A new word for an old arithmetic, shown on Platonic and Archimedean solids
by Grego (Gergely Földvári)
Summetry is symmetry that carries arithmetic: where symmetry says which sets of a figure's parts are equivalent, summetry requires them to sum alike, and because the groups exhaust a fixed total a known number of times the sum is not chosen but computed. This paper states that relation as S = mT/r, names the property, and attributes the underlying identity to the magic labelling literature, where it has been standard since 1963. It then turns the relation on the solids. All five Platonic and all thirteen Archimedean solids carry the same arrangement — twelve things, six groups of four, each thing in two groups — and therefore the same constant, 26, reached through the same 960 numberings; the arrangement is confirmed on several Catalan and Kepler-Poinsot solids as well. Of the twenty-four arrangements of that shape, exactly eight admit a numbering summing to 26, and two of those cannot occur as the faces of any polyhedron. Where the memberships are not uniform the constant is no longer determined but confined to an interval, the Summetry Range, bounded by the least and greatest weights W– and W+; the Formula is the Range with zero width. The Tabakgasse polyhedron of Budapest, which refuses summetry at three successive stages, is given as a counterexample.
summetry /ˈsʌmɪtri/ n. (pl. summetries) [blend of sum + symmetry, coined by G. Földvári, 2026]
1. The property of a figure whose symmetry compels a constant sum: a numbering of its parts such that every group of parts related by that symmetry adds to the same total. 2. That total itself; the value S in the relation S = mT/r.
S = mT / r
the sum a shape compels
| Letter | Meaning |
|---|---|
| S | the sum every group must show |
| m | how many groups each numbered part belongs to |
| T | the total of all the numbers used, that is 1 + 2 + 3 and so on to the last |
| r | how many groups there are, and so how many times the sum repeats |
S, m, T and r are all positive whole numbers, none of them zero. Add up all the groups and each part is counted once for every group that contains it; if every group is to show the same total, that grand total must divide evenly among the groups.
The gate. S must come out a whole number. If mT/r is not whole, no arrangement exists at all, and this is proved rather than merely unfound. The identity itself is classical, standard in magic labelling theory since Sedláček in 1963; no claim of priority is made for it here. What is new is what it finds on the solids.
Each solid was built from coordinates, its rotation group computed from its own vertices, and its features found by symmetry orbit. In every line m = 2, T = 78, r = 6, and so S = 26.
| Solid | The twelve | The six |
|---|---|---|
| tetrahedron | face-corners | edges |
| cube | edges | faces |
| octahedron | edges | vertices |
| dodecahedron | faces | inscribed cube |
| icosahedron | vertices | inscribed cube |
| truncated tetrahedron | hidden rectangles | edges of the parent |
| cuboctahedron | vertices | squares |
| truncated cube | faces | faces |
| truncated octahedron | edges | faces |
| rhombicuboctahedron | faces | faces |
| truncated cuboctahedron | edges | faces |
| snub cube | edges | faces |
| icosidodecahedron | faces | inscribed cube |
| truncated dodecahedron | faces | inscribed cube |
| truncated icosahedron | faces | inscribed cube |
| rhombicosidodecahedron | faces | inscribed cube |
| truncated icosidodecahedron | faces | inscribed cube |
| snub dodecahedron | faces | inscribed cube |
A magic star with n points carries 2n numbers on n lines of four, each number on two lines. The formula gives S = 4n + 2, the classical magic-star constant. The five-pointed star shows exactly how far the formula reaches: it gives 22, a whole number, so the gate opens — and yet no such star exists. The hexagram is the first magic star that can be built, and its constant is 26. That star is not merely a picture: it lives inside Escher's Solid, and the constant survives the lift.
Twelve things, six groups of four, each thing in two groups, is exactly a 4-regular multigraph on six nodes. There are twenty-four such arrangements. Exactly eight admit a numbering that makes every group sum to 26.
| Arrangement | Numberings | As faces of a solid |
|---|---|---|
| three fourfold shares | 2,654,208 | impossible |
| one fourfold share | 313,344 | impossible |
| six doubles, one ring | 64,512 | hexagonal prism |
| six doubles, two triples | 46,080 | prism–antiprism–prism stack |
| four doubles | 24,960 | arrangement exists, solid unnamed |
| three doubles | 11,040 | no planar arrangement found |
| two doubles | 5,760 | arrangement exists, solid unnamed |
| all single — the octahedron | 960 | the eighteen, and more |
Everything above assumes every part belongs to the same number of groups. Drop that and the sum stops being decided. Number all twenty-six elements of a cube from one pool and let each face take its own number, its four edges and its four corners: a face belongs to one group, an edge to two, a corner to three, and the constant is no longer computed but confined.
⌈ W– / r ⌉ ≤ S ≤ ⌊ W+ / r ⌋
the Summetry Range, where W– and W+ are the least and greatest weights of a numbering
For the cube that gives 100 to 143 — forty-four values, every one attainable. When every membership is equal the two weights coincide and the range collapses to a point. The Summetry Formula is the Summetry Range with zero width. The range is known in the literature as the spectrum of a magic labelling problem; the name Summetry Range is adopted here and no claim is made beyond the name.
A formula that only ever confirms is a coincidence, not a formula. The Tabakgasse polyhedron, the lantern of the Eternal Flame in the Dohány Street Synagogue in Budapest, refuses summetry at every stage and never twice for the same reason: its twelve pyramids fail on symmetry, its edges fail on arithmetic although the gate opens at S = 58, and the fully augmented solid fails on divisibility at 127.5.
Star Face is where summetry was first named. Masking Depth in Clinical Practice is its clinical follow-up. See also The English Cube and Metatron Numbers, The VIO Formula, The Tabakgasse Polyhedron and GooglyCube.