THE GRANT PROJECTION THEOREM
A New Geometric Proof Demonstrates All 3D Polyhedral Topology Is Generated Deterministically from 2D Right Triangles
Abstract
We present a mathematical framework demonstrating that three-dimensional polyhedral topology can be generated entirely from two-dimensional right triangles. Given a right triangle with legs a, b and hypotenuse c, we show that the complete topological structure of a corresponding polyhedron—vertex count, edge count, face count, and face type—emerges through deterministic harmonic cascade mechanics.
The central result is that Euler’s formula (V − E + F = 2) is not an external constraint but an automatic consequence of the generation. The triangle forces valid spherical topology without requiring verification.
The projection algorithm proceeds through three stages: (1) vertex count from harmonic equilibrium (V = a+2b+c), (2) face type from integer resonance (k = 6−integer count), and (3) face/edge counts from standard polyhedral relations. We identify a special consecutiveleg family of Pythagorean triples that generates an infinite sequence of polyhedra with triangular faces and quantized vertex counts.
The framework suggests that polyhedral geometry is discrete rather than continuous— valid topologies form quantized sequences, not a continuum.
We further demonstrate that the 120-cell, one of the six regular polytopes in fourdimensional space, is not an external orthogonal extension of the regular dodecahedron but rather the inward self-similar projection of the dodecahedron under the harmonic cascade generated by a single golden right triangle with sides φ−1, 1, and φ (the Kepler triangle). No additional orthogonal dimension is required; the fourth coordinate emerges purely as the depth of recursion.
