Holographic foliations: Self-similar quasicrystals from hyperbolic honeycombs

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 (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:math>)-dimensional hyperbolic space naturally admits a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi></c:math>-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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo stretchy="false">{</e:mo><e:mn>3</e:mn><e:mo>,</e:mo><e:mn>5</e:mn><e:mo>,</e:mo><e:mn>3</e:mn><e:mo stretchy="false">}</e:mo></e:math> 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.