Colour by divisibility — a fractal falls out.
Pascal's triangle is the lookup table for the binomial coefficients C(n, k). Each number is just the sum of the two above it. Reduce every entry modulo a prime and the resulting colour pattern is a perfect, infinite fractal. The reason hides inside base-p arithmetic.
Stack the binomial coefficients C(n, k) into a triangle: 1 on top, then 1 1, then 1 2 1, and so on, each entry being the sum of the two above. Now keep only the remainder modulo a prime p. The picture that emerges is fractal, exact, and infinite.
Take p = 2. Mark every odd entry, leave every even one blank. Step back. You are looking at the Sierpiński triangle — not an approximation, the real thing, generated by pure counting. Three self-similar copies at every scale, forever.
Kummer proved in 1852 that the power of p dividing C(n, k) equals the number of carries when k + (n−k) is added in base p. So the fractal is a picture of when base-p addition is clean — geometry hiding inside arithmetic.
Write 1 at the apex. Underneath, every entry is the sum of the two above it. The rows you get — 1 · 1 1 · 1 2 1 · 1 3 3 1 · 1 4 6 4 1 — are the binomial coefficients C(n, k), counting how many ways to choose k items from n. They are also the only ingredient needed to make a fractal.
Replace each entry with its remainder modulo 2 and colour only the odd cells. Step back and the picture is the Sierpiński triangle — the exact self-similar gasket, all the way down. Any block of 2^k rows is three copies of the 2^(k−1) block arranged in a triangle, with the centre hollow.
Pick a prime and the rendering snaps to a clean fractal sieve; toggle in composites and the symmetry softens. Crank N up to 256 rows to see deeper self-similarity.
Why does mod p factor so cleanly? Kummer's theorem says the highest power of p dividing C(n, k) is the number of carries when k and (n − k) are added in base p. So C(n, k) ≢ 0 mod p precisely when no carry happens — when every base-p digit of k stays ≤ the corresponding digit of n. The fractal is a portrait of carry-free addition.
Type any n and k. Pick a prime. The viewer stacks the base-p digits of k and (n − k); columns that carry glow amber. Zero carries ⇒ the cell is coloured. One carry ⇒ the cell is blank.
Zero carries means C(n, k) is non-zero mod p — a coloured cell in the gasket. Any carry sends C(n, k) to 0 mod p — a blank cell.
Switch p to 3 and the same rule produces a triangle of six copies; p = 5 gives a fifteen-piece micro-tile; p = 7 a twenty-eight-piece one. Lucas' theorem (1878) makes it explicit: C(n, k) mod p factors into the product of C(nᵢ, kᵢ) mod p over base-p digits. Different prime, different fractal — same machinery.
The mod-p gasket has Hausdorff dimension log((p+1)p/2) / log p. For p = 2 it equals log 3 / log 2 ≈ 1.585; for p = 3 about 1.631; for p = 7 already 1.712. As p grows the dimension creeps toward 2 — the triangle fills more and more of the plane, but never quite all of it.
Strictly between a curve (1) and a region (2). As p grows the gasket fills more — the limit is 2, never reached.
Reduce mod 6 and Kummer's clean carry count breaks: the prime-power structure tangles. The picture is still self-similar overall — Chinese Remainder Theorem stacks the mod-2 and mod-3 patterns — but local symmetry frays. Stephen Wolfram catalogued these in NKS as nearest cellular-automaton kin to Pascal mod p.
The Explorer lets you push the row count to half a thousand and step through every prime modulus from 2 to 23.
→ Open the ExplorerHow and where this technique lives in the world today.
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