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arxiv: 2601.07552 · v3 · submitted 2026-01-12 · 🧮 math.GT

An introduction to Coxeter polyhedra

Bruno Martelli This is my paper

Pith reviewed 2026-05-16 15:11 UTC · model grok-4.3

classification 🧮 math.GT MSC 52B1551M20
keywords Coxeter polyhedraWythoff constructionnon-obtuse polyhedrauniform polyhedratessellationshyperbolic geometryregular polyhedraAndreev theorem
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The pith

Coxeter polyhedra generate all regular, semiregular and uniform polyhedra and tessellations in spherical, Euclidean and hyperbolic spaces via the Wythoff construction.

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

The paper first introduces non-obtuse polyhedra in spherical, Euclidean and hyperbolic spaces and proves fundamental theorems on their angles and existence that originate with Andreev, Coxeter and Vinberg. It then defines Coxeter polyhedra and shows how the Wythoff construction applied to them systematically produces every regular, semiregular and uniform polyhedron and tessellation. This supplies a single geometric mechanism that works uniformly across the three geometries instead of requiring separate arguments for each curvature. A reader cares because the approach turns the classification of symmetric tilings into a matter of choosing appropriate reflection angles rather than enumerating cases by hand.

Core claim

Non-obtuse polyhedra satisfy the angle conditions needed for the classical theorems of Andreev, Coxeter and Vinberg to classify them in each geometry; Coxeter polyhedra are then the special case whose dihedral angles are submultiples of pi, and the Wythoff construction on these polyhedra yields all regular, semiregular and uniform polyhedra and tessellations.

What carries the argument

The Wythoff construction on a Coxeter polyhedron, which encodes reflections in the faces to produce the symmetry group and the resulting uniform figure.

If this is right

  • All uniform tessellations arise by reflecting the faces of a single Coxeter polyhedron.
  • Angle conditions on non-obtuse polyhedra determine exactly which Coxeter polyhedra exist in each geometry.
  • Semiregular figures are obtained by truncating or rectifying the regular ones inside the same reflection framework.
  • Hyperbolic space admits infinite families of such polyhedra whose volumes are controlled by the angle data.

Where Pith is reading between the lines

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

  • The same reflection data could enumerate uniform polytopes in higher-dimensional hyperbolic spaces.
  • Coxeter polyhedra supply explicit fundamental domains for studying discrete subgroups of isometries.
  • The construction gives a practical method for building symmetric meshes in spaces of constant curvature for numerical or physical models.

Load-bearing premise

The standard models of spherical, Euclidean and hyperbolic geometry are consistent and the classical theorems of Andreev, Coxeter and Vinberg apply without gaps to non-obtuse polyhedra.

What would settle it

An explicit regular or uniform polyhedron or tessellation that cannot be recovered from any Coxeter polyhedron by applying the Wythoff construction.

read the original abstract

This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev, Coxeter, and Vinberg. In the second we introduce Coxeter polyhedra and use them to describe regular, semiregular, and uniform polyhedra and tessellations, mostly via the Wythoff construction.

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

0 major / 3 minor

Summary. The manuscript is an expository introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It is divided into two independent parts: the first introduces non-obtuse polyhedra and proves fundamental theorems originating from Andreev, Coxeter, and Vinberg; the second introduces Coxeter polyhedra and employs them, primarily via the Wythoff construction, to classify and describe regular, semiregular, and uniform polyhedra and tessellations.

Significance. If the reproduced proofs faithfully follow the cited classical arguments without introducing gaps, the paper provides a structured, self-contained survey of key results on polyhedra of constant curvature. This could serve as a useful reference for students and researchers working in geometric group theory, hyperbolic geometry, and the classification of uniform tessellations, particularly by consolidating the Andreev–Coxeter–Vinberg theorems with the Wythoff construction in one place.

minor comments (3)
  1. [Abstract] The abstract and introduction should explicitly state the target audience (e.g., graduate students familiar with basic hyperbolic geometry) to clarify the level of prerequisite knowledge assumed.
  2. [Section 4] Several figures illustrating the Wythoff construction (e.g., for the icosahedral or hyperbolic cases) appear to be referenced but are not described in sufficient detail in the captions; adding coordinate labels or edge-coloring explanations would improve readability.
  3. [References] The bibliography lists the classical sources but omits page numbers or theorem numbers for the specific results being reproved (e.g., Andreev’s theorem on acute-angled polyhedra); cross-references to exact statements in the originals would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript as a structured survey of Coxeter polyhedra and related results. The recommendation for minor revision is noted, though no specific major comments were provided in the report.

Circularity Check

0 steps flagged

Expository survey with no circular derivations

full rationale

The paper is an introduction and survey that attributes its core theorems explicitly to external authors (Andreev, Coxeter, Vinberg) and reproduces them from standard axioms of spherical/Euclidean/hyperbolic geometry as developed in the cited classical works. No load-bearing step reduces by construction to a self-citation, fitted parameter, or internal definition; the Wythoff construction and polyhedral descriptions are presented as applications of prior results rather than novel derivations. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is an expository survey; all content rests on the standard axioms of constant-curvature geometries and the classical theorems of Andreev, Coxeter, and Vinberg. No free parameters, ad-hoc axioms, or invented entities are introduced by the paper itself.

axioms (2)
  • standard math Standard axioms of spherical, Euclidean, and hyperbolic 3-space
    Invoked throughout the definitions of non-obtuse and Coxeter polyhedra.
  • domain assumption Existence and classification theorems of Andreev, Coxeter, and Vinberg
    The paper states it proves these theorems, so they function as background results being re-derived or summarized.

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