arXiv subjects

Iván Rasskin

Publications and source records attributed to Iván Rasskin.

3 recordsLinked to original sources

arXiv2025

A polytopal generalization of Apollonian packings and Descartes' theorem

We present a generalization of Descartes' theorem for the family of polytopal sphere packings arising from uniform polytopes. The corresponding quadratic equation is expressed in terms of geometric invariants of uniform polytopes which are closely connected to canonical realizations of edge-scribable polytopes. We use our generalization to construct integral Apollonian packings based on the Platonic solids. Additionally, we also introduce and discuss a new spectral invariant for edge-scribable polytopes.

math.CO↗

arXiv2024

Regular polytopes, sphere packings and Apollonian sections

In this paper, we explore the geometry and the arithmetic of a family of polytopal sphere packings induced by regular polytopes in any dimension. We prove that every integral polytope is crystallographic, and we show that there are 11 crystallographic regular polytopes in any dimension. After introducing the notion of Apollonian section, we determine which Platonic crystallographic packings emerge as cross-sections of the Apollonian arrangements of the regular 4-polytopes. Additionally, we compute the Möbius spectrum of every regular polytope.

math.CO↗

arXiv2024

Links in orthoplicial Apollonian packings

In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation $x^4 + y^4 + z^4 = 2t^2$.

math.GT↗