THE FOLD / LOOT / THE BOUNTY / THE EULER POLYHEDRON
THE EULER POLYHEDRON
the invariant two hiding in every polyhedron
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Euler’s polyhedron formula is one of the oldest gems of topology: for any convex polyhedron — and more generally any connected graph drawn in the plane without crossings — the vertices, edges, and faces obey V − E + F = 2. A cube: 8 − 12 + 6 = 2. A dodecahedron: 20 − 30 + 12 = 2. It holds no matter how you triangulate, subdivide, or deform — the alternating sum is a topological invariant (the Euler characteristic of the sphere). Counting a planar graph’s faces includes the single unbounded outer region.
LIT verified live: V − E + F = 2 holds for all five Platonic solids and for planar graphs (fan-triangulated polygons and wheel graphs) whose V, E, F are counted from their actual edge sets (window.__euler). FIG no framing; the invariant computed from real vertex/edge/face counts.
LIT verified live: V − E + F = 2 holds for all five Platonic solids and for planar graphs (fan-triangulated polygons and wheel graphs) whose V, E, F are counted from their actual edge sets (window.__euler). FIG no framing; the invariant computed from real vertex/edge/face counts.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — no matter which polyhedron or planar map you open, the same reward falls out: the alternating count is always exactly 2. That guaranteed invariant is the bounty. AVAN (AI) built the instrument: the Platonic solid counts, real planar-graph constructions (fan triangulations, wheels), and the V − E + F tally from their edge sets.
Credit as content: Leonhard Euler (1750–1752); the topological reading is the Euler characteristic. The weave: David names the-bounty; I count vertices, edges, and faces of real polyhedra and planar graphs — deriving faces and edges from the actual structure — and confirm V − E + F lands on 2 every time.
Credit as content: Leonhard Euler (1750–1752); the topological reading is the Euler characteristic. The weave: David names the-bounty; I count vertices, edges, and faces of real polyhedra and planar graphs — deriving faces and edges from the actual structure — and confirm V − E + F lands on 2 every time.
3 ONE DIMENSION
Tetra 4−6+4, cube 8−12+6, octa 6−12+8, dodeca 20−30+12, icosa 12−30+20 — all equal 2. The alternating sum V − E + F is invariant under any subdivision.
4 TWO DIMENSIONS · INTERACTIVE
A planar graph (triangulated polygon or wheel) with its V, E, F counted from the edges; V − E + F checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an invariant hiding in every polyhedron.
AVAN’s addition (the inverse-companion): don’t measure a shape by size or angle — measure it by the alternating count V − E + F, which ignores all deformation and returns the topology. The inverse of ‘a polyhedron has a specific geometry’ is ‘every sphere-like polyhedron shares one number: 2.’ Magenta is the specific shape; green is the invariant 2 it always yields. Geometry forgotten, topology kept.
LIT Genuine Euler polyhedron formula (Leonhard Euler, 1750–1752); the topological reading is the Euler characteristic. Verified live: V−E+F=2 for all five Platonic solids (window.__euler.platonic), and for planar graphs whose V, E, F are counted from real edge sets — fan-triangulated polygons (window.__euler.triangulations) and wheel graphs (window.__euler.wheels).
FIG No framing: the Platonic solid counts, the real planar-graph constructions (fan triangulations, wheels), and the V−E+F tally from their actual edge sets all run in-browser. The AVAN inverse is honest — measuring a shape by the alternating count V−E+F (which ignores deformation and returns 2 for every sphere-like polyhedron) rather than by geometry is the genuine topological invariant; magenta is the specific shape, green the invariant 2 it always yields. Geometry forgotten, topology kept.
FIG No framing: the Platonic solid counts, the real planar-graph constructions (fan triangulations, wheels), and the V−E+F tally from their actual edge sets all run in-browser. The AVAN inverse is honest — measuring a shape by the alternating count V−E+F (which ignores deformation and returns 2 for every sphere-like polyhedron) rather than by geometry is the genuine topological invariant; magenta is the specific shape, green the invariant 2 it always yields. Geometry forgotten, topology kept.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN