A new class of constants in geometry
by Grego (Gergely Földvári)
The cube and the English alphabet share a count: the cube has 26 characteristic elements — 8 vertices, 12 edges and 6 faces — and the alphabet has 26 letters. Labelling the one with the other gives the English Cube. The 26 elements are joined by 253 non-overlapping segments whose total length is a constant, here called a Metatron Number. For the unit cube that grand constant, in a deliberately symmetric spelling, is
Cg()253cbs,12 =
24√1 + 36√2 + 16√3 + 7 + 16√4 + 36√5 + 24√6
≈ 280.9107
The paper sets out the counting, tabulates the eight distinct segment lengths, gives the constants for the surface, the centre and the body separately, proposes a notation, and extends the idea to other polyhedra.
The complete graph on 26 points has (26 × 25) / 2 = 325 connections. Of these, 72 are collinear with others and are absorbed: 24 vertex-to-edge, 24 vertex-to-face and 24 edge-to-face segments of length √2 / 2. That leaves 253 genuinely distinct segments.
253 is both a star number — the seventh, since 6 × 7 × 6 + 1 = 253 — and one of Euler's and Gauss' 65 idoneal numbers (numeri idonei, named here in Hungarian as hasznos számok for the first time in known Hungarian mathematical literature). It is also the number of pairings of 23 points, as used in the Witt Steiner system S(4, 7, 23).
| Code | Count | Length | ≈ | Joins | |
|---|---|---|---|---|---|
| 1 | CBD | 4 | √3 | 1.7321 | VV(B) |
| 2 | HCBD | 24 | 3/2 | 1.5000 | VE(B) |
| 3 | CFD | 18 | √2 | 1.4142 | VV(S1), EE(B1) |
| 4 | QCBD | 48 | √6/2 | 1.2247 | VF(B), EE(B2) |
| 5 | QCFD | 72 | √5/2 | 1.1180 | VE(S), EF(B1) |
| 6 | CE | 27 | 1 | 1.0000 | VV(12), EE(S1), FF(B1) |
| 7 | ECBD | 24 | √3/2 | 0.8660 | EF(B2) |
| 8 | ECFD | 36 | √2/2 | 0.7071 | EE(S2), FF(B2) |
| Σ | 253 | 280.9107 |
| Constant | Segments | Expansion | Value |
|---|---|---|---|
| C()96s | 96 | 12√1 + 24√2 + 24√5 | 99.6068 |
| C()108s,12 | 108 | 24√1 + 24√2 + 24√5 | 111.6068 |
| C()13c | 13 | 3√1 + 6√2 + 4√3 | 18.4134846 |
| C()25c,12 | 25 | 15√1 + 6√2 + 4√3 | 30.4134846 |
| C()132b | 132 | 36√1 + 6√2 + 12√3 + 12√5 + 24√6 | 150.8907 |
| C()144b,12 | 144 | 48√1 + 6√2 + 12√3 + 12√5 + 24√6 | 162.8907 |
| Cg()253cbs,12 | 253 | 24√1 + 36√2 + 16√3 + 7 + 16√4 + 36√5 + 24√6 | 280.9107 |
Every Metatron Number is written the same way. The sign carries four pieces of information around it, and each answers one question: of what solid, in what part, over how many segments, and between which kinds of element.
The anatomy of the notation, read on the cube's grand constant.
Before the bracket stands a capital letter for the polyhedron, and after it, in lower case, an optional attribute — g for the grand constant, taken over every part at once.
| C cube | T tetrahedron | O octahedron |
| D dodecahedron | I icosahedron | CO cuboctahedron |
| SnC snub cube | RhCO rhombicuboctahedron | RhD rhombic dodecahedron |
The closing bracket carries a superscript giving how many segments were counted, and a subscript giving what they run between. A trailing number in the subscript means the solid's own unit edges are counted in: the 12 of the cube, the 6 of the tetrahedron.
| s on the surface | b in the body | c through the centroid |
| v vertex to vertex | e edge to edge | f face to face |
| ve vertex to edge | vf vertex to face | ef edge to face |
So the constant above reads: of the cube, the grand constant, over 253 segments, through the centroid and the body and the surface, with the twelve unit edges counted in. The notation is meant to be read aloud, and to survive being written by hand.
For a regular tetrahedron of unit edge, with 14 elements: Tg()55vef,6 ≈ 35.7796 for all segments; T()24s ≈ 16.3923 on the surface; T()31b,6 ≈ 19.3873 inside the body with the edges; and T()7c ≈ 5.3873 for the seven rotational symmetry axes. The same partition check holds: 24 + 31 = 55 and 16.3923 + 19.3873 = 35.7796.
The symbol is a capital M with a capital N beneath it, sharing one baseline, with no stroke shared between the letters. Its two left stems, its two right stems, and the M's descending arm with the N's diagonal form three parallel pairs, and the perpendicular gap is identical for all three. That condition alone fixes the angle at 43.82° from the vertical. The full construction sheet is bound into the PDF.