Light in four dimensions, worked out and drawn beside our own. Everywhere below, 3D is warm and 4D is ice. Every number is computed, and a selftest checks them against what's known.
This is the light of ATOM⁴'s universe too. There, the vacuum that pins the force between charges at 1/r leaves light exactly as worked out here. What changes is how matter feels it: ATOM⁴ · light.
A source spreads its light over everything around it that it isn't already. A point in 3D has a sphere round it (area ∝ r²), so its light falls as 1/r². In 4D the sphere round a point is a 3-sphere (∝ r³): 1/r³. That's how KATABASIS's crystals are lit. A source of dimension k spreads into n − k − 1 directions:
Collimation is a narrow beam, in any number of dimensions. A 4D laser beam is a line whose cross-section is a small 3D ball, not a plane. It widens exactly as ours does, w(z) = w₀√(1 + (z/z_R)²), with z_R = πw₀²/λ. But its light is spread over a 3D cross-section, so on its axis it fades as (w₀/w)³ where ours fades as (w₀/w)². Far away, any laser is a point source. And a line meets your 3D slice in a single point, so unless the beam lies in your slice, a 4D laser is a dot hanging in the air that slides as you turn.
In 3D a flash is a shell of light that passes and is gone: behind the front it's dark again. That's Huygens' principle, and it holds only in odd dimensions. In even dimensions the response to an instant flash keeps going inside the light cone. On a pond (2D) the ripples ring on after a stone. In 4D the flash's front is followed by a negative wake, −1/4π²(t² − r²)^(3/2), dying like t⁻³. Here is a short pulse leaving a point. Gold is the field one way and violet the other, each row scaled to its brightest.
Light through a round hole of diameter D spreads into rings. In 3D the hole is a disc and the pattern is Airy's, (2J₁(x)/x)². In 4D the hole is a 3-ball and the pattern is (3 j₁(x)/x)², with j₁ the spherical Bessel function. That's spherical shells on a 3D screen, drawn here through their middle. The first dark ring moves out from 1.22 λ/D to 1.43 λ/D, so a 4D telescope of the same width resolves less.
Light's field is transverse, so it has one polarization fewer than space has dimensions: two in 3D, three in 4D. At a surface, one of them lies in the plane of incidence (p) and the rest across it (s). Fresnel's formulas are the same in any dimension, but 4D has two s directions to 3D's one. So unpolarized 4D light reflects (2R_s + R_p)/3 where ours reflects (R_s + R_p)/2. At Brewster's angle R_p = 0. Our glare is then polarized along a line, and 4D glare is polarized in a plane.
A hot body fills every mode of light with hν/(e^(hν/kT) − 1). The number of modes grows as ν^(n−1), so the spectrum is ν³/(eˣ − 1) in 3D and ν⁴/(eˣ − 1) in 4D. The peak moves to shorter wavelengths, and the power radiated grows as T⁵, not T⁴: double the temperature and a 4D star gives out 32 times the light, not 16. The swatches are the colour each would look to our eyes.
A rainbow is light turned back by raindrops, brightest about 42° from the point opposite the sun. Its geometry happens in one plane through each drop, so the angle is the same in 4D. But the rainbow is every direction at that angle, a cone, and you see a slice of directions. If the point opposite the sun leans β into w, the slice holds a circle of radius ρ, with cos 42° = cos β · cos ρ. Lean the sun into w and the bow shrinks. The violet goes first, at 40.3°, then the red at 42.3°. The sun itself is out of your view the moment it leans at all, yet its rainbow is still there.
In your slice the bow is a round cone of directions with its tip at your eye. A screen is a flat plane that misses the tip, and a plane cutting a cone off its tip draws a conic section. Look γ away from the point opposite the sun, and the bow on the screen has eccentricity e = sin γ / cos ρ. It's a circle head on, an ellipse, a parabola at γ = 90° − ρ, and past that one branch of a hyperbola (the other branch is behind you). Here, ρ = 42° for ever, and only your gaze changes the shape. In 4D, ρ shrinks as the sun leans into w, so holding your gaze still and leaning the sun turns a hyperbola into a parabola, an ellipse, a circle, a point, and nothing.
The sun leaning into w and you turning your slice through w are the same thing seen from two sides. Only where the point opposite the sun sits in your own frame matters: its parts a_F, a_R, a_U, a_N along your forward, right, up and w. Then:
the bow is in your slice if |a_N| ≤ sin 42°, and on your screen e = √(a_R² + a_U²) / cos 42°
So turning (forward toward right) and twisting (right toward w) change the bow's shape. Looking into w (forward toward w) can't: it leaves a_R and a_U alone, so the bow keeps its eccentricity while it shrinks and slides, until it's gone. On the map, every way of moving is a path. The bow is a hyperbola above the gold line and an ellipse below it, and there's none to the right of the grey line.
One level up: a 4D creature's retina is 3D, and its flat sensor cuts the 4D cone in a quadric: a spheroid, a paraboloid, or one sheet of a hyperboloid. Your screen is a slice of that.
To stand under it yourself and turn your slice: IRIS⁴.
Anything that happens in a plane is the same in any dimension. That covers refraction: the ray, the surface's normal and the bent ray share a plane, so Snell's law and total internal reflection are ours. A 4D ray can bend into w, though, and leave your slice. Slit diffraction spreads light only across the slit's narrow direction, so the sinc² fringes are ours. So are interference, Lambert's cosine law, and the conservation of a surface's brightness along a ray, though not its total light, which falls as 1/r³.
how it works · IRIS⁴: walk under the bow · KATABASIS · Foam⁴ · Four · the arcade