{"title":"A polytopal generalization of Apollonian packings and Descartes' theorem","authors":"Jorge L. Ramírez Alfonsín , Iván Rasskin","doi":"10.1016/j.jnt.2024.11.010","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 67-103"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X25000435","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.