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 is
Cg(MN)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(MN)96s | 96 | 12√1 + 24√2 + 24√5 | 99.6068 |
| C(MN)108s,12 | 108 | 24√1 + 24√2 + 24√5 | 111.6068 |
| C(MN)13c | 13 | 3√1 + 6√2 + 4√3 | 18.4134846 |
| C(MN)25c,12 | 25 | 15√1 + 6√2 + 4√3 | 30.4134846 |
| C(MN)132b | 132 | 36√1 + 6√2 + 12√3 + 12√5 + 24√6 | 150.8907 |
| C(MN)144b,12 | 144 | 48√1 + 6√2 + 12√3 + 12√5 + 24√6 | 162.8907 |
| Cg(MN)253cbs,12 | 253 | 24√1 + 36√2 + 16√3 + 7 + 16√4 + 36√5 + 24√6 | 280.9107 |
In front of the (MN) symbol's left bracket stands a capital letter code for the polyhedron: C cube, T tetrahedron, O octahedron, D dodecahedron, I icosahedron, CO cuboctahedron, SnC snub cube, RhCO rhombicuboctahedron, RhD rhombic dodecahedron. A lower case attribute may follow, such as g for the grand constant. The right bracket carries a superscript giving the number of segments counted and a subscript giving their composition: s on the surface, b in the body, c through the centroid, v between vertices, e between edges, f between faces, and ve, vf, ef between combinations of two. A further subscript number indicates the inclusion of the frame's unit edges.
For a regular tetrahedron of unit edge, with 14 elements: Tg(MN)55vef,6 ≈ 35.7796 for all segments; T(MN)24s ≈ 16.3923 on the surface; T(MN)31b,6 ≈ 19.3873 inside the body with the edges; and T(MN)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.