[Paper Review] Combinatorial Space Tiling

Egon Schulte|arXiv (Cornell University)|May 20, 2010

Quasicrystal Structures and Properties32 references4 citations

TL;DR

This paper surveys combinatorial space tiling in Euclidean and spherical spaces, focusing on monotypic tilings where tiles are combinatorially equivalent convex polytopes but not necessarily congruent. It highlights open problems in tiling theory, especially regarding space-fillers, monotypic tilings, and tilings by handlebodies, and proposes new directions in abstract polytope modeling via topological tilings of 3-space and the 3-sphere.

ABSTRACT

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.

Motivation & Objective

• To survey major developments in combinatorial space tiling, particularly focusing on monotypic and face-to-face tilings by convex polytopes.
• To highlight longstanding open problems in the classification of space-fillers and combinatorial prototiles in Euclidean and spherical spaces.
• To explore the potential of tilings by handlebodies of higher genus as topological models for abstract regular polytopes.
• To investigate whether universal locally toroidal regular 4-polytopes can be realized as tilings of E³ or S³ using toroidal handlebodies.
• To examine the relationship between compactness of ambient spaces and finiteness of abstract polytopes in topological tiling models.

Proposed method

• Uses combinatorial equivalence as a relaxation of isometry to define monotypic tilings, where tiles are not required to be congruent but must be combinatorially isomorphic to a single prototile.
• Applies face-to-face tiling conditions to ensure topological regularity and avoid pathological configurations in tilings by topological polytopes.
• Employs abstract polytope theory and boundary complex structures to model tilings of S³ and E³ by handlebodies with genus g ≥ 1.
• Utilizes the chamber complex of the regular cubical tessellation in E³ to construct monotypic tilings by topological polytopes via symmetries and boundary decompositions.
• Relies on known results from abstract polytope theory (e.g., McMullen & Schulte, Brehm et al.) to model universal locally toroidal regular 4-polytopes as topological tilings.
• Analyzes the structure of the face lattice and combinatorial automorphism groups (e.g., S₆ × C₂) to verify realizability of abstract polytopes as tilings.

Experimental results

Research questions

• RQ1 Can every universal locally toroidal regular 4-polytope of type {4,4,m}, m ≥ 4, be realized as a tiling of E³ by toroidal handlebodies?
• RQ2 Which finite universal locally toroidal regular 4-polytopes with spherical vertex-figures can be modeled as tilings of S³ by handlebodies?
• RQ3 Do all convex d-polytopes serve as combinatorial prototiles for some monotypic tiling of E^d, even if not face-to-face?
• RQ4 Is the classification of isometric space-fillers complete in dimensions d ≥ 3, particularly for pentagons in the plane?
• RQ5 Can abstract regular polytopes of types {6,3,3}, {6,3,4}, and {6,3,5} be realized via tilings of E³ or S³ with hexagonal-faced handlebodies?

Key findings

• Every convex d-polytope can serve as the combinatorial prototile for a monotypic tiling of E^d by topological d-polytopes, though such tilings are generally not face-to-face.
• The 3-sphere S³ admits a face-to-face tiling by 20 toroidal handlebodies, each with a 3×3 square boundary complex, modeling the universal locally toroidal regular 4-polytope { {4,4}_{(3,0)}, {4,3} }.
• This tiling has 20 facets, 30 vertices, and a combinatorial automorphism group isomorphic to S₆ × C₂, confirming its abstract regularity.
• The existence of such a tiling demonstrates that abstract regular 4-polytopes can be realized as topological models in S³ via handlebody tilings.
• The paper shows that the face-to-face condition is essential to avoid degeneracy, as non-face-to-face tilings can be constructed for any convex polytope but fail to preserve structural integrity.
• The classification of isometric space-fillers remains open in the plane, particularly for pentagons, and is expected to be incomplete.