Every semiprime — a number that is exactly two primes multiplied — has a natural triangle. Plant the two feet of its base at the two prime factors, p and q. Then send the apex climbing straight up. As it rises, the peak angle narrows — more and more acute, halving with every doubling of height, approaching zero but never touching it.
Drag up and down on the diagram to climb the apex · or use ↑ ↓ when focused
What the collapse tells you
The only thing that keeps the peak open is the base — the distance q − p between the two prime factors. At any height the angle is, near enough, that base divided by how high you've climbed. So the triangle is a magnifying glass for one thing: the gap between a number's two primes.
Try 121 = 11 × 11. Its two primes are equal, so the gap is zero, the base has no width, and the triangle is already flat — collapsed before it begins. Now try 166 = 2 × 83. The primes are far apart, the base is wide, and the peak stays stubbornly open, resisting the climb. The more lopsided the factorization, the slower the collapse. The more balanced, the faster it drills to nothing.
And that is the whole shape of the difficulty. A number with a small factor — like 166, with its 2 — keeps a wide base: there is a big, obvious signal, and such numbers fall to simple trial division in moments. A number whose two primes are nearly equal keeps almost no base at all: its triangle collapses instantly, there is no signal left to read, and it is exactly these balanced semiprimes that resist factoring. RSA chooses its two primes to be enormous and nearly equal on purpose — the flattest possible triangle, the one with nothing left to see.
An honest note: this is a way to see the difficulty, not a way to defeat it. The collapse rate reads the gap q − p — but knowing that gap, together with the number itself, is already to have the factors. The picture displays the factor relationship beautifully once you have it, and offers no shortcut to recover it when you don't. What it does show, plainly, is why the hard cases are hard: they are the triangles that have already collapsed into a line.