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Wythoﬃan Skeletal Polyhedra in Ordinary Space, I

Egon Schulte and Abigail Williams

Department of Mathematics

Northeastern University, Boston, MA 02115, USA

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Abstract Skeletal polyhedra are discrete structures made up of ﬁnite, ﬂat or skew, or inﬁnite, helical or

zigzag, polygons as faces, with two faces on each edge and a circular vertex-ﬁgure at each vertex. When a

variant of Wythoﬀ’s construction is applied to the forty-eight regular skeletal polyhedra (Gr¨unbaum-Dress

polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as “truncations” of the original

polyhedra. These Wythoﬃans are vertex-transitive and often feature vertex conﬁgurations with an attractive

mix of diﬀerent face shapes. The present paper describes the blueprint for the construction and treats the

Wythoﬃans for distinguished classes of regular polyhedra. The Wythoﬃans for the remaining classes of

regular polyhedra will be discussed in Part II, by the second author. We also examine when the construction

produces uniform skeletal polyhedra.

Key words. Uniform polyhedron, Archimedean solids, regular polyhedron, maps on surfaces, Wythoﬀ’s

construction, truncation

MSC 2010. Primary: 51M20. Secondary: 52B15.

1 Introduction

Since ancient times, mathematicians and scientists have been studying polyhedra in ordinary Euclidean

3-space E3. With the passage of time, various notions of polyhedra have attracted attention and have

brought to light exciting new classes of highly symmetric structures including the well-known Platonic and

Archimedean solids, the Kepler-Poinsot polyhedra, the Petrie-Coxeter polyhedra, and the more recently

discovered Gr¨unbaum-Dress polyhedra (see [5,7,18, 19, 22]). Over time we can observe a shift from the clas-

sical approach of viewing a polyhedron as a solid, to topological and algebraic approaches focussing on

the underlying maps on surfaces (see Coxeter-Moser [12]), to graph-theoretical approaches highlighting the

combinatorial incidence structures and featuring a polyhedron as a skeletal ﬁgure in space.

The skeletal approach to polyhedra in E3was pioneered by Gr¨unbaum in [22] and has had an enormous

impact on the ﬁeld. Skeletal polyhedra are discrete geometric structures made up of convex or non-convex,

ﬂat (planar) or skew, ﬁnite or inﬁnite (helical or zigzag) polygons as faces, with a circular vertex-ﬁgure at

each vertex, such that every edge lies in exactly two faces. There has been a lot of recent activity in this area:

Supported by NSA-grant H98230-14-1-0124. Email: schulte@neu.edu

Email: abigail.williams13@gmail.com

Address(es) of author(s) should be given

2 Schulte and Williams

the skeletal regular polyhedra were enumerated by Gr¨unbaum [22] and Dress [18, 19] (for a simpler approach

to the classiﬁcation see McMullen & Schulte [40,41]); the skeletal chiral polyhedra were classiﬁed in [50,51]

(see also Pellicer & Weiss [47]); the regular polygonal complexes, a more general class of discrete skeletal

structures than polyhedra, were classiﬁed in Pellicer & Schulte [45, 46]; and corresponding enumerations

for certain classes of regular polyhedra, polytopes, or apeirotopes (inﬁnite polytopes) in higher-dimensional

Euclidean spaces were achieved by McMullen [35, 37, 38] (see also Arocha, Bracho & Montejano [1] and

Bracho [2]). All these skeletal structures are relevant to the study of crystal nets in crystal chemistry (see [16,

43,44,52, 57]).

The present paper and its successor [58] by the second author are inspired by the quest for a deeper

understanding of the uniform skeletal polyhedra in E3, that is, the skeletal analogues of the Archimedean

solids (see also [59]). There is a large body of literature on the traditional uniform polyhedra and higher-

dimensional polytopes (see [6, 8,9,32, 34]). Recall that a convex polyhedron in E3is said to be uniform if

its faces are convex regular polygons and its symmetry group is transitive on the vertices. The uniform

convex polyhedra are precisely the Archimedean solids and the prisms and antiprisms. The classiﬁcation for

the ﬁnite, convex or non-convex, uniform polyhedra with planar faces was essentially obtained in a classical

paper by Coxeter, Longuet-Higgins and Miller [11], but the completeness of the enumeration was only proved

years later, independently, by Skilling [55,56] and Har’El [30]. The classiﬁcation of arbitrary uniform skeletal

polyhedra is a challenging open problem. Even the ﬁnite polyhedra with skew faces have not been classiﬁed.

The Wythoﬃans of the regular skeletal polyhedra studied in this paper represent a tractable class of

skeletal polyhedra that contains a wealth of new examples of uniform polyhedra with non-planar faces. In

fact, our study actually goes a long way in classifying all the uniform skeletal polyhedra in E3. The name

“Wythoﬃan” is derived from Wythoﬀ’s construction (see [7, 41]). Our approach takes a geometrically regular

polyhedron Pin E3as input and then produces from it up to seven diﬀerent kinds of geometric Wythoﬃans

by an analogue of Wythoﬀ’s construction. The procedure applies to all forty-eight geometrically regular

polyhedra in E3and often produces amazing ﬁgures as output. Our goal is to analyze these Wythoﬃans.

The paper is organized as follows. In Section 2 we begin by reviewing the basic concept of a regular

polyhedron, both geometric and abstract, and discussing realizations as a means to connect the abstract

theory with the geometric theory. In Section 3 we introduce the seven Wythoﬃans at the abstract level

and then in Section 4 provide the blueprint for the realization as geometric Wythoﬃans in E3. Finally, in

Section 5 we describe the geometric Wythoﬃans of various distinguished classes of regular polyhedra. The

subsequent paper [58] treats the geometric Wythoﬃans for the remaining classes of regular polyhedra.

2 Geometric and abstract polyhedra

We begin by deﬁning a geometric polyhedron as a discrete structure in Euclidean 3-space E3rather than as

a realization of an abstract polyhedron.

Given a geometric ﬁgure in E3, its (geometric) symmetry group consists of all isometries of its aﬃne hull

that map the ﬁgure to itself. When a ﬁgure is linear or planar we sometimes view this group as a subgroup

of the isometry group of E3, with the understanding that the elements of the group have been extended

trivially from the aﬃne hull of the ﬁgure to the entire space E3.

2.1 Geometric polyhedra

Informally, a geometric polyhedron will consist of a family of vertices, edges, and ﬁnite or inﬁnite polygons,

all ﬁtting together in a way characteristic for traditional convex polyhedra (see [22] and [41, Ch. 7E]). For

two distinct points uand u0of E3we let (u, u0) denote the closed line segment with ends uand u0.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 3

Aﬁnite polygon, or simply an n-gon, (v1, v2, . . . , vn) in E3is a ﬁgure formed by distinct points v1, . . . , vn,

together with the line segments (vi, vi+1), for i= 1, . . . , n −1, and (vn, v1). Similarly, an inﬁnite polygon con-

sists of an inﬁnite sequence of distinct points (...,v−2, v−1, v0, v1, v2, . . .) and of the line segments (vi, vi+1)

for each i, such that each compact subset of E3meets only ﬁnitely many line segments. In either case the

points are the vertices and the line segments the edges of the polygon.

A polygon is geometrically regular if its geometric symmetry group is a (ﬁnite or inﬁnite dihedral) group

acting transitively on the ﬂags, that is, the 2-element sets consisting of a vertex and an incident edge.

Deﬁnition 1 Ageometric polyhedron, or simply polyhedron (if the context is clear), Pin E3consists of a

set of points, called vertices, a set of line segments, called edges, and a set of polygons, called faces, such

that the following properties are satisﬁed.

(a) The graph deﬁned by the vertices and edges of P, called the edge graph of P, is connected.

(b) The vertex-ﬁgure of Pat each vertex of Pis connected. By the vertex-ﬁgure of Pat a vertex vwe mean

the graph whose vertices are the neighbors of vin the edge graph of Pand whose edges are the line

segments (u, w), where (u, v) and (v, w ) are adjacent edges of a common face of P.

(c) Each edge of Pis contained in exactly two faces of P.

(d) Pis discrete, meaning that each compact subset of E3meets only ﬁnitely many faces of P.

Note that the discreteness assumption in Deﬁnition 1(d) implies that the vertex-ﬁgure at every vertex of

a polyhedron Pis a ﬁnite polygon. Thus vertices have ﬁnite valency in the edge graph of P. The edge graph

is often called the 1-skeleton of P.

Aﬂag of a geometric polyhedron Pis a 3-element set containing a vertex, an edge, and a face of P, all

mutually incident. Two ﬂags of Pare called adjacent if they diﬀer in precisely one element. An apeirohedron

is an inﬁnite geometric polyhedron.

A geometric polyhedron Pin E3is said to be (geometrically)regular if its symmetry group G(P) is

transitive on the ﬂags of P. The symmetry group G(P) of a regular polyhedron Pis transitive, separately,

on the vertices, edges, and faces of P. In particular, the faces are necessarily regular polygons, either ﬁnite,

planar (convex or star-) polygons or non-planar, skew, polygons, or inﬁnite, planar zigzags or helical polygons

(see [10, Ch. 1] or [22]). Linear apeirogons do not occur as faces of regular polyhedra.

We also brieﬂy touch on chiral polyhedra. These are nearly regular polyhedra. A geometric polyhedron

Pis called (geometrically)chiral if its symmetry group has two orbits on the ﬂags of P, such that adjacent

ﬂags are in distinct orbits.

The geometric polyhedra in E3which are regular or chiral all have a vertex-transitive symmetry group

and regular polygons as faces. They are particular instances of uniform polyhedra. A geometric polyhedron

Pis said to be (geometrically )uniform if Phas a vertex-transitive symmetry group and regular polygons as

faces. The uniform polyhedra with planar faces have attracted a lot attention in the literature. Our methods

will provide many new examples of uniform skeletal polyhedra with nonplanar faces.

At times we encounter geometric ﬁgures which are not polyhedra but share some of their properties.

Examples are the polygonal complexes described in [45, 46]. Roughly speaking, a polygonal complex Kin E3

is a structure with the deﬁning properties (a), (b) and (d) of Deﬁnition 1 for polyhedra, but with property

(c) replaced by the more general property, (c’) say, requiring that each edge of Kbe contained in exactly r

faces of K, for a ﬁxed number r>2. The polygonal complexes with r= 2 are just the geometric polyhedra.

The vertex-ﬁgures of polygonal complexes need not be simple polygons as for polyhedra; they even can be

graphs with double edges (edges of multiplicity 2). A polygonal complex is regular if its geometric symmetry

group is transitive on the ﬂags.

The Wythoﬃans we construct from geometrically regular polyhedra Pin E3will usually be generated

from the orbit of a single point inside the fundamental region of the symmetry group of P. Given a discrete

group Gof isometries of an n-dimensional Euclidean space En, an open subset Dof Enis called a fundamental

4 Schulte and Williams

region for Gif r(D)∩D=∅for r∈G\ {1}and En=Sr∈Gr(cl(D)), where cl(D) denotes the closure of D

(see [21]). Note that our notion of fundamental region is not quite consistent with the notion of a fundamental

simplex used in the theory of Coxeter groups or related groups (see [41, Ch. 3]), where a fundamental simplex

by deﬁnition is a closed simplex (its interior is a fundamental region according to our deﬁnition).

