is deeper when you create the phenomenon yourself.
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