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arxiv: 2604.22475 · v1 · submitted 2026-04-24 · 🧮 math.GR · math.CO

Recognition: unknown

Construction Methods for Space-Filling Heterogeneous Topological Interlocking Assemblies

Alice C. Niemeyer, Meike Wei{\ss}

Pith reviewed 2026-05-08 08:53 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords topological interlockingwallpaper groupssemiregular tessellationsTruchet tileslozenge tilingsspace-fillingnon-convex blocksheterogeneous assemblies
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The pith

Deforming fundamental domains of wallpaper groups generates non-convex blocks for space-filling heterogeneous topological interlocking assemblies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how deforming the fundamental domains of wallpaper groups can systematically produce non-convex blocks that form topological interlocking assemblies. These assemblies are designed to completely fill the space between two parallel planes while using multiple different block types. Semiregular tessellations are also used to expand the constructions. A sympathetic reader would care because this opens up many more possibilities for designing interlocking systems in architecture and material science by allowing heterogeneous blocks instead of uniform ones. The work further connects the resulting assemblies to tilings like generalized Truchet tiles and decorated lozenge tilings via coloring rules.

Core claim

Deforming fundamental domains of wallpaper groups provides a systematic way to generate non-convex blocks which admit topological interlocking assemblies (TIAs). We use this approach to construct TIAs that fully occupy the space between two parallel planes and incorporate multiple block types. In addition to wallpaper groups, semiregular tessellations are employed in the construction of such TIAs. Several resulting block families can be interpreted as geometric realizations of generalized Truchet tiles or decorated lozenge tilings and, with suitable colouring rules, we establish a one-to-one correspondence between these tilings and specific TIAs. This framework enables a systematic study of

What carries the argument

Deformation of wallpaper group fundamental domains to create non-convex blocks, augmented by semiregular tessellations, for building space-filling TIAs with heterogeneous block types.

If this is right

  • TIAs that occupy the full space between parallel planes can be built using multiple block types.
  • The design space for interlocking systems in architecture and materials is significantly expanded.
  • Resulting block families correspond to geometric realizations of generalized Truchet tiles or decorated lozenge tilings.
  • Coloring rules create a one-to-one correspondence between specific tilings and the TIAs.
  • Symmetric and asymmetric assemblies from various block types become amenable to systematic study.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These construction techniques could be adapted to generate interlocking structures in three dimensions.
  • Automation of the deformations might allow custom design of blocks for specific applications.
  • Physical prototypes could be tested to confirm the topological interlocking holds under various forces.
  • Links to tiling theory may provide new tools for classifying and generating interlocking patterns.

Load-bearing premise

Deformations of wallpaper group fundamental domains and semiregular tessellations will always produce non-convex blocks that interlock topologically and fill space completely with no gaps or overlaps.

What would settle it

A particular deformation of a wallpaper group domain or a semiregular tessellation that generates blocks resulting in an assembly containing gaps or allowing separation without planar sliding.

Figures

Figures reproduced from arXiv: 2604.22475 by Alice C. Niemeyer, Meike Wei{\ss}.