Some of the groups we encounter have complicated fundamental regions. The following procedure pro-

duces a possible fundamental region for any given discrete group Gof isometries of En. Let u∈Enbe a

point that is not held invariant under any non-identity transformation in G. For r∈Gdeﬁne H[r(u)] as

the open half space containing ubounded by the hyperplane which perpendicularly bisects the line segment

between uand r(u). Then D:= Tr∈GH[r(u)] is a fundamental region of Gin En. In other words, Dis the

open Dirichlet-Voronoi region, centered at u, of the orbit of uunder Gin En(see [3]).

2.2 Abstract polyhedra

While our focus is on geometric polyhedra it is often useful to view a geometric polyhedron as a realization

of an abstract polyhedron in Euclidean space. We begin with a brief review of the underlying abstract theory

(see [41, Ch. 2]).

An abstract polyhedron, or abstract 3-polytope, is a partially ordered set Pwith a strictly monotone rank

function with range {−1,0,1,2,3}. The elements of rank jare the j-faces of P. For j= 0, 1 or 2, we also

call j-faces vertices,edges and facets, respectively. When there is little chance of confusion, we use standard

terminology for polyhedra and reserve the term “face” for “2-face” (facet). There is a minimum face F−1(of

rank −1) and a maximum face F3(of rank 3) in P; this condition is included for convenience and is often

omitted as for geometric polyhedra. The ﬂags (maximal totally ordered subsets) of Peach contain, besides

F−1and F3, exactly one vertex, one edge and one facet. In practice, when listing the elements of a ﬂag we

often suppress F−1and F3. Further, Pis strongly ﬂag-connected, meaning that any two ﬂags Φand Ψof P

can be joined by a sequence of ﬂags Φ=Φ0, Φ1, . . . , Φk=Ψ, where Φi−1and Φiare adjacent (diﬀer by one

face), and Φ∩Ψ⊆Φifor each i. Finally, if Fand Gare a (j−1)-face and a (j+ 1)-face with F < G and

06j62, then there are exactly two j-faces Hsuch that F < H < G. As a consequence, for 0 6j62,

every ﬂag Φof Pis adjacent to just one ﬂag, denoted Φj, diﬀering in the j-face; the ﬂags Φand Φjare said

to be j-adjacent to each other.

When Fand Gare two faces of an abstract polyhedron Pwith F6G, we call G/F := {H|F6H6G}

asection of P. We usually identify a face Fwith the section F/F−1. The section F3/F is the co-face of P

at F, or the vertex-ﬁgure at Fif Fis a vertex.

If all facets of an abstract polyhedron Pare p-gons for some p, and all vertex-ﬁgures are q-gons for some

q, then Pis said to be of (Schl¨aﬂi )type {p, q}; here pand qare permitted to be inﬁnite. We call an abstract

polyhedron locally ﬁnite if all its facets and all its vertex-ﬁgures are ﬁnite polygons.

An automorphism of an abstract polyhedron Pis an incidence preserving bijection of P(that is, if ϕis

the bijection, then F6Gin Pif and only if ϕ(F)6ϕ(G) in P.) By Γ(P) we denote the (combinatorial)

automorphism group of P.

We call an abstract polyhedron Pregular if Γ(P) is transitive on the ﬂags of P. Let Φ:= {F0, F1, F2}

be a base ﬂag of P. The automorphism group Γ(P) of a regular polyhedron Pis generated by distinguished

generators ρ0, ρ1, ρ2(with respect to Φ), where ρjis the unique automorphism which ﬁxes all faces of Φbut

the j-face. These generators satisfy the standard Coxeter-type relations

ρ2

0=ρ2

1=ρ2

2= (ρ0ρ1)p= (ρ1ρ2)q= (ρ0ρ2)2= 1 (1)

determined by the type {p, q}of P(when p=∞or q=∞the corresponding relation is superﬂuous and

hence is omitted); in general there are also other independent relations. Note that, in a natural way, the

automorphism group of the facet of Pis hρ0, ρ1i, while that of the vertex-ﬁgure is hρ1, ρ2i.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 5

An abstract polyhedron Pis said to be chiral if Γ(P) has two orbits on the ﬂags, such that adjacent ﬂags

are in distinct orbits. Note that the underlying abstract polyhedron of a geometrically chiral (geometric)

polyhedron must be (combinatorially) chiral or (combinatorially) regular.

In analogy with the geometric case we could deﬁne an abstract polyhedron Pto be (combinatorially)

“uniform” if Phas regular facets and Γ(P) acts transitively on the vertices of P. However, the facets of

any abstract polyhedron trivially are combinatorially regular, so being uniform just reduces to being vertex-

transitive under the automorphism group.

The Petrie dual of a (geometric or abstract) regular polyhedron Phas the same vertices and edges as P;

its facets are the Petrie polygons of P, which are paths along the edges of Psuch that any two successive

edges, but not three, belong to a facet of P.

2.3 Realizations

The abstract theory is connected to the geometric theory through the concept of a realization. Let Pbe an

abstract polyhedron, and let Fjdenote its set of j-faces for j= 0,1,2. Following [41, Sect. 5A], a realization

of Pis a mapping β:F0→Eof the vertex-set F0into some Euclidean space E. Then deﬁne β0:= βand

V0:= β(P0), and write 2Xfor the family of subsets of a set X. The realization βrecursively induces two

surjections: a surjection β1:F1→V1, with V1⊂2V0consisting of the elements

β1(F) := {β0(G)|G∈ F0and G6F}

for F∈ F1; and a surjection β2:F2→V2, with V2⊂2V1consisting of the elements

β2(F) := {β1(G)|G∈ F1and G6F}

for F∈ F2. Even though each βjis determined by β, it is helpful to think of the realization as given by all

the βj. A realization βis said to be faithful if each βjis a bijection; otherwise, βis degenerate . Note that

not every abstract polyhedron admits a realization in a Euclidean space. (In diﬀerent but related contexts,

a realization is sometimes called a representation [27,49].)

In our applications, E=E3and all realizations are faithful (and discrete). In this case, the vertices, edges

and facets of Pare in one-to-one correspondence with certain points, line segments, and (ﬁnite or inﬁnite)

polygons in E3, and it is safe to identify a face of Pwith its image in E3. The resulting family of points,

line segments, and polygons then is a geometric polyhedron in E3and is denoted by P; it is understood that

Pinherits the partial ordering of P. We frequently identify Pand P. Note that the symmetry group of a

faithful realization is a subgroup of the automorphism group.

Conversely, all geometric polyhedra as deﬁned above arise as realizations of abstract polyhedra. In par-

ticular, the geometrically regular polyhedra in E3are precisely the 3-dimensional realizations of abstract

regular polyhedra which are discrete and faithful and have a ﬂag-transitive symmetry group. These polyhe-

dra have been extensively studied (see [41, Sect. 7E]). We brieﬂy review them in Section 4 as they form the

basis of our construction.

For geometrically regular polyhedra Pin E3we prefer to denote the distinguished generators of G(P) by

r0, r1, r2. Thus, if Φ={F0, F1, F2}is again a base ﬂag of P, and rjthe involutory symmetry of Pﬁxing all

faces of Φbut the j-face, then G(P) = hr0, r1, r2iand the Coxeter-type relations

r2

0=r2

1=r2

2= (r0r1)p= (r1r2)q= (r0r2)2= 1 (2)

hold, where again {p, q}is the type of P. Here qmust be ﬁnite since Pis discrete; however, pstill can

be inﬁnite. When Pis geometrically regular the groups Γ(P) and G(P) are isomorphic; in particular, the

mapping ρj7→ rj(j= 0,1,2) extends to an isomorphism between the groups.

6 Schulte and Williams

Two realizations of an abstract regular polyhedron Pcan be combined to give a new realization of Pin a

higher-dimensional space. Suppose we have two (not necessarily faithful) realizations of Pin two Euclidean

spaces, say Pwith generators r0, r1, r2in Eand P0with generators r0

0, r0

1, r0

2in E0(possibly some rj= 1 or

r0

j= 1 if Por P0is not faithful). Then their blend, denoted P#P0, is a realization of Pin E×E0obtained

by Wythoﬀ’s construction as an orbit structure as follows (see [7] and [41, Ch. 5A]). Write Rjand R0

jfor the

mirror (ﬁxed point set) of a distinguished generator rjin Eor r0

jin E0, respectively. The cartesian products

R0×R0

0,R1×R0

1and R2×R0

2, respectively, then are the mirrors for involutory isometries s0,s1and s2of

E×E0which generate the symmetry group G(P#P0) of the blend. Indeed, if v∈R1∩R2and v0∈R0

1∩R0

2

are the base (initial) vertices of the two realizations, then the point w:= (v, v0) in E×E0can be chosen

as the base (initial) vertex for the blend P#P0. Then the base edge and base face are determined by the

orbits of wunder the subgroups hs0iand hs0, s1i, respectively. Finally, the vertices, edges, and faces of the

entire polyhedron P#P0are the images of the base vertex, base edge, or base face under the entire group

hs0, s1, s2i. A realization which cannot be expressed as a blend in a non-trivial way is called pure.

3 Wythoﬃans of abstract polyhedra

Every abstract polyhedron Pnaturally gives rise to generally seven new abstract polyhedra, the abstract

Wythoﬃans of P. These Wythoﬃans have appeared in many applications, usually under diﬀerent names (see

[3,48]); they are often called truncations of the respective polyhedron or map (see [7]). (In the literature, the

word “Wythoﬃan” is mostly used as an adjective, not a noun, to describe a ﬁgure obtained by Wythoﬀ’s

construction. The use of “Wythoﬃan” in [17] is similar to ours.)

3.1 Wythoﬃans from the order complex

It is convenient to construct the Wythoﬃans from the order complex of P. The order complex C:= C(P)

of an abstract polyhedron Pis the 2-dimensional abstract simplicial complex, whose vertices are the proper

faces of P, and whose simplices are the chains (totally ordered subsets) of Pwhich only contain proper faces

of P(see [41, Ch. 2C]). The maximal simplices in Care in one-to-one correspondence with the ﬂags of P, and

are 2-dimensional. The type of a vertex of Cis its rank as a face of P. More generally, the type of a simplex

Ωof Cis the set of types of the vertices of Ω. Thus every 2-simplex has type {0,1,2}. Two 2-simplices of

Care j-adjacent if and only if they diﬀer in their vertices of type j. With this type function on chains, the

order complex acquires the structure of a labelled simplicial complex.

The deﬁning properties of Ptranslate into strong topological properties of C. In particular, each 2-

simplex of Cis j-adjacent to exactly one other 2-simplex, for j= 0,1,2. When rephrased for C, the strong

ﬂag-connectedness of Psays that, for any two 2-simplices Φand Ψof Cwhich intersect in a face Ω(a simplex

or the empty set) of C, there exists a sequence Φ=Φ0, Φ1, . . . , Φk−1, Φk=Ψof 2-simplices of C, all containing

Ω, such that Φi−1and Φiare adjacent for i= 1, . . . , k.

Recall that the star of a face Ωin a simplicial complex is the subcomplex consisting of all the simplices

which contain Ω, and all their faces. The link of Ωis the subcomplex consisting of all the simplices in the

star of Ωwhich do not intersect Ω. For an abstract polyhedron P, the structure of the link of a vertex in

its order complex Cdepends on the number of 2-simplices it is contained in. Every vertex of Cof type 1 has

a link isomorphic to a 4-cycle. If a vertex Fof Cis of type 2, and the 2-face F/F−1of Pis a p-gon, then

the link of Fin Cis a 2p-cycle if pis ﬁnite, or an inﬁnite path (an inﬁnite 1-dimensional simplicial complex

in which every vertex lies in exactly two 1-simplices) if pis inﬁnite. Similarly, if Fis a vertex of Cof type

0, and the vertex-ﬁgure F3/F of Pat Fis a q-gon, then the link of Fin Cis a 2q-cycle if qis ﬁnite, or an

inﬁnite path if qis inﬁnite.

