Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs

TL;DR

This paper explores how regular tessellations of hyperbolic space induce self-similar quasicrystalline boundary geometries, extending previous work to higher dimensions and introducing the concept of holographic foliations.

Contribution

It refines substitution rules for boundary tilings, extends the framework to higher-dimensional hyperbolic spaces, and introduces holographic foliations linking bulk and boundary geometries.

Findings

Boundary geometries are described by self-similar substitution tilings.

Extended the description to all regular 3D hyperbolic tessellations.

Identified a 2D quasicrystal with 5-fold symmetry not matching Penrose tiling.

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.