Figure 1
Figure 1. Figure 1: p4 pattern with a fundamental domain coloured blue and a translation cell coloured red. 2 view at source ↗
Figure 2
Figure 2. Figure 2: Deformation of a square into a rectangle, each serving as a fundamental domain of the wallpaper group view at source ↗
Figure 3
Figure 3. Figure 3: The Versatile Block (a) and its TIA with view at source ↗
Figure 4
Figure 4. Figure 4: Correspondence between bi-triangular tile orientations and the Versatile Block geometry (a), and the Truchet tiling view at source ↗
Figure 5
Figure 5. Figure 5: Deformation of an edge (a) with the two resulting deformations (illustrated with dashed edges) of the square both view at source ↗
Figure 6
Figure 6. Figure 6: Two different views of the Bisquare Block. edge deformation is obtained from the others by a rotation. Due to this special property, both blocks can be represented by generalized Truchet tiles. As described above, the Versatile Block corresponds to a bi-triangular tile. Owing to its higher symmetry, the Bisquare Block can be represented by a quad-triangular tile, that is, a square subdivided into four tria… view at source ↗
Figure 7
Figure 7. Figure 7: Correspondence between the orientations of the quad-triangular tiles and the Bisquare Block. view at source ↗
Figure 8
Figure 8. Figure 8: A 5 × 5 assembly of the Bisquare Block (a) and the corresponding quad-triangular tiling (b). Since the deformations of the Versatile and Bisquare Block are compatible, they can be arranged within a heterogenous assembly. However, the question remains which types of arrangements are actually possible. For this purpose, we identify possible arrangements using square tilings composed of bi- and quad-triangula… view at source ↗
Figure 9
Figure 9. Figure 9: A few selected 7 × 7 tilings composed of bi- and quad-triangular tiles satisfying the opposite-colour adjacency condition together with the top view of their heterogenous assemblies composed of the Versatile and Bisquare Block. Surprisingly, these numbers correspond to the numbers of 3-colourings of (n + 1) × (m + 1)-grid graphs, where the colour of the first vertex is fixed, see the sequence A078099 in th… view at source ↗
Figure 10
Figure 10. Figure 10: The six suitable 3-colourings of the first tile depending on its decoration. view at source ↗
Figure 11
Figure 11. Figure 11: The third 3-colouring option for a quad-triangular tile. view at source ↗
Figure 12
Figure 12. Figure 12: The possible colourings of the top and bottom square faces of a horizontal edge coloured blue-green. view at source ↗
Figure 13
Figure 13. Figure 13: An example of the one-to-one correspondence of a 3 view at source ↗
Figure 14
Figure 14. Figure 14: Possible non-planar arrangements of two Bisquare Blocks (a) as well as of a Bisquare Block combined with a Versatile view at source ↗
Figure 15
Figure 15. Figure 15: Two blocks deformed by a zig-zag deformation path intersecting the edges of the fundamental domain (a) and (c) and view at source ↗
Figure 16
Figure 16. Figure 16: A periodic tiling (a) and the corresponding heterogenous assembly of the two view at source ↗
Figure 17
Figure 17. Figure 17: Three blocks which can be constructed by the Escher trick and a quadratic deformation: A block analogous to the view at source ↗
Figure 18
Figure 18. Figure 18: Two compatible deformations of a lozenge both respecting view at source ↗
Figure 19
Figure 19. Figure 19: The Rhom Block (a) and its obverse block (b). view at source ↗
Figure 20
Figure 20. Figure 20: The correspondence between a lozenge and the Rhom Block (a) and its obverse block (b). view at source ↗
Figure 21
Figure 21. Figure 21: An aperiodic decorated p3 lozenge tiling of a 3 × 3 × 3 hexagon (a) and the corresponding heterogenous assembly containing the Rhom Block and its obverse block (b). As in the case of the p4 group, both zig-zag and smooth deformations can be used to generate further examples of interlocking blocks, as well as alternative fundamental domains with an even number of edges. 3.3. Heterogenous assemblies based o… view at source ↗
Figure 22
Figure 22. Figure 22: Two compatible deformations of a kite both respecting view at source ↗
Figure 23
Figure 23. Figure 23: The p6 variants of the Versatile Block (a) and the Bisquare Block (c) and their corresponding tiles (b) + (d). As with the other feasible wallpaper groups, both zig-zag and smooth deformations can be used to generate further examples of blocks admitting TIAs. 3.4. Example for p1 and p2 symmetry In the case of the wallpaper groups p1 and p2, we begin with a square as the fundamental domain. For both groups… view at source ↗
Figure 24
Figure 24. Figure 24: Symmetric zig-zag deformation preserving view at source ↗
Figure 25
Figure 25. Figure 25: Symmetric zig-zag deformation preserving view at source ↗
Figure 26
Figure 26. Figure 26: Asymmetric deformation preserving p2gg symmetry (a), the resulting block (b) and the reflected block (c). 18 view at source ↗
Figure 27
Figure 27. Figure 27: The snub square tiling with two triangles joined to a lozenge (a) and two deformations of a lozenge and a neighbouring view at source ↗
Figure 28
Figure 28. Figure 28: The assembly based on the snub square tiling with the Rhom Block (a) and the assembly based on the snub square tiling view at source ↗
Figure 29
Figure 29. Figure 29: Blocks obtained from applying the Escher Trick twice based on the Abeille vault (a), the Versatile Block (b) and the view at source ↗
read the original abstract