If Pis a locally ﬁnite abstract polyhedron with order complex C, then Pcan be viewed as a face-

to-face tessellation on a (compact or non-compact) closed surface Sby topological polygons, and Cas a

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 7

triangulation of Sreﬁning Pin the manner of a “barycentric subdivision”. If Phas faces or vertex-ﬁgures

which are apeirogons, then the link of the corresponding vertices in Cis not a 1-sphere and so Pis not

supported by a closed surface; however, Cstill has the structure of a 2-dimensional pseudo-manifold, which

we again denote by S. (A 2-dimensional pseudo-manifold is a topological space Xwith a 2-dimensional

triangulation Ksuch that the following three conditions hold: ﬁrst, Xis the union of all triangles of K;

second, every edge of Klies in exactly two triangles of K; and third, any two triangles of Kcan be joined by

a ﬁnite sequence of triangles of Ksuch that successive triangles in the sequence intersect in an edge [53].) In

our applications, the vertex-ﬁgures of Pare always ﬁnite polygons, whereas the faces are often apeirogons.

If Pis regular and ρ0, ρ1, ρ2are the generators of Γ(P) associated with a base ﬂag Φof P, then Γ(P)

acts on Sas a group of homeomorphisms of Sand the distinguished generators appear as “combinatorial

reﬂections” in the sides of the 2-simplex Φof C. The 2-simplex Φis a fundamental triangle for the action of

Γ(P) on S, meaning that the orbit of every point of Sunder Γ(P) meets Φin exactly one point. (Recall

our previous remark about the notion of fundamental simplex, or in this case, fundamental triangle.) In

fact, every 2-simplex of Cis a fundamental triangle for Γ(P) on S, with a conjugate set of distinguished

generators occurring as “combinatorial reﬂections” in its sides. For a regular polyhedron, the order complex

can be completely described in terms of Γ(P) since this is already true for Pitself (see [41, Sect. 2C]).

For example, consider the regular tessellation P={4,4}of the Euclidean plane by squares, four coming

together at a vertex. Here Cappears as the actual barycentric subdivision of the tessellation, and any triangle

in Ccan serve as the fundamental region for the symmetry group, which in this case is a Euclidean plane

reﬂection group.

An alternative approach to abstract Wythoﬃans using dissections of fundamental triangles is described

in Pisanski & Zitnik [49] (see also [15]).

3.2 Wythoﬃans of abstract regular polyhedra

The Wythoﬃans of an abstract regular polyhedron Pare derived as orbit structures from the order complex

C, or equivalently, from the underlying surface or pseudo-manifold S. The construction can be carried out

at a purely combinatorial level for arbitrary abstract polyhedra without reference to automorphism groups

(by exploiting the action of the monodromy groups [42]). However, as we are mainly interested in geometric

Wythoﬃans derived from geometrically regular polyhedra, we will concentrate on regular polyhedra and

exploit their groups. The method employed is known as Wythoﬀ’s construction (see [7, 41]).

Now let Pbe an abstract regular polyhedron with order complex Cand surface or pseudo-manifold S.

Suppose the base ﬂag Φ={F0, F1, F2}of Pis realized as a fundamental triangle on Swith vertices F0, F1, F2.

This fundamental triangle Φnaturally partitions into seven subsets: its vertices, the relative interiors of its

edges, and its relative interior. The closure of each of these subsets is a simplex in Cand hence has a type

I. Thus each subset is naturally associated with a nonempty subset Iof {0,1,2}specifying the generators

which move the points of Φin the subset; more precisely, the subset belonging to Iconsists of the points of

Φwhich are transient under the generators ρiwith i∈Ibut invariant under the generators ρiwith i /∈I.

More explicitly, for the vertex Fiof Φthe type Iis {i}; for the relative interior of the edge joining Fiand

Fjit is {i, j}; and for the relative interior it is {0,1,2}.

The abstract Wythoﬃans to be deﬁned will be in one-to-one correspondence with the seven subsets

in the partition of Φand hence be parametrized by subsets Iof {0,1,2}. Each subset in the partition is

characterized as the set of possible locations for the initial vertex of the corresponding Wythoﬃan; diﬀerent

choices of initial vertices within each subset will produce isomorphic Wythoﬃans. Thus a subset Iindexing

an abstract Wythoﬃan of Pspeciﬁes precisely the generators, namely the generators ρiwith i∈I, under

which the corresponding initial vertex is transient. The generators ρiwith i /∈Ithen leave the initial vertex

invariant; in fact, the choice of initial vertex within the fundamental triangle is such that its stabilizer in

Γ(P) is precisely given by the subgroup hρi|i /∈Ii. It then follows that the vertices of the Wythoﬃan are

8 Schulte and Williams

IInitial vertex Ringed diagram Wythoﬃan Vertex Symbol

{0}P0(pq)

{1}P1(p.q.p.q)

{2}P2(qp)

{0,1}P01 (p.q.q)

{0,2}P02 (p.4.q.4)

{1,2}P12 (2p.2p.q)

{0,1,2}P012 (2p.2q.4)

Table 1 Notation for the Wythoﬃan, PI, based on choice of I.

in one-to-one correspondence with the left cosets of this subgroup in Γ(P). We write PIfor the Wythoﬃan

associated with I.

Table 1 indicates the seven possible placements for the initial vertex inside the fundamental triangle. We

have adopted an analogue of Coxeter’s [7] diagram notation for truncations of regular convex polyhedra,

and have included the corresponding diagrams in the third column; following Coxeter’s convention, a node

of the diagram for PIis ringed if and only if its label belongs to I. We also refer to these diagrams as ringed

diagrams. The fourth column shows the Wythoﬃan (in Latin letters, for the realizations), where we have

written Pi,Pij and Pijk in place of P{i},P{i,j}or P{i,j,k}, respectively. The ﬁnal column gives the vertex

symbol for each Wythoﬃan. These symbols will discussed in further detail in Section 4.2. Note that in the

present context the basic, “unringed” Coxeter diagrams are not generally representing Coxeter groups [41,

Ch. 3] as for regular convex polyhedra. Here they are representing symmetry groups of arbitrary regular

polyhedra in E3. For a regular convex polyhedron, the seven Wythoﬃans correspond to the seven possible

ways of “truncating” the given polyhedron [7].

The abstract Wythoﬃan PIfor a given subset I⊆ {0,1,2}then is constructed as follows. Choose a

point v, the initial vertex of PI, inside the fundamental chamber Φon Ssuch that ρi(v)6=vfor i∈Iand

ρi(v) = vfor i /∈I. We ﬁrst generate the base faces for PI. Here we need to broaden the term base l-face to

include any l-face of PIincident with vwhose vertex set on Sis the orbit of vunder precisely ldistinguished

generators of Γ(P). Unlike in the case of regular polyhedra we now can have up to three diﬀerent kinds

of base l-face for l= 1,2. For instance, when I={0,1,2}there is a base 1-face corresponding to each

distinguished generator in Γ(P). To fully deﬁne the poset of faces obtained by Wythoﬀ’s construction we

will give explicit deﬁnitions of each type of base l-face for l= 0,1,2.

There is only one base 0-face, namely

F0:= v. (3)

For each i∈Iwe deﬁne the base 1-face

Fi

1:= {ρ(v)|ρ∈ hρii} ={v, ρi(v)}.(4)

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 9

Thus the number of base 1-faces in PIis |I|. The base 2-faces of PIwill be parametrized by the set I2of

2-element subsets of {0,1,2}given by

I2:=

{{i, j} | j=i±1}if I={i}, i = 0,1,2,

{{0,1},{1,2}} if I={0,1},{1,2},

{{0,1},{1,2},{0,2}} if I={0,2},{0,1,2}.

(5)

Now for each {i, j} ∈ I2we can deﬁne a base 2-face

Fij

2:= Fji

2:= {ρ(Fk

1)|ρ∈ hρi, ρji, k ∈I∩ {i, j}}.(6)

The full Wythoﬃan PIthen is the union (taken over l= 0,1,2) of the orbits of the base l-faces under

Γ(P). In PI, vertices are points on S, edges are 2-element subsets consisting of vertices, and 2-faces are sets

of edges. The partial order between faces of consecutive ranks is given by containment (meaning that the

face of lower rank is an element of the face of higher rank), and the full partial order then is the transitive

closure. When a least face (of rank −1) and a largest face (of rank 3) are appended PIbecomes an abstract

polyhedron. Note that when Pis locally ﬁnite the abstract Wythoﬃan PIcan be realized as a face-to-face

tessellation on the surface Sin much the same way in which the geometric Wythoﬃans of the regular plane

tessellation {4,4}were derived; then edges are simple curves and faces are topological polygons on S.

The Wythoﬃan P{0}is isomorphic to Pitself, and P{2}is isomorphic to the dual P∗of P. Thus both are

regular. The Wythoﬃan P{1}is isomorphic to the medial of P; it has p-gonal faces and q-gonal faces, if P

is of type {p, q}, and its vertices have valency 4. Recall that the medial of a polyhedron is a new polyhedron

(on the same surface), with vertices at the “midpoints” of the old edges and with edges joining two new

vertices if these are the midpoints of adjacent edges in an old face (see [48,49]).

The two Wythoﬃans P{0,1}and P{1,2}each have two base 2-faces and thus two kinds of 2-face: P{0,1}

has 2p-gons and q-gons, and P{1,2}has p-gons and 2q-gons. Each has 3-valent vertices. On the other hand,

P{0,2}and P{0,1,2}each have three base 2-faces and thus three kinds of 2-face: P{0,2}has p-gons, 4-gons

and q-gons, and P{0,1,2}has 2p-gons, 4-gons and 2q-gons. The vertices of P{0,2}are 4-valent and those of

P{0,1,2}3-valent.

Observe that the exchange of indices 0 ↔2 on an index set Ifor a Wythoﬃan, results in the index set

for the Wythoﬃan of the dual polyhedron P∗; that is, if I∗={2−i|i∈I}then PI∗= (P∗)I. Thus the dual

P∗has the same set of seven Wythoﬃans as the original polyhedron P. Moreover, if Pis self-dual then the

Wythoﬃans PIand PI∗are isomorphic for each I.

It is worth noting that the abstract Wythoﬃans PIdescribed in this section can be described purely

combinatorially without any explicit reference to the underlying surface. This is of little interest when Pis

locally ﬁnite, since then the 2-faces of the Wythoﬃans are topological polygons with ﬁnitely many edges.

However, if Phas apeirogonal 2-faces or vertex-ﬁgures, respectively, the base 2-face of PIgenerated from

the subgroup hρ0, ρ1ior hρ1, ρ2iof Γ(P) is an apeirogon and does not bound a disk in S. In our applications,

while the 2-faces of Pmay be inﬁnite, the vertex-ﬁgures of Pwill always be ﬁnite. In this case, if Phas

apeirogons as 2-faces then PIalso has an apeirogonal 2-face, except when I={2}and PI=P∗.

