ADEG-Polyhedra: Finite Hyperbolic Coxeter Polyhedra

• ADEG-polyhedra are finite-volume hyperbolic Coxeter polyhedra characterized by mutually intersecting facets and dihedral angles exclusively in {π/2, π/3, π/6} with at least one π/6 angle.
• They extend the ADE family by incorporating the G₂ angle type, resulting in a finite set of 24 explicitly classified polyhedra with dimensions capped at 11.
• The classification employs Coxeter diagrams, affine subdiagrams, and admissible configurations, highlighting exceptional cases like the 9-dimensional Pₛₜₐᵣ with 14 facets.

ADEG-polyhedra are a class of finite-volume hyperbolic Coxeter polyhedra $P\subset \mathbb H^n$ characterized by three conditions: all facets are mutually intersecting, all dihedral angles belong to $\{\pi/2,\pi/3,\pi/6\}$, and at least one dihedral angle is $\pi/6$. They extend Prokhorov’s ADE-polyhedra by adjoining the $G_2$ angle type, and the resulting family is unexpectedly rigid: the classification in hyperbolic space is finite, explicit, and bounded in dimension. The current classification theorem states that there are exactly $24$ ADEG-polyhedra, all of finite volume, all with $n\le 11$, and combinatorially each is either a triangle or tetrahedron, a doubly-truncated simplex in $\mathbb H^5$, a pyramid, or the exceptional polyhedron $P_\star\subset\mathbb H^9$ with $14$ facets (Bredon, 7 Jul 2025).

1. Definition and Coxeter-hyperbolic framework

The ambient model is the hyperboloid model

$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$

with Lorentzian form

$\{\pi/2,\pi/3,\pi/6\}$0

A finite-volume hyperbolic polyhedron is written as

$\{\pi/2,\pi/3,\pi/6\}$1

where each $\{\pi/2,\pi/3,\pi/6\}$2 is a unit spacelike outward normal to a bounding hyperplane. Its Gram matrix is

$\{\pi/2,\pi/3,\pi/6\}$3

with diagonal entries $\{\pi/2,\pi/3,\pi/6\}$4, and off-diagonal entries determined by whether the corresponding facets meet at angle $\{\pi/2,\pi/3,\pi/6\}$5, are parallel, or are ultraparallel (Bredon, 7 Jul 2025).

For ADEG-polyhedra, the diagrammatic condition is especially simple. In the Coxeter diagram, nodes correspond to facets, unlabeled simple edges encode angle $\{\pi/2,\pi/3,\pi/6\}$6, absence of an edge encodes $\{\pi/2,\pi/3,\pi/6\}$7, and an edge labeled $\{\pi/2,\pi/3,\pi/6\}$8 encodes $\{\pi/2,\pi/3,\pi/6\}$9. Because every pair of facets intersects in $\pi/6$0, ADEG diagrams contain no $\pi/6$1-edges. The phrase “mutually intersecting facets” is understood in $\pi/6$2, not only in the interior of $\pi/6$3. This excludes both parallel and ultraparallel facet pairs.

The name “ADEG” reflects the root-system types appearing in the spherical and affine subdiagrams of the corresponding Coxeter diagrams: only $\pi/6$4, $\pi/6$5, $\pi/6$6, and $\pi/6$7 occur. The defining $\pi/6$8 angle is what distinguishes ADEG-polyhedra from the earlier ADE case, where only $\pi/6$9 and $G_2$0 are allowed (Bredon, 7 Jul 2025).

2. Angle restrictions in high dimension

A central structural result is that large-dimensional hyperbolic Coxeter polyhedra cannot have arbitrarily small non-right dihedral angles. More precisely, if $G_2$1 and $G_2$2 is a finite-volume Coxeter polyhedron, then every non-zero dihedral angle is of the form $G_2$3 with $G_2$4. Under the stronger hypothesis that all facets are mutually intersecting, the same conclusion already holds for all $G_2$5. There is also an ideal analogue: if $G_2$6 and $G_2$7 is an ideal Coxeter polyhedron, then every non-zero dihedral angle is $G_2$8 with $G_2$9 (Bredon, 7 Jul 2025).

These propositions isolate the angle set $24$0 as a natural extremal regime. The restriction is not merely combinatorial: it arises from the interaction between high-dimensional affine subdiagrams, finite-volume criteria, and the behavior of $24$1-faces. In particular, once a diagram contains an edge $24$2 with $24$3, the codimension-$24$4 face associated to that $24$5 subdiagram forces affine structure incompatible with finite volume in the relevant dimensions.

The paper is careful about terminology here. “Non-zero dihedral angle” includes the right angle $24$6; the excluded phenomenon is the limiting zero-angle behavior associated with ideal tangency or parallelism. This distinction matters because ADEG-polyhedra, by definition, have no parallel or disjoint facets at all (Bredon, 7 Jul 2025).

