Every number contains a universe of divisors. The divisor lattice reveals their hidden structure — a geometric map where every path from 1 to n traces a different way to build the number from primes.
The divisor lattice is a Hasse diagram — a visualization of a partially ordered set. Two divisors are connected if one divides the other with no divisor in between. The result is a lattice structure where you can trace paths from 1 up to n.
Every path from 1 to n corresponds to a different way of building n through multiplication: if n = 60 = 2² × 3 × 5, one path might be 1 → 2 → 4 → 12 → 60, another might be 1 → 3 → 15 → 60. Each path represents a particular ordering of the prime factors.
The width of the lattice (the maximum number of divisors at any level) corresponds to how many "parallel" ways you can be partway through building n. Numbers with many small prime factors (highly composite numbers) have wide, complex lattices. Prime powers like 2⁸ have thin, linear lattices with a single path.
In tree terms, divisors aren't subtrees of n's tree (the Matula bijection doesn't preserve divisibility that directly). But the lattice reveals a different structure: how the multiplicative building blocks fit together. Each node in the lattice represents a number with its own tree structure, and the paths between them show how trees grow through prime multiplication.