Note that the Wythoﬃans of an abstract polyhedron Pcan also described in terms of the monodromy

group of the polyhedron (see [42]). In the case of a regular polyhedron P, the monodromy group and

automorphism group are isomorphic and either can be chosen to deﬁne the Wythoﬃans PI. However, as we

will work in a geometric context where automorphisms become isometries, we have adopted an automorphism

based approach to Wythoﬃans.

4 Wythoﬃans of geometric polyhedra

In this section we discuss Wythoﬃans for geometrically regular polyhedra Pin E3. In particular, we explain

how an abstract Wythoﬃan associated with Pas an abstract polyhedron, can often itself be realized faithfully

10 Schulte and Williams

in E3in such a way that all combinatorial symmetries of Pare realized as geometric symmetries, and thus

be viewed as a geometric Wythoﬃan of P. In fact, whenever a realization exists there are generally many

such realizations. For a point u∈E3we let Gu(P) denote the stabilizer of uin G(P).

The key idea is to place the initial vertex for the realization inside a speciﬁed fundamental region of the

symmetry group G(P) in E3and then let Wythoﬀ’s construction applied with the generating reﬂections of

G(P) produce the desired geometric Wythoﬃan. The precise construction is detailed below. The fundamental

region of G(P) can be quite complicated and is generally not a simplicial cone as for the Platonic solids.

The generating symmetries r0, r1, r2of G(P) corresponding to the abstract symmetries ρ0, ρ1, ρ2of Γ(P)

are involutory isometries in E3and therefore are point reﬂections, halfturns (line reﬂections), or plane

reﬂections, with mirrors of dimension 0, 1 or 2, respectively. In order to realize an abstract Wythoﬃan PI

(with I⊆ {0,1,2}) of the given geometric polyhedron Pin E3, the initial vertex vmust be chosen such that

Gv(P) = hri|i /∈Ii.(7)

This initial placement condition will allow us to construct a faithful realization of PI. In fact, (7) is a

necessary and suﬃcient condition for the existence of a faithful realization of PIin E3which is induced by

the given realization of Pin the sense that all geometric symmetries of Pare also geometric symmetries of

PI. Note that condition (7) implies the more easily veriﬁable condition

ri(v)6=v(i∈I), ri(v) = v(i /∈I),(8)

which for speciﬁc points vusually is equivalent to (7).

The shape of the geometric Wythoﬃans will vary greatly with the choice of initial vertex. Our assumption

that vbe chosen inside the fundamental region for G(P) is, strictly speaking, not required. The initial

placement condition for valone guarantees that a faithful realization of PIcan be found by Wythoﬀ’s

construction. However, if the initial vertex vis chosen inside the fundamental region, then the original

polyhedron Pand its Wythoﬃan PIare similar looking in shape and so their intrinsic relationship is

emphasized.

By the very nature of the construction, Wythoﬃans are vertex-transitive and have vertex-transitive faces.

If the faces are actually regular polygons, then the Wythoﬃan is a geometrically uniform polyhedron in E3.

4.1 Regular polyhedra in E3

We brieﬂy review the classiﬁcation of the geometrically regular polyhedra in E3following the classiﬁcation

scheme of [41, Sect. 7E] (or [40]). There are 48 such regular polyhedra, up to similarity and scaling of

components (if applicable): 18 ﬁnite polyhedra, 6 planar apeirohedra, 12 blended apeirohedra, and 12 pure

(non-blended) apeirohedra. They are also known as the Gr¨unbaum-Dress polyhedra.

The ﬁnite regular polyhedra comprise the ﬁve Platonic solids {3,3},{3,4},{4,3},{3,5},{5,3}and the

four Kepler-Poinsot star-polyhedra {3,5

2},{5

2,3},{5,5

2},{5

2,5}, where the faces and vertex-ﬁgures are planar

but are permitted to be star polygons (the entry 5

2indicates pentagrams as faces or vertex-ﬁgures); and the

Petrie-duals of these nine polyhedra.

The planar regular apeirohedra consist of the three regular plane tessellations {4,4},{3,6}and {6,3},

and their Petrie-duals {∞,4}4,{∞,6}3and {∞,3}6, respectively.

There are twelve regular apeirohedra that are “reducible” and have components that are lower-dimensional

regular ﬁgures. These apeirohedra are blends of a planar regular apeirohedron Pand a line segment { } or

linear apeirogon {∞}. The notion of a blend used in this context is a variant of the notion of a blend of two

realizations of abstract polyhedra described earlier (but is technically not the same). The formal deﬁnition

is as follows. We let P0denote the line segment { } or the linear apeirogon {∞}.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 11

Suppose the symmetry groups of Pand P0are, respectively, G(P) = hr0, r1, r2iand G(P0) = hr0

0ior

hr0

0, r0

1i. For our purposes, G(P) acts on a plane in E3while G(P0) acts on a line perpendicular to that plane;

in particular, these two groups commute at the level of elements. The blending process requires us ﬁrst to

take the direct product of the groups, G(P)×G(P0), viewed as a subgroup of the full isometry group of

E3. The new regular apeirohedron, the blend P#P0, then is obtained from the subgroup of G(P)×G(P0)

generated by the set of involutions

(r0, r0

0),(r1,1),(r2,1)

or

(r0, r0

0),(r1, r0

1),(r2,1),

respectively; this subgroup is the symmetry group of the blend, and the involutions are the distinguished

generators. Thus G(P#P0) is a subgroup of G(P)×G(P0). In particular, if the plane of Pand line of

P0meet at the origin, and vand v0are the initial vertices of Pand P0for Wythoﬀ’s construction, then

the point (v, v0) in E3is the initial vertex of the blend. More explicitly, the blend, P#{ }, of Pand { }

has symmetry group h(r0, r0

0),(r1,1),(r2,1)iwhile the blend, P#{∞}, of Pand {∞} has symmetry group

h(r0, r0

0),(r1, r0

1),(r2,1)i. Throughout we will simplify the notation from (r, r0) to rr0for an element of

G(P)×G(P0).

For example, the blend of the standard square tessellation {4,4}and the linear apeirogon {∞}, denoted

{4,4}#{∞}, is an apeirohedron in E3whose faces are helical apeirogons (over squares), rising as “spirals”

above the squares of {4,4}such that 4 meet at each vertex; the orthogonal projections of {4,4}#{∞} onto

their component subspaces recover the original components, that is, the square tessellation and the linear

apeirogon. Each blended apeirohedron represents an entire family of apeirohedra of the same kind, where the

apeirohedra in a family are determined by a parameter describing the relative scale of the two component

ﬁgures; our count of 12 refers to the 12 kinds rather than the individual apeirohedra.

Finally there are twelve regular apeirohedra that are “irreducible”, or pure (non-blended). These are listed

in Table 2 (see [41, p. 225]). The ﬁrst column gives the mirror vector of an apeirohedron; its components,

in order, are the dimensions of the mirrors of the generating symmetries r0,r1and r2of G(P) (this is the

dimension vector of [41, Ch. 7E]). The last two columns say whether the faces and vertex-ﬁgures are planar,

skew, or helical regular polygons. In the second, third, and fourth columns, the (rotation or full) symmetry

group of the Platonic solid at the top indexing that column is closely related to the special group of each

apeirohedron listed below it; the special group is the quotient of the symmetry group by the translation

subgroup. The three polyhedra in the ﬁrst row are the well-known Petrie-Coxeter polyhedra (see [5]), which

along with those in the third row comprise the pure regular polyhedra with ﬁnite faces. The pure polyhedra

with helical faces are listed in the second and last row. Inﬁnite zigzag polygons do not occur as faces of pure

polyhedra.

The ﬁne Schl¨aﬂi symbol used to designate a polyhedron signiﬁes extra deﬁning relations for the symmetry

group (see [41, Ch. 7E]). For example, the parameters l,mand nin the symbols {p, q}l,{p, q}l,m and {p, q |n}

indicate the relations (r0r1r2)l= 1, (r0(r1r2)2)m= 1, or (r0r1r2r1)n= 1, respectively; together with the

standard Coxeter relations they form a presentation for the symmetry group of the corresponding polyhedron.

Note that l,mand n, respectively, give the lengths of the Petrie polygons (1-zigzags), the 2-zigzags (paths

traversing edges where the new edge is chosen to be the second on the right, but reversing orientation on

each step, according to some local orientation on the underlying surface), and the holes (paths traversing

edges where the new edge is chosen to be the second on the right on the surface).

4.2 Geometric Wythoﬃans of regular polyhedra

Let Pbe a geometrically regular polyhedron in E3with symmetry group G(P) = hr0, r1, r2i, and let I⊆

{0,1,2}. If we write Rifor the mirror of a distinguished generator riin E3, and Xfor the complement in E3

12 Schulte and Williams

mirror vector {3,3} {3,4} {4,3}faces vertex-ﬁgures

(2,1,2) {6,6|3} {6,4|4} {4,6|4}planar skew

(1,1,2) {∞,6}4,4{∞,4}6,4{∞,6}6,3helical skew

(1,2,1) {6,6}4{6,4}6{4,6}6skew planar

(1,1,1) {∞,3}(a){∞,4}·,∗3{∞,3}(b)helical planar

Table 2 The pure apeirohedra in E3

of a subset Xof E3, then the weaker form (8) of the initial placement condition in (7) for the initial vertex

vis equivalent to requiring that vlies in

MI:= \

i∈I

Ri∩\

i /∈I

Ri.(9)

Thus, if c

MIdenotes the set of permissible choices of initial vertices v, then c

MIis a subset of MI. In general

we would expect the complement of c

MIin MIto be “small”. If I6={0,1,2}the aﬃne hull of MIis a proper

aﬃne subspace of E3given by Ti/∈IRi; in this case, MImust lie in a plane. If I={0,1,2}the aﬃne hull of

MIis E3.

Now suppose the initial vertex vis chosen in c

MI(and also lies in a speciﬁed fundamental region of G(P)

in E3). To construct the geometric Wythoﬃans of Pwe follow the same pattern as in (3), (4), (5) and (6).

We often write PI(v) in place of PIin order to emphasize the fact that PIis generated from v. The base

0-face of PI(v) is again given by

F0:= v. (10)

For each i∈Ithere is a base 1-face,

Fi

1:= (v, ri(v)),(11)

which is a line segment; and for each {i, j} ∈ I2there is a base 2-face,

Fij

2:= Fji

2:= {r(Fk

1)|r∈ hri, rji, k ∈I∩ {i, j}},(12)

which forms a ﬁnite or inﬁnite polygon according as hri, rjiis a ﬁnite or inﬁnite dihedral group. The full

geometric Wythoﬃan PI(v) then is the union (taken over l= 0,1,2) of the orbits of the base l-faces under

G(P).

We often write Pi(v), Pij (v), Pijk (v) in place of P{i}(v), P{i,j}(v) or P{i,j,k }(v), respectively, and

similarly without vas qualiﬁcation.

Observe that our construction of geometric Wythoﬃans always uses a geometrically regular polyhedron

in E3as input and then produces from it a realization of its abstract Wythoﬃan. Thus the pair of abstract

polyhedra (P,PI) is simultaneously realized in E3as a pair of geometric polyhedra (P, P I). The following

lemma shows that the geometric Wythoﬃans are indeed faithful realizations of the abstract Wythoﬃans.

Lemma 1 Let Pbe a geometrically regular polyhedron in E3, and let I⊆ {0,1,2}. Then for each v∈c

MI

the geometric Wythoﬃan PI(v)is a faithful realization of the abstract Wythoﬃan PI.

Proof The initial placement condition in (7) for vimplies that there is a one-to-one correspondence between

the vertices of PIand PI(v). In fact, by construction, the vertices of PIand PI(v), respectively, are in

one-to-one correspondence with the left cosets of the stabilizers of the initial vertices in Γ(P) or G(P),

which are given by hρi|i /∈Iiand Gv(P). But the group isomorphism between Γ(P) and G(P) naturally

takes the vertex stabilizer hρi|i /∈Iito hri|i /∈Ii, and by (7) the latter subgroup of G(P) coincides with

Gv(P). Thus there is a bijection between the two vertex sets, and the number of vertices is the index of

hri|i /∈Iiin G(P).

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 13

As for both the abstract and geometric Wythoﬃan the base edges and base faces are entirely determined

by their vertices, and the overall construction method for the two polyhedra is the same, the one-to-one

correspondence between the two vertex sets extends to an isomorphism between the two polyhedra. 2

By construction, the Wythoﬃan PI(v) inherits all geometric (and combinatorial) symmetries of Pand

is (trivially) vertex-transitive under G(P). Thus all vertices are surrounded alike (and in particular in the

same way as v). Following standard notation for classical Archimedean solids and tilings we will introduce

a vertex symbol for PI(v) that describes the neighborhood of a vertex and hence collects important local

data.

Let ube a vertex of PI(v) of valency k, let G1, . . . , Gk(in cyclic order) be the 2-faces containing u, and

let Gjbe a qj-gon for j= 1, . . . , k (with qj=∞if Gjis an apeirogon). Then we call (G1, G2, G3, . . . , Gk)

and (q1.q2.q3. . . qk) the vertex conﬁguration and vertex symbol of PI(v) at u, respectively. The vertex

conﬁguration and vertex symbol at a vertex are determined up to cyclic permutation and reversal of order.

By the vertex-transitivity, the vertex symbols of PI(v) at diﬀerent vertices are the same and so we can

safely call the common symbol the vertex symbol of PI(v) (or PI). If a vertex symbol contains a string of

midentical entries, q, we simply shorten the string to qm.

As we will see there are several instances where certain abstract Wythoﬃans of geometrically regular

polyhedra cannot be realized as geometric Wythoﬃans. This is already true for the geometrically regular

polyhedra themselves. Not every geometrically regular polyhedron has a geometrically regular polyhedron

as a dual. There are several possible obstructions to this. If the original polyhedron has inﬁnite faces, then

the dual would have to have vertices of inﬁnite valency, which is forbidden by our discreteness assumption.

Thus local ﬁniteness is a necessary condition for pairs of geometric duals to exist. However, local ﬁniteness

is not a suﬃcient condition. For example, the (abstract) dual of the Petrie dual of the cube, {3,6}4, cannot

be realized as a geometric polyhedron in E3while the Petrie dual of the cube itself, {6,3}4, is one of the

ﬁnite regular polyhedra in E3. The abstract polyhedron {3,6}4is a triangulation of the torus; since its edges

have multiplicity 2, they cannot be geometrically represented by straight line segments in E3. Geometric

polyhedra must necessarily have a simple edge graph.

In practice we often employ a padded vertex symbol to describe the ﬁner geometry of the vertex-

conﬁguration. We use symbols like pc,ps,∞k, or t∞2, respectively, to indicate that the faces are (not

necessarily regular) convex p-gons, skew p-gons, helical polygons over k-gons, or truncated planar zigzag.

(The k= 2 describes a planar zigzag viewed as a helical polygon over a “2-gon”, where here a 2-gon is a

line segment traversed in both directions. A truncated planar zigzag is obtained by cutting oﬀ the vertices

of a planar zigzag, while maintaining segments of the old edges as new edges.) There are other shorthands

that we introduce when they occur. For example, a symbol like (82

s.6c.32.6s) would say that each vertex is

surrounded (in cyclic order) by a skew octagon, another skew octagon, a convex hexagon, a triangle, another

triangle, and a skew hexagon.

We should point out that there are uniform skeletal polyhedra that cannot occur as geometric Wythoﬃans

of regular polyhedra in E3. The simplest example is the snub cube, which is an Archimedean solid whose

symmetry group is the octahedral rotation group and hence does not contain plane reﬂections. On the

other hand, all 18 ﬁnite regular polyhedra and thus their geometric Wythoﬃans have reﬂection groups as

symmetry groups. Note that the snub cube can be derived by Wythoﬀ’s construction from the octahedral

rotation group rather than the full octahedral group.

The geometrically chiral polyhedra in E3are also examples of uniform skeletal polyhedra that cannot arise

as geometric Wythoﬃans of regular polyhedra (see [50, 51]). This immediately follows from a comparison of

the structure of the faces and the valencies of the vertices for the Wythoﬃans and chiral polyhedra, except

possibly when I={0}or {2}. In these two cases the Wythoﬃans are regular and thus cannot coincide with

a chiral polyhedron.

We have not yet fully explored the “snub-type” polyhedra that arise from regular skeletal polyhedra

Pvia Wythoﬀ’s construction applied to the “rotation subgroup” G+(P) of the symmetry group G(P).

14 Schulte and Williams

This subgroup is generated by the symmetries r0r1, r1r2and consists of all symmetries of Pthat realize

combinatorial rotations of P; that is, G+(P) is the image of the combinatorial rotation subgroup Γ+(P) :=

hρ0ρ1, ρ1ρ2iof Γ(P) = hρ0, ρ1, ρ2iunder the representation Γ(P)7→ G(P) in E3. Note that r0r1and r1r2

may not actually be proper isometries and hence G+(P) may not only consist of proper isometries.

A similar remark also applies to possible geometric “snub-type” Wythoﬃans of the chiral polyhedra

in E3.

5 The Wythoﬃans of various regular polyhedra

In this section we treat the Wythoﬃans of a number of distinguished classes of regular polyhedra in E3,

including in particular the four ﬁnite polyhedra with octahedral symmetry and various families of apeirohedra

(the two planar and the two blended apeirohedra derived from the square tiling, as well as the three Petrie-

Coxeter polyhedra). As the fundamental regions of the symmetry groups vary greatly between the various

kinds of polyhedra, we address the possible choices of initial vertices in the subsections. The geometric shape

of a geometric Wythoﬃan will greatly depend on the choice of initial vertex, and diﬀerent choices may lead

to geometric Wythoﬃans in which corresponding faces look quite diﬀerently and may be planar versus skew.

Figures of distinguishing features of the resulting Wythoﬃans are included; the pictures show the base faces

in diﬀerent colors.

We leave the analysis of the Wythoﬃans for the remaining classes of regular polyhedra to the subsequent

paper [58] by the second author.

5.1 Finite polyhedra with octahedral symmetry

There are four regular polyhedra in E3with an octahedral symmetry group: the octahedron {3,4}and cube

{4,3}, and their Petrie-duals {6,4}3and {6,3}4, respectively. The octahedron and cube produce familiar

ﬁgures as Wythoﬃans each related to an Archimedean solid (see [54]), but already their Petrie duals produce

interesting new structures. The sets of distinguished generators for the four individual symmetry groups can

all be expressed in terms of the set for the octahedron {3,4}. We write G({3,4}) = hs0, s1, s2i, where s0, s1, s2

are the distinguished generators. All of the initial vertices used for Wythoﬃans with octahedral symmetry

are chosen from within the standard fundamental region of the octahedral group, which is a closed simplicial

cone bounded by the reﬂection planes of s0,s1and s2. (Recall our previous remark about the notion of

fundamental simplex, or in this case, fundamental simplicial cone.) Each of the Wythoﬃans in this section is

related to an Archimedean solid, as the ﬁgures will show. In fact, for the Wythoﬃans of the convex regular

polyhedra we can choose the initial vertex so that the resulting polyhedron is uniform. That is not the case

with all skeletal regular polyhedra. For example, the Wythoﬃans P02 and P012 derived from {6,4}3cannot

be uniform, though we can still see a relationship between them and the Archimedean solids.

The Wythoﬃans of the octahedron are shown in Figure 1. The ﬁrst Wythoﬃan, P0, is the regular

octahedron {3,4}itself. The Wythoﬃan P1is a uniform cuboctahedron. Examining P2we get the dual to

the octahedron, the regular cube. The Wythoﬃan P01 is a polyhedron which is isomorphic to the truncated

octahedron. For a particular choice of initial vertex P01 is the uniform truncated octahedron. The polyhedron

P02 is isomorphic to the rhombicuboctahedron, and for a carefully chosen initial vertex P02 is the uniform

rhombicuboctahedron. For P12 Wythoﬀ’s construction yields a polyhedron isomorphic to the truncated cube

which for a speciﬁcally chosen initial vertex is the uniform truncated cube. The Wythoﬃan P012 is isomorphic

to the truncated cuboctahedron, and for a certain initial vertex is the uniform truncated cuboctahedron.

For the Wythoﬃans of the cube {4,3}we can exploit the duality between the cube and the octahedron

(using the generators s2, s1, s0for {4,3}). In fact, interchanging 0 and 2 in the superscripts from the Wythof-

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 15

P0P1P2P01

P02 P12 P012

Fig. 1 The Wythoﬃans derived from {3,4}.

ﬁans of the octahedron (of Figure 1) results in the Wythoﬃans of the cube, and vice versa. We will not

reproduce the results for the cube in detail.

The Petrie dual {6,4}3of {3,4}has a group of the form G({6,4}3) = hr0, r1, r2i, where r0=s0s2,

r1:= s1,r2:= s2and s0, s1, s2are as above. Given these generators, we are limited in our choice of initial

vertex. As the rotation axis of the halfturn r0lies in the reﬂection plane of r2, any point invariant under r0

is also invariant under r2. Thus there is no point that is invariant under only r0or under both r0and r1

and not r2. As such there is no polyhedron P2nor a polyhedron P12 . For pictures of the Wythoﬃans, see

Figure 2.

The ﬁrst Wythoﬃan, P0, is the regular polyhedron {6,4}3itself which has four regular, skew hexagonal

faces which all meet at each vertex; the vertex symbol is (64

s). The vertex ﬁgure is then a convex square, as

for the octahedron with which P0shares an edge graph.

The Wythoﬃan P1shares its edge graph with the cuboctahedron. The faces are four convex, regular

hexagons of type F{0,1}

2(the equatorial hexagons of the cuboctahedron) and six convex squares of type

F{1,2}

2. The vertex symbol is (4c.6c.4c.6c). The hexagons all intersect leading to a vertex ﬁgure which is a

crossed quadrilateral (like a bowtie). This is a uniform polyhedron with planar faces, in the notation of [11]

it is 4

34|3.

The polyhedron P01 shares an edge graph with a polyhedron which is isomorphic to a truncated octa-

hedron. It has four skew dodecagons of type F{0,1}

2(truncations of skew hexagonal faces of {6,4}3) and six

convex squares of type F{1,2}

2. The vertex symbol is (4c.122

s) with an isosceles triangle as the vertex ﬁgure.

The Wythoﬃan P02 shares a vertex set with a polyhedron which is isomorphic to a rhombicuboctahedron.

There are four skew hexagons of type F{0,1}

2, six convex squares of type F{1,2}

2, and twelve crossed quadri-

laterals of type F{0,2}

2. At each vertex a crossed quadrilateral, a square, a crossed quadrilateral, and a skew

hexagon occur in cyclic order yielding a convex quadrilateral vertex ﬁgure with vertex symbol (4 .4c.4.6s),

where 4 indicates a crossed quadrilateral.