3. Constructive method via $24$7-faces and admissible configurations

The classification is driven by a constructive procedure tailored to the angle set $24$8. Every ADEG-polyhedron of dimension $24$9 contains a $n\le 11$0-subdiagram $n\le 11$1, because at least one $n\le 11$2-angle is present. By Allcock’s theorem on faces of Coxeter polyhedra, the corresponding codimension-$n\le 11$3 face $n\le 11$4 is itself a Coxeter polyhedron, and in the ADEG setting $n\le 11$5 is necessarily an ADE- or ADEG-polyhedron in dimension $n\le 11$6 (Bredon, 7 Jul 2025).

A key lemma states that if $n\le 11$7 and $n\le 11$8 is ADEG, then its Coxeter diagram contains an affine subdiagram of type $n\le 11$9. More precisely, there is an affine rank-$\mathbb H^5$0 subdiagram

$\mathbb H^5$1

corresponding to a non-simple ideal vertex $\mathbb H^5$2. Since facets are pairwise intersecting, there are no affine components of rank $\mathbb H^5$3, so no $\mathbb H^5$4 components occur.

The remaining facets are encoded by vectors $\mathbb H^5$5 of squared norm $\mathbb H^5$6. Their incidence with the affine components is recorded by coefficients $\mathbb H^5$7 and $\mathbb H^5$8, constrained by the allowed angles. For each affine component one defines

$\mathbb H^5$9

and the collinearity of the ideal vectors forces

$P_\star\subset\mathbb H^9$0

The decisive compatibility formula is Prokhorov’s relation

$P_\star\subset\mathbb H^9$1

where each $P_\star\subset\mathbb H^9$2 is an explicit quadratic expression in the coefficients $P_\star\subset\mathbb H^9$3 with constants determined by the relevant root system. An admissible pair $P_\star\subset\mathbb H^9$4 is then one for which the common-$P_\star\subset\mathbb H^9$5 condition holds and

$P_\star\subset\mathbb H^9$6

corresponding exactly to the permitted angles $P_\star\subset\mathbb H^9$7 (Bredon, 7 Jul 2025).

This reduces classification to a finite search. One starts from lower-dimensional ADE or ADEG $P_\star\subset\mathbb H^9$8-faces, chooses a possible affine rank-$P_\star\subset\mathbb H^9$9 subdiagram containing $14$0, solves the admissibility constraints for extra vectors, and then discards candidates obstructed by superhyperbolicity, forbidden affine configurations, failure of Vinberg’s finite-volume criterion, or inconsistency of the induced $14$1-faces. The procedure is inductive but finite because the lower-dimensional ADE and ADEG cases are already classified.

4. Complete classification

The classification theorem states that every ADEG-polyhedron is one of the $14$2 Coxeter polyhedra listed in the paper’s table. In particular, every ADEG-polyhedron is non-compact for $14$3, non-simple for $14$4, and satisfies $14$5. Combinatorially, it is either a triangle or a tetrahedron, a doubly-truncated simplex in $14$6, a pyramid, or the exceptional polyhedron $14$7 (Bredon, 7 Jul 2025).

The dimension-by-dimension discussion highlights the following families:

Dimension	Families singled out
$14$8	hyperbolic Coxeter triangles with at least one angle $14$9
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$0	non-compact ADEG-tetrahedra
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$1	one pyramid and two doubly-truncated $$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$2-simplices
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$3	a single pyramid
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$4	pyramids over products of two or three simplices
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$5	one pyramid and $$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$6
$$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$7	a single pyramid

The paper also states that no ADEG-polyhedra exist in dimensions $$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$8, $$\mathbb H^n=\{x\in\mathbb R^{n+1}\mid \langle x,x\rangle=-1,\ x_{n+1}>0\},$$9, $\{\pi/2,\pi/3,\pi/6\}$00, or $\{\pi/2,\pi/3,\pi/6\}$01. Combined with the angle bounds, this gives a notably rigid picture: once the pairwise-intersection condition and the angle set $\{\pi/2,\pi/3,\pi/6\}$02 are imposed, hyperbolic Coxeter polyhedra become a finite exceptional family rather than an open-ended classification problem.

Simple ADEG-polyhedra occur only in the lowest dimensions. Using prior classification results for simple hyperbolic Coxeter polyhedra with mutually intersecting facets, the paper concludes that the simple ADEG examples are exactly the triangles and tetrahedra. All higher-dimensional ADEG-polyhedra are non-simple (Bredon, 7 Jul 2025).

5. Exceptional cases and the polyhedron $\{\pi/2,\pi/3,\pi/6\}$03

Besides the well-known simplices and pyramid families, the classification contains three exceptional polyhedra: two doubly-truncated simplices in $\{\pi/2,\pi/3,\pi/6\}$04, already known from Im Hof’s work, and the new polyhedron $\{\pi/2,\pi/3,\pi/6\}$05 (Bredon, 7 Jul 2025).

