{"title":"Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs","authors":"Latham Boyle, Justin Kulp","doi":"arxiv-2408.15316","DOIUrl":null,"url":null,"abstract":"Discrete geometries in hyperbolic space are of longstanding interest in pure\nmathematics and have come to recent attention in holography, quantum\ninformation, and condensed matter physics. Working at a purely geometric level,\nwe describe how any regular tessellation of ($d+1$)-dimensional hyperbolic\nspace naturally admits a $d$-dimensional boundary geometry with self-similar\n''quasicrystalline'' properties. In particular, the boundary geometry is\ndescribed by a local, invertible, self-similar substitution tiling, that\ndiscretizes conformal geometry. We greatly refine an earlier description of\nthese local substitution rules that appear in the 1D/2D example and use the\nrefinement to give the first extension to higher dimensional bulks; including a\ndetailed account for all regular 3D hyperbolic tessellations. We comment on\nglobal issues, including the reconstruction of bulk geometries from boundary\ndata, and introduce the notion of a ''holographic foliation'': a foliation by a\nstack of self-similar quasicrystals, where the full geometry of the bulk (and\nof the foliation itself) is encoded in any single leaf in a local invertible\nway. In the $\\{3,5,3\\}$ tessellation of 3D hyperbolic space by regular\nicosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold\nsymmetry which is not the Penrose tiling, and record and comment on a related\nconjecture of William Thurston. We end with a large list of open questions for\nfuture analytic and numerical studies.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15316","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Discrete geometries in hyperbolic space are of longstanding interest in pure
mathematics and have come to recent attention in holography, quantum
information, and condensed matter physics. Working at a purely geometric level,
we describe how any regular tessellation of ($d+1$)-dimensional hyperbolic
space naturally admits a $d$-dimensional boundary geometry with self-similar
''quasicrystalline'' properties. In particular, the boundary geometry is
described by a local, invertible, self-similar substitution tiling, that
discretizes conformal geometry. We greatly refine an earlier description of
these local substitution rules that appear in the 1D/2D example and use the
refinement to give the first extension to higher dimensional bulks; including a
detailed account for all regular 3D hyperbolic tessellations. We comment on
global issues, including the reconstruction of bulk geometries from boundary
data, and introduce the notion of a ''holographic foliation'': a foliation by a
stack of self-similar quasicrystals, where the full geometry of the bulk (and
of the foliation itself) is encoded in any single leaf in a local invertible
way. In the $\{3,5,3\}$ tessellation of 3D hyperbolic space by regular
icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold
symmetry which is not the Penrose tiling, and record and comment on a related
conjecture of William Thurston. We end with a large list of open questions for
future analytic and numerical studies.