Geometry of the 120-cell revealed by golden number dissections
The 120-Cell Dissected by the Hyperplanes of Its Final Stellation: the golden ring Z[phi] as foundation, from Kramer's seven cells to 2,455,201 cells in 338 orbits
Computational Geometry
Summary
The paper studies a complex four-dimensional shape known as the 120-cell, dividing it into many smaller pieces using certain geometric planes. The authors use the properties of the 'golden ring,' a special number system based on the golden ratio, as the mathematical foundation. They carefully count and classify over two million resulting pieces, linking these to symmetry groups and smaller shapes inside the 120-cell. This work explores deep connections between geometry, number theory, and symmetry in higher dimensions.
What this means in practice
- •For mathematical software developers: Develop precise computational tools for manipulating four-dimensional polytopes based on golden ratio arithmetic and symmetry classifications presented here.
- •For visualization engineers: Use the detailed dissection and classification of the 120-cell to create accurate visual models of complex higher-dimensional objects for educational or design purposes.
Authors
Miguel E. Pérez Ibarra, Jaziel Flores Rodríguez
Abstract
The diagonals of a regular pentagon dissect it into eleven pieces and draw a pentagram; the dodecahedron dissected by the twelve face planes of its inner core gives Kramer's seven "elementary cells" of 1982. We define the stellation dissection of a golden regular polytope in any dimension as the dissection of the convex hull of its final regular stellation by the facet hyperplanes of the core, show that the core has phi^(-d) of the inradius of the hull for d = 2, 3, 4, and carry the dissection out for the first time on the 120-cell. The foundation is the arithmetic of the golden ring Z[phi], the ring of integers of Q(sqrt 5): vertices and hyperplanes live in Z[phi]^d, every orientation test is the exact sign of an element of Z[phi], decided with integers, and the pieces are named by sign vectors. We develop this foundation from scratch (conjugation, norm, units, the primes sqrt 5, 2, 3, the exact sign test, the lattice x -> (x, conj x), the vertex modules of the dodecahedron and the 120-cell, the icosian ring) and let it carry the geometry. The definition bifurcates in dimension four: the star cut at depth phi^(-4), a unit, has 2,455,201 cells in 338 orbits of H4; the hereditary cut at depth 1 - phi^(-4) = sqrt 5 / phi^2, a unit times the ramified prime, induces the three-dimensional dissection on every facet and has 8,161 cells in 8 orbits. New here: all vertices of the dodecahedral and of the hereditary dissection lie in the vertex module of the polytope being cut, whereas the star cut needs the denominators 2, 3 and 5; the seven kinds of pieces of the dodecahedron are exactly the I_h-orbits of sign vectors, with stellation layer equal to the number of plus signs. We catalogue the four-dimensional pieces and treat the circumscribed disc, ball and 4-ball. Every number quoted is recomputed exactly by a script whose output is reproduced in an appendix.