For P012 the resulting polyhedron shares a vertex set with a polyhedron which is isomorphic to the trun-

cated cuboctahedron. The ﬁgure has four skew dodecagonal faces of type F{0,1}

2(truncated skew hexagons),

six convex octagons of type F{1,2}

2(truncated squares), and twelve crossed quadrilaterals of type F{0,2}

2. The

vertex symbol is (4 .8c.12s) with a triangular vertex ﬁgure.

16 Schulte and Williams

P0P1P01 P02 P012

Fig. 2 The Wythoﬃans derived from {6,4}3.

The ﬁnal geometrically regular polyhedron with octahedral symmetry is the Petrie-dual of the cube,

{6,3}4. Its symmetry group is G({6,3}4) = hr0, r1, r2i, where r0=s2s0,r1:= s1,r2:= s0and s0, s1, s2

are as above. The duality between the octahedron and the cube can again be seen here. The generators

s2s0, s1,and s0of G({6,3}4) are obtained from the generators of G({6,4}3) by interchanging s0and s2.

The Wythoﬃans of {6,3}4also share many similarities with the Wythoﬃans of {6,4}3. As with {6,4}3,

every vertex which is stabilized by r0is also stabilized by r2. Thus there is no point which is stabilized by r0

alone, nor is there one which is stabilized by both r0and r1. Consequently, there is no polyhedron P2and

no polyhedron P12 . For pictures of the Wythoﬃans, see Figure 3.

The ﬁrst Wythoﬃan, P0, is the regular polyhedron {6,3}4itself. It shares its edge graph with the cube

and thus has eight vertices and twelve edges. The four faces are the Petrie polygons of the cube which are

regular, skew hexagons. Three faces meet at each vertex, with a vertex symbol (63

s) and a regular triangle

as the vertex ﬁgure.

The Wythoﬃan P1has the same edge graph as the cuboctahedron. There are four intersecting, regular,

convex hexagons of type F{0,1}

2(the equatorial hexagons of the cuboctahedron) and eight regular triangles

of type F{1,2}

2. The vertex symbol is (3.6c.3.6c) with a vertex ﬁgure of a crossed quadrilateral. This is a

uniform polyhedron with planar faces, in the notation of [11] it is 3

23|3.

When the initial vertex is stabilized by r2alone then the resulting polyhedron, P01, shares its edge graph

with a polyhedron which is isomorphic to the truncated cube. Then there are four skew dodecagons of type

F{0,1}

2(truncations of the skew hexagonal faces of {6,3}4) and eight regular triangles of type F{1,2}

2. The

vertex symbol is (3.122

s) and the polyhedron has an isosceles triangle as a vertex ﬁgure.

The Wythoﬃan P02 shares its vertex set with a polyhedron which is isomorphic to a rhombicuboctahe-

dron. The faces are four regular hexagons (convex or skew depending on the exact choice of initial vertex)

of type F{0,1}

2, eight regular triangles of type F{1,2}

2, and twelve crossed quadrilaterals of type F{0,2}

2. The

vertex symbol is (3.4.6.4 ) and the vertex ﬁgure is a convex trapezoid.

For P012 the resulting polyhedron shares its vertex set with a polyhedron which is isomorphic to the

truncated cuboctahedron. It has four dodecagons (which may be skew or convex depending on the choice

of initial vertex) of type F{0,1}

2(truncated hexagons), eight convex hexagons of type F{1,2}

2(truncated

triangles), and twelve crossed quadrilaterals of type F{0,2}

2. The vertex ﬁgure is a triangle and the vertex

symbol is (4 .6c.12).

5.2 Planar polyhedra derived from the square tiling

The square tiling of the plane is the (self-dual) regular geometric apeirohedron {4,4}, with symmetry group

G({4,4}) = hs0, s1, s2i. The second regular apeirohedron we investigate is its Petrie dual, {∞,4}4. All initial

vertices for Wythoﬃans of these two polyhedra are chosen from the fundamental triangle of {4,4}. Pictures

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 17

P0P1P01 P02 P012

Fig. 3 The Wythoﬃans derived from {6,3}4.

of the Wythoﬃans are in Figures 4 and 5, with base faces indicated in color. The Wythoﬃans for {4,4}are

well-known but those for {∞,4}4certainly have not received much attention (however, see [29, Sect. 12.3]).

Beginning with the Wythoﬃans of P={4,4}we ﬁrst note that P0is the regular apeirohedron {4,4}

itself. All faces are convex squares of type F{0,1}

2. Four squares meet at each vertex, giving a vertex symbol

(44

c) and a convex square vertex ﬁgure. By the self-duality of {4,4}this is also the Wythoﬃan P2(which is

P0for the dual of {4,4}).

In P1the apeirohedron has two types of face: convex squares of type F{0,1}

2and congruent convex squares

of type F{1,2}

2. The vertex ﬁgures are convex squares since the vertex symbol is (44

c). This is again a regular

apeirohedron, a similar copy of the original square tessellation.

The apeirohedron P01 has two distinct types of 2-faces. The ﬁrst type of base face is a convex octagon

of type F{0,1}

2(truncated square), and the second type is a convex square of type F{1,2}

2. Two octagons

and one square meet at each vertex yielding an isosceles triangle for a vertex ﬁgure with vertex symbol

(4c.82

c). The initial vertex can be chosen so that the octagons are regular in which case the Wythoﬃan is a

uniform apeirohedron, the Archimedean tessellation (4.8.8). Again by the self-duality of {4,4}this is also

the Wythoﬃan P1,2(which is P0,1for the dual of {4,4}).

The apeirohedron P02 has three diﬀerent types of 2-faces. The ﬁrst is a convex square face of type F{0,1}

2,

the second is a convex square of type F{1,2}

2, and the ﬁnal type of face is a convex rectangle of type F{0,2}

2.

At each vertex there is a square of the ﬁrst kind, a rectangle, a square of the second kind, and a rectangle,

giving a vertex symbol (4c.4c.4c.4c). The resulting vertex ﬁgure is convex quadrilateral. When the initial

vertex is chosen so that the base edges have the same length, the rectangles are squares and the Wythoﬃan

is a congruent copy of the original tessellation.

For P012, the apeirohedron has two diﬀerent octagonal faces and a rectangular face. The ﬁrst type of

convex octagons are of type F{0,1}

2(truncated squares), the second type of convex octagons are of type

F{1,2}

2(truncated squares), and the convex rectangles are of type F{0,2}

2. One octagon of each type and a

rectangle come together at each vertex to make a triangular vertex ﬁgure with vertex symbol (4c.82

c). When

the initial vertex is chosen to make the base edges have equal length then the faces are regular polygons and

the Wythoﬃan is again the Archimedean tessellation (4.8.8) with an isosceles triangle as the vertex ﬁgure.

The symmetry group of the regular apeirohedron {∞,4}4is given by G({∞,4}4) = hr0, r1, r2i, where

r0=s0s2,r1:= s1,r2:= s2and s0, s1, s2are the generators of G({4,4}). Since the center of the point

reﬂection r0lies on the reﬂection line of r2, every point held invariant by r0is also invariant under r2so

there is no polyhedron P2or P12 in this case. For pictures of the Wythoﬃans of {∞,4}4see Figure 5.

The Wythoﬃan P0is the regular apeirohedron {∞,4}4itself. Its 2-faces are apeirogons which appear as

inﬁnite zigzags whose consecutive edges meet at an angle of π

2. Four apeirogons meet at each vertex, giving

a square vertex ﬁgure with vertex symbol (∞4

2).

18 Schulte and Williams

P0P1P2P01

P02 P12 P012

Fig. 4 The Wythoﬃans derived from {4,4}.

The apeirohedron P1only has two types of base faces. They are linear apeirogons of type F{0,1}

2and

convex squares of type F{1,2}

2. About each vertex there is an apeirogon, a square, an apeirogon, and a square,

with vertex symbol (∞.4c.∞.4c). The apeirogons dissect the plane into squares, exactly half of which are

the square faces of type F{1,2}

2. The vertex ﬁgure is a crossed quadrilateral. All faces of this Wythoﬃan are

regular polygons so this shape is a uniform apeirohedron with squares and linear apeirogons as faces.

The apeirohedron P01 has ﬁnite and inﬁnite faces. The apeirogonal faces are of type F{0,1}

2, each of which

is a truncated zigzag. The ﬁnite faces of this apeirohedron are convex squares of type F{1,2}

2. The vertex sym-

bol is (4c.t∞2.t∞2) and the resulting vertex ﬁgure is an isoceles triangle (recall that tindicates truncation).

The truncated zigzags are not regular apeirogons so this Wythoﬃan is not a uniform apeirohedron.

The Wythoﬃan P02 is an apeirohedron whose faces are regular zigzags of type F{0,1}

2where the angle

between consecutive edges is greater than π

2, convex squares of type F{1,2}

2, and crossed quadrilaterals of

type F{0,2}

2. The vertex ﬁgure is a convex quadrilateral with vertex symbol (4 .4c.4.∞2). The crossed

quadrilaterals are not regular so this is not a uniform apeirohedron.

The ﬁnal Wythoﬃan is P012. There are apeirogonal faces of type F{0,1}

2which are truncated zigzags.

There are also convex octagonal faces of type F{1,2}

2(truncated squares) and crossed quadrilaterals of type

F{0,2}

2. There is one apeirogon, one octagon, and one quadrilateral at each vertex yielding a triangular

vertex ﬁgure with vertex symbol (4 .8c.t∞2). The truncated zigzags and crossed quadrilaterals are not

regular polygons so the polyhedron is not uniform.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 19

P0P1P01

P02 P012

Fig. 5 The Wythoﬃans derived from {∞,4}4.

5.3 Blended polyhedra derived from the square tiling

Next we investigate the Wythoﬃans of the regular polyhedra {4,4}#{ } and {4,4}#{∞}, the blends of the

square tiling {4,4}with a line segment { } or linear apeirogon {∞}, respectively, as well as their Petrie

duals {∞,4}4#{ } and {∞,4}4#{∞}. Suppose the symmetry groups of {4,4},{ } and {∞} are given by

G({4,4}) = hs0, s1, s2i,G({ }) = ht0iand G({∞}) = ht0, t1i, each with all generators viewed as plane

reﬂections in E3. Note that the reﬂection planes for s0,s1,s2are perpendicular to the reﬂection planes for

t0or t0, t1(which are parallel to one another), respectively.

In general the projection of the Wythoﬃans in this section onto the reﬂection plane of t0is congruent

to a Wythoﬃan of {4,4}or {∞,4}4. In some instances if the initial vertex is chosen from the boundary of

the fundamental region, the projection of the Wythoﬃan of the blended polyhedron will no longer appear

as a Wythoﬃan of {4,4}or {∞,4}4. Speciﬁcally, the Wythoﬃans P01,P02, and P012 of {4,4}#{ } and

{∞,4}4#{ } will not project onto the reﬂection plane of t0as the Wythoﬃans of {4,4}and {∞,4}4, respec-

tively, if the initial vertex lies in the reﬂection plane of s0. For {4,4}#{∞} and {∞,4}4#{∞}, if the initial

vertex lies in the reﬂection plane of s0then P01,P02, and P012 will not pro ject onto Wythoﬃans of {4,4}

and {∞,4}4, respectively. Similarly, for these two blends, if the initial vertex lies in the reﬂection plane of s1

then P01,P12 , and P012 will not project onto Wythoﬃans of {4,4}and {∞,4}4, respectively. In all other

cases discussed below the Wythoﬃans project onto Wythoﬃans of {4,4}or {∞,4}4.

