Summetry: A New Word for an Old Arithmetic, Shown on Platonic and Archimedean Solids

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. Abstract. 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.