Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs

TL;DR

This work shows that every regular tessellation of hyperbolic space $\mathbb{H}^{d+1}$ induces a $d$-dimensional boundary with self-similar quasicrystalline structure, encoded by invertible local inflation/deflation rules. It introduces the concept of holographic foliations, constructing higher-dimensional generalizations and providing detailed analyses of the $\{3,5,3\}$ case, including a 2D boundary quasicrystal with genuine 5-fold symmetry that is not Penrose. The authors develop a refined half-step inflation framework that unifies $\{p,q\}$ and $\{q,p\}$ boundaries, extend the construction to 3D and beyond, and show both the global potential for bulk reconstruction from a leaf and concrete counterpoints to Thurston’s conjecture about horospherical Penrose tilings. The paper closes with open questions linking discrete hyperbolic geometry to AdS/CFT, quantum error correction, and higher-dimensional holographic structures, offering a new geometric pathway to holography in discrete hyperbolic settings.

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.

Figures (14)

• Figure 1: Left, the Poincaré Ball presentation of $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ with a $\{4,5\}$ tessellation drawn on top. In the picture, an $O(2)$ subgroup is manifest (but broken by the tessellation) and the boundary of $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ is clear. Center, the same tessellation of $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ is drawn in the Upper Half-Space model. Right, $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ is depicted as a hyperboloid (red) in $\mathop{\mathrm{\mathrm{Mink}}}\nolimits_{2+1}$ (with lightcone depicted in yellow). The Lorentz group acts naturally on the hyperboloid, and the boundary is the "celestial circle" which lives at future timelike infinity. A blue plane through the origin in Minkowski space cuts the hyperboloid and appears as a geodesic (dark blue curve) in $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$. Drawn in units of $L$.
• Figure 2: Left, a green triangle with interior angles $(\tfrac{\pi}{2},\tfrac{\pi}{3},\tfrac{\pi}{6})$ forms the fundamental domain for mirror planes corresponding to the Coxeter-Dynkin diagram in Equation \ref{['eq:CDD236']}. Reflections of this fundamental domain in the mirrors triangulates $\mathop{\mathrm{\mathbb{E}}}\nolimits^2$. Tracking the node (red) opposite the "6-mirror" under these reflections generates the $\{6,3\}$ tiling. Right, a $(\tfrac{\pi}{2},\tfrac{\pi}{3},\tfrac{\pi}{7})$ triangle (green) forms the fundamental domain for the mirrors in Equation \ref{['eq:CDD237']}. Reflections of this fundamental domain in the mirrors triangulates $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$. Tracking the node (red) opposite the "7-mirror" generates the $\{7,3\}$ tiling. In both cases, one can see the dual $\{3,6\}$ and $\{3,7\}$ triangle honeycombs whose vertices are the centers of the hexagons and heptagons respectively.
• Figure 3: Left, a patch of Penrose tiling given by rhombs in pink and overlaid with its inflation in purple. Right top, the "thin" Penrose rhomb ${\bf{{T}}}$ (coral) and the "thick" or "fat" Penrose rhomb ${\bf{{F}}}$ (light blue) are presented with their matching rules on edges and their inflation rules. Right bottom, the darts (coral) and kites (light blue) are given with their inflation rules. Unlike the rhombs, the kites and darts do not require matching rules to form an admissible Penrose tiling. The two presentations are equivalent since tilings built from kites and darts are mutually locally derivable from tilings by rhombs.
• Figure 4: The half-step growth procedures between the $\{3,7\}$ and $\{7,3\}$ tessellations. Left, a patch with 1 triangle (red) in the $\{7,3\}$ tiling sits inside the triangulation of $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ by $(\tfrac{\pi}{2},\tfrac{\pi}{3},\tfrac{\pi}{7})$ triangles. The associated boundary string is $[{\bf{\bar{{1}}}} {\bf{\bar{{1}}}} {\bf{\bar{{1}}}}]$. Middle, a patch of 3 heptagons (black) appear in the $\{7,3\}$ tiling, with associated boundary string $[{\bf{{2}}} {\bf{{1}}}^4 {\bf{{2}}} {\bf{{1}}}^4 {\bf{{2}}} {\bf{{1}}}^4]$. Right, another half-step inflation to the $\{3,7\}$ tiling with boundary string $[{\bf{\bar{{3}}}}{\bf{\bar{{2}}}}^3{\bf{\bar{{3}}}}{\bf{\bar{{2}}}}^3{\bf{\bar{{3}}}}{\bf{\bar{{2}}}}^3]$.
• Figure 5: A holographic foliation of $\mathop{\mathrm{\mathbb{H}}}\nolimits^2$ by $\{7,3\}$/$\{3,7\}$ quasicrystals. Orange layers mark the $\{7,3\}$ tiling and blue layers mark the $\{3,7\}$ tiling. The same local substitution rules defined in Section \ref{['sec:3773Example']} describe the evolution from layer to layer. This holographic foliation in particular can be thought of as given by (approximately) concentric horospheres, i.e. slices of constant height.

Theorems & Definitions (2)

• Conjecture 1: Thurston, rough
• Conjecture 2: Thurston, modified