The ﬁrst apeirohedron we examine is {4,4}#{ }, which is isomorphic to {4,4}and combinatorially self-

dual. Its symmetry group is G({4,4}#{ }) = hr0, r1, r2iwith r0:= s0t0,r1:= s1and r2:= s2. Here, the

generator r0is a half-turn and the generators r1and r2are plane reﬂections. Note that a generic apeirohedron

{4,4}#{ } is not geometrically self-dual; in fact, reversing the order of the generators of the group and running

Wythoﬀ’s construction does not generally produce an apeirohedron similar to the original one.

20 Schulte and Williams

Some care will have to be taken in our choice of initial vertex to ensure an interesting Wythoﬃan. If

a point, v, is invariant under t0then the Wythoﬃan of {4,4}#{ } with initial vertex vis the same as the

(planar) Wythoﬃan of {4,4}with initial vertex v. For the following Wythoﬃans assume that none of the

initial vertex choices are invariant under t0, and consequently we will not look at any initial vertices which

are invariant under r0. This excludes P1,P2, and P12 as geometric Wythoﬃans. (Note, however, that by

the combinatorial self-duality of {4,4}#{ } there are abstract Wythoﬃans of these types isomorphic to P1,

P0, and P01, respectively.) All initial vertices are chosen from the fundamental region corresponding to

{4,4}#{ } which is a one-sided inﬁnite cylinder over a triangle formed as the union of a pair of 0-adjacent

triangles in the barycentric subdivision of {4,4}. For pictures of the Wythoﬃans, see Figure 6.

The ﬁrst Wythoﬃan, P0, is {4,4}#{ } itself. Its 2-faces are all skew squares, {4}#{ }, of type F{0,1}

2.

Four faces meet at each vertex, yielding a vertex symbol (44

s) and a convex square as the vertex ﬁgure. The

projection of this Wythoﬃan, that is, of {4,4}#{ }, onto the reﬂection plane of t0appears as {4,4}.

In the next apeirohedron, P01 , the faces of type F{0,1}

2are skew octagons (truncated skew squares) and

the faces of type F{1,2}

2are convex squares. Two octagons and one convex square meet at each vertex giving

an isosceles triangle as a vertex ﬁgure with vertex symbol (4.82

s). The truncated skew squares are not regular

so this is not a uniform apeirohedron. The projection of this Wythoﬃan onto the reﬂection plane of t0

appears as the Wythoﬃan P01 of {4,4}.

In the apeirohedron P02 , the faces of type F{0,1}

2are skew squares, the faces of type F{1,2}

2are convex

squares, and the faces of type F{0,2}

2are convex rectangles. Cyclically, about each vertex, there is a skew

square, a rectangle, a square, and a rectangle, giving the vertex symbol (4s.4c.4c.4c). The resulting vertex

ﬁgure is a convex quadrilateral. For a speciﬁcally chosen initial vertex the faces of type F{0,2}

2are squares

and the Wythoﬃan is a uniform apeirohedron with one kind of planar square and one kind of non-planar

square. The projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan P02 of

{4,4}.

For the Wythoﬃan P012 the faces of type F{0,1}

2are skew octagons (truncated skew squares), the faces of

type F{1,2}

2are convex octagons (truncated squares), and the faces of type F{0,2}

2are convex rectangles. At

each vertex there is one face of each type, yielding a vertex symbol (4c.8s.8c) and a triangular vertex ﬁgure.

The truncated squares are not regular so the Wythoﬃan is not a uniform apeirohedron. The projection of

this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan P012 of {4,4}.

P0P01 P02 P012

Fig. 6 The Wythoﬃans derived from {4,4}#{ }.

The next regular apeirohedron we examine is {∞,4}4#{ }, the Petrie-dual of {4,4}#{ }, which is iso-

morphic to {∞,4}4. The symmetry group is G({∞,4}4#{ }) = hr0, r1, r2iwith r0:= s0t0s2,r1:= s1,

r2:= s2, and s0, s1, s2, t0as above. Here r0is a point reﬂection (through the midpoint of the base edge of

the underlying plane tessellation {4,4}) and r1and r2are plane reﬂections. Individually s0, s1, s2, and t0

are plane reﬂections in E3.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 21

The initial vertices we use come from the same fundamental region as for {4,4}#{ }. As with {4,4}#{ },

any initial vertex left invariant by t0will result in the Wythoﬃan being the same as the corresponding

Wythoﬃan derived from the planar {∞,4}4. Assume all choices of initial vertex are transient under t0, and

consequently we will not look at any initial vertices which are invariant under r0. This excludes P1,P2, and

P12. For pictures of the Wythoﬃans, see Figure 7.

The ﬁrst Wythoﬃan, P0, is the regular apeirohedron {∞,4}4#{ } itself whose faces are regular zigzag

apeirogons, {∞}#{ }, such that each edge is bisected by the reﬂection plane of t0. Four of these apeirogons

meet at each vertex resulting in a convex, square vertex ﬁgure with vertex symbol (∞4

2). The projection of

this Wythoﬃan onto the reﬂection plane of t0appears as {∞,4}4.

In the apeirohedron P01 , the faces of type F{0,1}

2are apeirogons which appear as truncations of the faces

of {∞,4}4#{ }, while the faces of type F{1,2}

2are convex squares which lie parallel to the reﬂection plane of

t0. Two apeirogons and one square meet at each vertex, yielding the vertex symbol (4.t∞2.t∞2). The vertex

ﬁgure is an isosceles triangle. The truncated zigzags are not regular so this Wythoﬃan is not a uniform

apeirohedron. The projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan

P01 of {∞,4}4.

With the apeirohedron P02 the faces of type F{0,1}

2are regular zigzag apeirogons which are bisected by

the reﬂection plane of t0, the faces of type F{1,2}

2are convex squares parallel to the reﬂection plane of t0, and

the faces of type F{0,2}

2are planar crossed quadrilaterals which intersect the reﬂection plane of t0. Cyclically

at each vertex there is an apeirogon, a crossed quadrilateral, a square, and a crossed quadrilateral, resulting

in the vertex-symbol (4 .4c.4 .∞2). The vertex ﬁgure is a convex quadrilateral. The crossed quadrilaterals

are not regular so this apeirohedron is not uniform. The projection of this Wythoﬃan onto the reﬂection

plane of t0appears as the Wythoﬃan P02 of {∞,4}4.

For the ﬁnal apeirohedron, P012 , the faces of type F{0,1}

2are truncated zigzag apeirogons. The faces of

type F{1,2}

2are convex octagons (truncated squares) which lie parallel to the reﬂection plane of t0, and the

faces of type F{0,2}

2are crossed quadrilaterals which intersect the reﬂection plane of t0at their centers. There

is one face of each type at each vertex, so that the vertex symbol is (4 .8c.t∞2) and the vertex ﬁgure is a

triangle. The truncated zigzags and crossed quadrilaterals are not regular so this Wythoﬃan is not a uniform

apeirohedron. The projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan

P012 of {∞,4}4.

P0P01 P02 P012

Fig. 7 The Wythoﬃans derived from {∞,4}4#{ }.

Now we consider the blended regular apeirohedron {4,4}#{∞} with symmetry group G({4,4}#{∞}) =

hr0, r1, r2i, where r0:= s0t0,r1:= s1t1,r2:= s2, and s0, s1, s2, t0, t1are as above. Here r0and r1are

half-turns and r2is a plane reﬂection.

In E3the reﬂection planes corresponding to s0,s1, and s2are orthogonal to the reﬂection planes corre-

sponding to t0and t1which are parallel to one another. There is no point which is invariant under t0and

22 Schulte and Williams

t1so this will limit the choice of initial vertex and consequently we will not look at P2. In all cases, any

edge of type F{2}

1lies parallel to the reﬂection planes of t0and t1. All initial vertices are chosen from the

fundamental region of {4,4}#{∞}. This fundamental region is a right prism over the fundamental triangle

of {4,4}. For pictures of the Wythoﬃans see Figure 8.

The ﬁrst Wythoﬃan, P0, is the regular apeirohedron {4,4}#{∞} itself. Its faces are helical apeirogons

spiraling around a cylinder with a square base. Each edge is incident to two helices which spiral upward

in opposite orientations. Four helices meet at each vertex resulting in an antiprismatic square vertex ﬁgure

with vertex symbol (∞4

4). The projection of this Wythoﬃan onto the reﬂection plane of t0appears as {4,4}.

In the apeirohedron P1the faces of type F{0,1}

2are regular helices over square bases while the faces of

type F{1,2}

2are antiprismatic squares. At each vertex, in alternating order, there are two helices and two

antiprismatic squares, yielding a convex rectangle as a vertex ﬁgure with a vertex symbol (4s.∞4.4s.∞4). The

faces are all regular polygons so the Wythoﬃan is a uniform apeirohedron. The projection of this Wythoﬃan

onto the reﬂection plane of t0appears as the Wythoﬃan P1of {4,4}.

For P01 the faces of type F{0,1}

2are helices over an octagon (for initial vertices which lie on a base edge

of {4,4}#{∞} these helices are truncations of the helical faces of {4,4}#{∞}). The faces of type F{1,2}

2are

regular squares, for some initial vertices they are skew and for some initial vertices they are convex. There

are two helices and one quadrilateral at each vertex resulting in an isosceles triangle vertex ﬁgure with vertex

symbol (4.∞2

8). For a carefully chosen initial vertex the helices are regular helices about octagonal bases and

the Wythoﬃan is uniform. The projection of this Wythoﬃan onto the reﬂection plane of t0appears as the

Wythoﬃan P01 of {4,4}.

In the apeirohedron P02 the faces of type F{0,1}

2are regular helices over squares, the faces of type

F{1,2}

2are convex squares lying parallel to the plane of t0, and the faces of type F{0,2}

2are convex rectangles

which are not parallel to this plane. Cyclically, about each vertex there is a square, a rectangle, a helix, and a

rectangle, with vertex symbol (4c.4c.∞4.4c). The resulting vertex ﬁgure is a skew quadrilateral. For a carefully

chosen initial vertex the faces of type F{0,2}

2are regular and the Wythoﬃan is a uniform apeirohedron. The

projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan P02 of {4,4}.

For the Wythoﬃan P12 the faces of the apeirohedron of type F{0,1}

2are regular helices over squares and

the faces of type F{1,2}

2are skew octagons (truncated antiprismatic squares). There are two octagons and

one helix meeting at each vertex, yielding (82

s.∞4) as the vertex symbol and an isosceles triangle as the

vertex ﬁgure. The truncated antiprismatic squares are not regular so P12 is not a uniform apeirohedron. The

projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan P12 of {4,4}.

In the apeirohedron P012 the faces of type F{0,1}

2are apeirogons and appear as helices over octagons

(truncated helices over squares). The ﬁnite faces are skew octagons of type F{1,2}

2(truncated antiprismatic

squares) and convex rectangles of type F{0,2}

2. One face of each type meets at each vertex yielding a triangular

vertex ﬁgure with vertex symbol (4c.8s.∞8). The truncated antiprismatic squares are not regular so this

Wythoﬃan is not a uniform apeirohedron. The projection of this Wythoﬃan onto the reﬂection plane of t0

appears as the Wythoﬃan P012 of {4,4}.