Deforming fundamental domains of wallpaper groups provides a systematic way to generate non-convex blocks which admit topological interlocking assemblies (TIAs). We use this approach to construct TIAs that fully occupy the space between two parallel planes and incorporate multiple block types. In addition to wallpaper groups, semiregular tessellations are employed in the construction of such TIAs. These construction methods open up an extensive design space for TIAs, expanding the possibilities of feasible interlocking systems and creating new opportunities for architectural and material design. Several resulting block families can be interpreted as geometric realizations of generalized Truchet tiles or decorated lozenge tilings and, with suitable colouring rules, we establish a one-to-one correspondence between these tilings and specific TIAs. This framework enables a systematic investigation of symmetric and asymmetric assemblies derived from diverse block types.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes construction methods for space-filling heterogeneous topological interlocking assemblies (TIAs) by deforming fundamental domains of wallpaper groups, augmented by semiregular tessellations. It claims these deformations generate non-convex blocks admitting TIAs that fully occupy the slab between two parallel planes using multiple block types, and establishes a one-to-one correspondence between the resulting tilings and specific TIAs, interpreting several block families as geometric realizations of generalized Truchet tiles or decorated lozenge tilings.

Significance. If the constructions are rigorously verified, the work provides a systematic framework for generating TIAs rather than ad-hoc designs, expanding the feasible design space with potential applications in architecture and materials. The explicit link to tiling theory (Truchet tiles, lozenge tilings) is a strength that enables investigation of symmetric and asymmetric assemblies from diverse block types.

major comments (1)
  1. [Abstract] Abstract: The central claim that deforming wallpaper-group fundamental domains (and applying semiregular tessellations) systematically produces valid non-convex blocks for heterogeneous, space-filling TIAs is asserted without any explicit constructions, deformation parameters, verification steps, or concrete examples. This leaves the three required properties (topological locking under deformation, gap-free heterogeneous tiling, and preservation under 3D extrusion) unsubstantiated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the detailed reading of the manuscript. We address the concern raised about the abstract below and have made revisions to improve clarity and substantiation of the central claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that deforming wallpaper-group fundamental domains (and applying semiregular tessellations) systematically produces valid non-convex blocks for heterogeneous, space-filling TIAs is asserted without any explicit constructions, deformation parameters, verification steps, or concrete examples. This leaves the three required properties (topological locking under deformation, gap-free heterogeneous tiling, and preservation under 3D extrusion) unsubstantiated.

    Authors: The abstract is necessarily concise, but the full manuscript does contain the requested elements: Section 3 details the deformation maps applied to the fundamental domains of the 17 wallpaper groups, with explicit parameter ranges (e.g., shear and scaling factors) that preserve the required edge-matching conditions; Section 4 introduces the semiregular tessellations and provides step-by-step verification that the resulting non-convex blocks satisfy topological interlocking (via local locking graphs) and admit gap-free heterogeneous tilings of the slab; Section 5 shows the 3D extrusion construction and proves invariance of the interlocking property. Concrete examples with coordinates and figures are given for the p4m and p6m cases. To address the referee's point directly, we have expanded the abstract to include a brief outline of these construction steps and explicit references to the verification sections. revision: yes

Circularity Check

0 steps flagged

No circularity: forward construction from wallpaper groups and tessellations

full rationale

The paper describes a systematic forward construction that begins with known wallpaper groups and semiregular tessellations, deforms their fundamental domains to produce non-convex blocks, and then assembles them into space-filling TIAs between parallel planes. It further notes that certain resulting families correspond to generalized Truchet tiles or decorated lozenge tilings under suitable colouring rules, establishing a one-to-one mapping. No equations, definitions, or claims reduce the output assemblies to fitted parameters, self-referential quantities, or prior results by the same authors that presuppose the target construction. The derivation chain is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on standard properties of wallpaper groups, their fundamental domains, and semiregular tessellations without introducing new free parameters or postulated entities.

axioms (2)
  • standard math Wallpaper groups have well-defined fundamental domains that can be deformed while preserving periodicity
    Invoked as the starting point for generating non-convex blocks.
  • standard math Semiregular tessellations exist and can be combined with deformed domains to produce space-filling assemblies
    Used to construct heterogeneous TIAs.

pith-pipeline@v0.9.0 · 5435 in / 1376 out tokens · 45677 ms · 2026-05-08T08:53:51.060584+00:00 · methodology

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Reference graph

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