An algorithmic approach for computing fundamental domains of crystallographic groups

Alice C. Niemeyer; Lukas Schnelle; Reymond Akpanya

arxiv: 2607.02130 · v1 · pith:GF53I3WF · submitted 2026-07-02 · math.MG · cs.CG· math.GR

An algorithmic approach for computing fundamental domains of crystallographic groups

Reymond Akpanya , Alice C. Niemeyer , Lukas Schnelle This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-03 02:06 UTCgrok-4.3pith:GF53I3WF record.json open to challenge →

classification math.MG cs.CGmath.GR

keywords crystallographic groupsDirichlet cellsfundamental domainsalgorithmic computationEuclidean groupword lengthtopological interlocking assemblies

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The pith

The half-spaces defining Dirichlet cells of crystallographic groups come from group elements expressible as words of bounded length in a generating set.

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

Crystallographic groups are infinite discrete subgroups of the Euclidean group that still possess compact fundamental domains. Direct computation of these domains is difficult because any enumeration over the group must stop at some point. The paper shows that the half-spaces bounding a Dirichlet cell, which can serve as a fundamental domain, arise only from those group elements that appear as words of bounded length in a suitable generating set. The existence of such a bound converts the problem into a finite search, allowing an explicit algorithm to list the necessary half-spaces and construct the cell. The same procedure is applied to generate examples of topological interlocking assemblies.

Core claim

The half-spaces defining such a Dirichlet cell can be derived from elements of Γ acting on R^n that can be expressed as words of bounded length in a suitable generating set. Based on these results, an algorithm for the computation of fundamental domains of crystallographic groups is designed and used to study topological interlocking assemblies.

What carries the argument

Elements of the crystallographic group expressed as words of bounded length whose isometries determine the half-spaces of the Dirichlet cell.

If this is right

Only finitely many words need to be checked to obtain all half-spaces of the Dirichlet cell.
Fundamental domains become computable for every crystallographic group despite the group being infinite.
The resulting cells can be used directly as fundamental domains in geometric constructions.
The algorithm supplies explicit domains for the study of topological interlocking assemblies.

Where Pith is reading between the lines

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

If an explicit method to compute the bound from the generating set is found, the algorithm becomes fully automatic.
Similar length bounds might make fundamental domains computable for other discrete groups acting on Euclidean space.
Software built on the algorithm could generate large families of interlocking structures for materials or architectural design.

Load-bearing premise

There exists a finite bound on word length such that every half-space of the Dirichlet cell is produced by some element whose word representation is at most that long.

What would settle it

An explicit crystallographic group together with a generating set for which at least one bounding half-space of its Dirichlet cell requires a group element whose shortest word representation exceeds every finite candidate bound.

Figures

Figures reproduced from arXiv: 2607.02130 by Alice C. Niemeyer, Lukas Schnelle, Reymond Akpanya.

**Figure 1.** Figure 1: A visualisation of how the group p2 acts on R 2 . Each of the two rectangles with one blue edge is a fundamental domain of the group. A set of points forming a letter β is placed into one fundamental domain and acted on with the group. We say that v ∈ R n is in special position for a crystallographic group Γ ≤ E(n), if StabΓ(v) ̸= {Id}. If the stabiliser of v is trivial, then v is said to be in general pos… view at source ↗

**Figure 2.** Figure 2: Different views of the fundamental domain [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Different views of the deformed fundamental domain [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: The original Stanford bunny (a) and a corresponding approximation given by [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Different views of the curved deformed fundamental domain [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $\Gamma$ is infinite, computing fundamental domains of $\Gamma$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $\Gamma$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $\Gamma$ acting on $\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.

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.

Desk Editor's Note private letter to a colleague

The paper reduces Dirichlet domain computation to a search over bounded-length words in the generators, which is a practical step if the bound is handled effectively.

read the letter

The main new element is the claim that half-spaces for Dirichlet cells in crystallographic groups come from words of bounded length in a suitable generating set, which lets them turn an infinite search into a finite one and build an algorithm around it. They then apply the method to examples in topological interlocking assemblies.

This is useful because it gives a concrete way to compute fundamental domains for these groups, where the infinite nature of the group has always made direct enumeration tricky. The reduction itself is a reasonable algorithmic device on top of the standard fact that only finitely many half-spaces matter due to discreteness.

The soft spot is whether the bound is made effective. Discreteness guarantees some finite length exists, but an algorithm needs either an explicit, computable upper bound in terms of the input generators or a termination test that works without knowing the length ahead of time. The abstract leaves this open, so the paper stands or falls on how clearly the full text supplies one or the other. If that part is only sketched or relies on an ad-hoc search limit, the algorithm is not fully rigorous for arbitrary input.