The polyhedron $\{\pi/2,\pi/3,\pi/6\}$06 has dimension $\{\pi/2,\pi/3,\pi/6\}$07, $\{\pi/2,\pi/3,\pi/6\}$08 facets, and $\{\pi/2,\pi/3,\pi/6\}$09 vertices, of which $\{\pi/2,\pi/3,\pi/6\}$10 are ideal. Its $\{\pi/2,\pi/3,\pi/6\}$11-vector is

$\{\pi/2,\pi/3,\pi/6\}$12

Its Coxeter diagram has a highly symmetric form: four disjoint copies of the $\{\pi/2,\pi/3,\pi/6\}$13-node chain $\{\pi/2,\pi/3,\pi/6\}$14 are arranged in parallel, a left extra node is joined by simple edges to the leftmost node of each row, and a right extra node is joined similarly to the rightmost node of each row. This makes $\{\pi/2,\pi/3,\pi/6\}$15 neither a simplex nor a pyramid, and it is not one of the two known doubly-truncated $\{\pi/2,\pi/3,\pi/6\}$16-simplices.

The paper further notes that all $\{\pi/2,\pi/3,\pi/6\}$17-faces of $\{\pi/2,\pi/3,\pi/6\}$18 have the same combinatorial structure, namely that of a pyramid over a product of three simplices of type $\{\pi/2,\pi/3,\pi/6\}$19. In arithmetic terms, the associated reflection group $\{\pi/2,\pi/3,\pi/6\}$20 is arithmetic over $\{\pi/2,\pi/3,\pi/6\}$21, and is commensurable with both the reflection group of Prokhorov’s $\{\pi/2,\pi/3,\pi/6\}$22-dimensional ADE-polyhedron $\{\pi/2,\pi/3,\pi/6\}$23 and the minimal-covolume cusped hyperbolic Coxeter simplex group in dimension $\{\pi/2,\pi/3,\pi/6\}$24. Its volume is of the form

$\{\pi/2,\pi/3,\pi/6\}$25

The two doubly-truncated $\{\pi/2,\pi/3,\pi/6\}$26-simplices are also exceptional in a precise sense. They are the only higher-dimensional ADEG-polyhedra besides $\{\pi/2,\pi/3,\pi/6\}$27 that are neither pyramids nor simplices, and both arise from admissible configurations recovered in the $\{\pi/2,\pi/3,\pi/6\}$28 stage of the inductive construction (Bredon, 7 Jul 2025).

6. Relation to ADE-polyhedra and terminological scope

ADEG-polyhedra are best understood as a strict enlargement of the ADE family. In the ADE case, only $\{\pi/2,\pi/3,\pi/6\}$29 and $\{\pi/2,\pi/3,\pi/6\}$30 occur, and the relevant spherical and affine diagram types are $\{\pi/2,\pi/3,\pi/6\}$31, $\{\pi/2,\pi/3,\pi/6\}$32, and $\{\pi/2,\pi/3,\pi/6\}$33. ADEG-polyhedra add the $\{\pi/2,\pi/3,\pi/6\}$34 component, equivalently the angle $\{\pi/2,\pi/3,\pi/6\}$35, while preserving the requirement that all facets intersect pairwise. This produces a class that is still finite-volume and classifiable, but admits genuinely new non-simple behavior, including the $\{\pi/2,\pi/3,\pi/6\}$36-dimensional example $\{\pi/2,\pi/3,\pi/6\}$37 (Bredon, 7 Jul 2025).

The classification also depends heavily on earlier structure theorems. Vinberg’s finite-volume criterion is used in diagrammatic form to control spherical and affine subdiagrams; Allcock’s theorem identifies codimension-$\{\pi/2,\pi/3,\pi/6\}$38 $\{\pi/2,\pi/3,\pi/6\}$39-faces as Coxeter polyhedra in their own right; and prior classifications of simple polyhedra with mutually intersecting facets and of ADE-polyhedra supply the lower-dimensional input for the induction. A plausible implication is that ADEG-polyhedra occupy a boundary position between the tractable ADE regime and the much less classifiable general theory of hyperbolic Coxeter polyhedra.

The term “ADEG-polyhedra” is specific to this hyperbolic Coxeter setting. Nearby polyhedral literatures in the same source set do not define such a class: the almost-regular spherical-polyhedra literature does not use the term (Rasheed et al., 2015), and the quasi-Euclidean classification of alcoved polyhedra likewise does not identify a named ADEG subclass (Puente, 2020). Within current usage, the direct technical meaning of ADEG-polyhedra is therefore the hyperbolic one given above.