The rest of the work looks standard: the application examples are concrete but secondary, and the citations follow the usual geometric group theory and crystallography literature without obvious omissions.

This is for readers who need working code or methods for fundamental domains in discrete geometry or symmetry-based design. A specialist in computational crystallography would get the most out of the reduction and the implementation details. It is worth sending to referees who can check the termination argument and verify the examples run correctly.

Referee Report

2 major / 1 minor

Summary. The manuscript asserts that for a crystallographic group Γ ≤ E(n), the half-spaces defining a Dirichlet fundamental domain arise from group elements expressible as words of bounded length in a suitable generating set; it uses this to design an algorithm for computing such domains and applies the algorithm to examples in topological interlocking assemblies.

Significance. A fully effective version of the claimed algorithm would supply a practical computational tool for enumerating fundamental domains of infinite discrete subgroups of the Euclidean group, where naive orbit enumeration is impossible; this would be a concrete advance in computational crystallography and geometric group theory.

major comments (2)

[Abstract] Abstract: the central claim that 'words of bounded length' suffice for all relevant half-spaces is asserted on the basis of discreteness of the orbit, but the manuscript supplies neither an explicit computable upper bound on that length (in terms of the input generators) nor an independent termination test; without one of these the algorithm is not shown to be effective for arbitrary input data.
[Algorithm description] Algorithm section: the termination argument for the enumeration procedure is load-bearing for the main result, yet the text does not demonstrate that the procedure halts after finitely many steps when only the generators are given; this gap prevents verification that the method is fully algorithmic.

minor comments (1)

The notation for the generating set S and the word-length function should be introduced with explicit definitions before the statement of the main theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying the key points where the effectiveness of the algorithm requires further justification. We address each major comment below and will revise the manuscript to strengthen the presentation of the algorithmic claims.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'words of bounded length' suffice for all relevant half-spaces is asserted on the basis of discreteness of the orbit, but the manuscript supplies neither an explicit computable upper bound on that length (in terms of the input generators) nor an independent termination test; without one of these the algorithm is not shown to be effective for arbitrary input data.

Authors: The manuscript establishes the existence of a finite bound on word length via the discreteness of the orbit and compactness of any fundamental domain, which guarantees that only finitely many half-spaces are needed. However, the proof does not yield an explicit, computable expression for this bound directly from the input generators, nor does it supply an independent termination test. We agree that this leaves the algorithm short of being fully effective for arbitrary input data. In the revision we will add an explicit discussion of this limitation in both the abstract and the theoretical section, together with practical heuristics used in the examples. revision: yes
Referee: [Algorithm description] Algorithm section: the termination argument for the enumeration procedure is load-bearing for the main result, yet the text does not demonstrate that the procedure halts after finitely many steps when only the generators are given; this gap prevents verification that the method is fully algorithmic.

Authors: The termination of the enumeration is intended to follow from the finite bound whose existence is proved earlier. As the referee correctly observes, the current text does not demonstrate that this bound is computable from the generators alone, so the procedure is not shown to halt after finitely many steps for arbitrary input. We will revise the algorithm section to state the precise conditions under which termination is guaranteed, to separate the existence result from the question of computability, and to indicate where additional geometric tests (e.g., checking that all orbit points beyond a certain radius lie outside the current cell) could serve as a practical stopping criterion. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on group discreteness and finiteness, not self-reference or fitted inputs.

full rationale

The abstract states that half-spaces derive from group elements of bounded word length in a generating set, but supplies no equations, fitted parameters, or self-citations that reduce the claim to its own inputs by construction. Discreteness of crystallographic groups guarantees finitely many relevant elements, so existence of some bound follows from standard facts about discrete subgroups of E(n) without requiring the paper to presuppose the algorithm's output. No load-bearing step matches any enumerated circularity pattern; the result is self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; the approach implicitly relies on standard facts from geometric group theory and Euclidean geometry without introducing new free parameters or invented entities visible at this level.

axioms (2)

domain assumption Crystallographic groups are discrete subgroups of E(n) with compact fundamental domains (standard definition).
Invoked in the first sentence of the abstract as the object of study.
domain assumption Dirichlet cells can serve as fundamental domains for such groups.
Stated as the target of computation.

pith-pipeline@v0.9.1-grok · 5651 in / 1305 out tokens · 23865 ms · 2026-07-03T02:06:27.259486+00:00 · methodology

discussion (0)

Reference graph

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