The last geometrically regular blended apeirohedron based on the square tessellation of the plane is

{∞,4}4#{∞}. Letting s0, s1, s2and t0, t1be as above, the symmetry group is given by G({∞,4}4#{∞}) =

hr0, r1, r2iwith r0:= s0s2t0,r1:= s1t1and r2:= s2. Here r0is a point reﬂection, r1is a half-turn, and r2

is a plane reﬂection. Again there is no point which is invariant under t0and t1so this will limit the choice

of initial vertex and prevent there being a P2. Additionally, any point which is invariant under r0is also

invariant under r2, so this excludes P2and P12. The initial vertices all come from the fundamental region

of {4,4}#{∞}. We further restrict the choice of initial vertex to lie in a base face of {∞,4}4#{∞} when

applicable (P01 and P012). This will ensure that the faces of the Wythoﬃans will have similar planarity to

the faces of {∞,4}4#{∞}. For pictures of the Wythoﬃans see Figure 9.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 23

P0P1P01

P02 P12 P012

Fig. 8 The Wythoﬃans derived from {4,4}#{∞}.

The ﬁrst Wythoﬃan, P0, is the regular apeirohedron {∞,4}4#{∞} itself. The faces of type F{0,1}

2are

regular zigzag apeirogons. Each apeirogon lies in a plane which crosses through both the reﬂection planes

of t0and t1. When projected onto the plane of t0they appear as planar zigzags. Four zigzags meet at each

vertex, yielding the vertex symbol (∞4

2) and making the vertex ﬁgure an antiprismatic square. The projection

of this Wythoﬃan onto the reﬂection plane of t0appears as {∞,4}4.

In the apeirohedron P1, the faces of type F{0,1}

2are linear apeirogons. Each apeirogon corresponds to

a zigzag of {∞,4}4#{∞} and tessellates the line connecting the midpoints of the edges of the zigzag. The

faces of type F{1,2}

2are antiprismatic squares. There are two squares and two lines alternating about each

vertex, giving the vertex symbol (4s.∞.4s.∞). The vertex ﬁgure is a crossed quadrilateral. This is a uniform

apeirohedron. The projection of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan

P1of {∞,4}4.

For the apeirohedron P01 the faces of type F{0,1}

2are apeirogons. Each one corresponds to a face of P0

such that the orthogonal projection of F{0,1}

2onto the plane of the base face of {∞,4}4#{∞} appears as a

truncated zigzag. When the initial vertex lies in the base face of {∞,4}4#{∞}, the faces of type F{0,1}

2is

a truncation of that base face. The faces of type F{1,2}

2are antiprismatic squares. There are two apeirogons

and one square at each vertex, with vertex symbol (4s.t∞2.t∞2) (where here we use t∞2to indicate the

apeirogon’s relationship with truncated zigzags). The resulting vertex ﬁgure is an isosceles triangle. The

truncated zigzags are not regular polygons so this Wythoﬃan is not a uniform apeirohedron. The projection

of this Wythoﬃan onto the reﬂection plane of t0appears as the Wythoﬃan P01 of {∞,4}4.

24 Schulte and Williams

In the apeirohedron P02 the faces of type F{0,1}

2are regular zigzags, the faces of type F{1,2}

2are convex

squares lying parallel to the plane of t0, and the faces of type F{0,2}

2are crossed quadrilaterals. At each

vertex, in cyclic order, there is a crossed quadrilateral, a square, a crossed quadrilateral, and an apeirogon,

giving (4 .4c.4 .∞2) as vertex symbol. The resulting vertex ﬁgure is a skew quadrilateral. The crossed

quadrilateral faces are not regular so this is not a uniform apeirohedron. The projection of this Wythoﬃan

onto the reﬂection plane of t0appears as the Wythoﬃan P02 of {∞,4}4.

Lastly, we will look at the apeirohedron P012 . Similar to P01 the face F{0,1}

2is an apeirogon which

orthogonally projects as a truncated zigzag onto the plane of the base face of {∞,4}4#{∞}. For some initial

vertices this apeirogon is planar and for other choices it is not. The faces of type F{1,2}

2are skew octagons

(truncated antiprismatic squares) and the faces of type F{0,2}

2are planar crossed quadrilaterals. There is one

face of each type meeting at each vertex yielding a triangular vertex ﬁgure with vertex symbol (4 .8s.t∞2).

None of the faces are regular so P012 is not a uniform apeirohedron. The projection of this Wythoﬃan onto

the reﬂection plane of t0appears as the Wythoﬃan P012 of {∞,4}4.

P0P1P01 P02 P012

Fig. 9 The Wythoﬃans derived from {∞,4}4#{∞}.

5.4 Petrie-Coxeter polyhedra

In this ﬁnal section we examine the Wythoﬃans of the Petrie-Coxeter polyhedra, the three most prominent

examples of pure regular apeirohedra. These have convex faces and skew vertex ﬁgures. The symmetry group

of each of them can be derived from the symmetry group of the cubical honeycomb, {4,3,4}. We take this

symmetry group in the form G({4,3,4}) = ht0, t1, t2, t3i, where t0, t1, t2, t3are the distinguished generators

(as in [41, p. 231]). The fundamental region of G({4,3,4}) in E3is a simplex with vertices at the centers of

the faces in a ﬂag of {4,3,4}, and each generator tjis the reﬂection in the plane bounding the simplex and

opposite to the vertex corresponding to the j-face in the ﬂag.

We begin with the Petrie-Coxeter polyhedron {4,6|4}. From [41, p. 231] we know that G({4,6|4}) =

hr0, r1, r2i, where r0:= t0,r1:= t1t3and r2:= t2. Note that r0and r2are plane reﬂections, and that r1

is a halfturn. For the Wythoﬃans the initial vertices have all been chosen so that they are points of the

fundamental region within the convex hull of the base face of {4,6|4}. This choice leads to the resulting

ﬁgures being more geometrically similar to {4,6|4}. Other points in the fundamental region which belong

to the same Wythoﬃan class generate combinatorially isomorphic ﬁgures, but previously planar faces may

become skew, or vice versa. For instance, in the cases where the initial vertex is transient under r1, choosing

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 25

a point outside of the convex hull of a face of {4,6|4}destroys the planarity of the faces of type F{0,1}

2but

does preserve the isomorphism type of the apeirohedron. For pictures of the Wythoﬃans see Figure 10.

The ﬁrst apeirohedron, P0, is {4,6|4}itself. It has convex square faces, six of which meet at each vertex,

and so the vertex symbol is (46). The vertex ﬁgure is a regular, antiprismatic hexagon.

For P1, the apeirohedral Wythoﬃan has convex square faces of type F{0,1}

2while the faces of type F{1,2}

2

are regular, antiprismatic hexagons. Cyclically, at each vertex, there is a hexagon, a square, a hexagon, and

a square. Thus the vertex symbol is (4c.6s.4c.6s) and the vertex ﬁgure is a rectangle. The faces are all regular

polygons so this is a uniform apeirohedron.

The Wythoﬃan P2is the dual of {4,6|4}, the regular apeirohedron {6,4|4}. Each face is a regular, convex

hexagon of type F{1,2}

2. Four come together at each vertex yielding skew quadrilateral as the vertex ﬁgure

with a vertex symbol (64

c).

For the next apeirohedral Wythoﬃan, P01 , the faces of type F{0,1}

2are convex octagons and the faces of

type F{1,2}

2are regular, antiprismatic hexagons. The vertex symbol is (6s.82

c) and an isosceles triangle is the

vertex ﬁgure. For a speciﬁc choice of initial vertex the octagons are regular and the Wythoﬃan is uniform.

Note that for an initial vertex chosen outside of the convex hull of the base face of {4,6|4}, the octagon

would become a truncated antiprismatic quadrilateral which can not be made regular and so in this case the

Wythoﬃan is not uniform.

In the apeirohedral Wythoﬃan P02 the faces of type F{0,1}

2are convex squares; the faces of type F{1,2}

2

are regular, convex hexagons; and the faces of type F{0,2}

2are convex rectangles. Cyclically, at each vertex,

there is a square, a rectangle, a hexagon, and a second rectangle, giving a vertex symbol (4c.4c.6c.4c). The

vertex ﬁgure is a skew quadrilateral. For certain initial vertex choices the rectangles can be made into squares

making the Wythoﬃan uniform.

In the apeirohedron P12 the faces of type F{0,1}

2are convex squares and the faces of type F{1,2}

2are

skew dodecagons (truncated antiprismatic hexagons). The vertex symbol is (4.122

s) which corresponds to an

isosceles triangle as the vertex ﬁgure. The skew dodecagons cannot be made regular by any vertex choice

and thus this Wythoﬃan is not a uniform apeirohedron for any initial vertex choice.

Finally, consider P012. In this apeirohedron, the faces of type F{0,1}

2are convex octagons, the faces of

type F{1,2}

2are skew dodecagons (truncated antiprismatic hexagons), and the faces of type F{0,2}

2are convex

rectangles. As with P12 the skew dodecagons are never regular so the apeirohedron is not uniform. There is

one face of each type at each vertex, yielding (4c.8c.12s) as a vertex symbol and a triangular vertex ﬁgure.

Next we investigate the Wythoﬃans of the Petrie-Coxeter polyhedron {6,4|4}, the dual of {4,6|4}. As

such, its symmetry group is G({6,4|4}) = hr0, r1, r2i, where r0:= t2,r1:= t1t3and r2:= t0; these are

generators of G({4,6|4}) in reverse order. Note that the duality of {6,4|4}and {4,6|4}is geometric: we can

produce one polyhedron from the other by reversing the order of the generators of its symmetry group and

then applying Wythoﬀ’s construction with the new generators. As with {4,6|4}we will only consider initial

vertices which are contained within the convex hull of the base face and the fundamental region of {6,4|4}.

As before, choosing the vertices in this way makes the faces of the Wythoﬃans more geometrically similar

to the faces of {6,4|4}. Choosing an initial vertex within the base face versus an initial vertex from outside

of the base face (but still within the fundamental region) will only aﬀect the planarity of the faces but not

the combinatorial properties. For pictures of the Wythoﬃans see Figure 11.

Due to the geometric duality between {6,4|4}and {4,6|4}we can interchange 0 and 2 in the superscripts

of the Wythoﬃans of {6,4|4}and get the Wythoﬃans of {4,6|4}, and vice versa. Note, however, that an initial

vertex chosen in the base face of one of {6,4|4}or {4,6|4}will generally not also lie in the base face of the

other. This explains why some of the Wythoﬃans in Figures 10 and 11 that correspond to each other under

the interchange of the subscripts 0 and 2 look quite diﬀerent (although they are isomorphic). For example,

P01 of Figure 10 has convex octagons and skew hexagons as faces, while the corresponding polyhedron P12

26 Schulte and Williams

P0P1P2P01

P02 P12 P012

Fig. 10 The Wythoﬃans derived from {4,6|4}.

of Figure 11 has skew octagons and convex hexagons as faces. The geometry of the Wythoﬃans of {6,4|4}

with initial vertices in the base face of {6,4|4}is as follows.

The initial Wythoﬃan, P0, is the regular apeirohedron {6,4|4}itself whose faces are convex, regular

hexagons. Four such hexagons meet at each vertex yielding a regular, skew quadrilateral as the vertex ﬁgure

with vertex symbol (64

c).

In the apeirohedron P1the faces of type F{0,1}

2are convex hexagons and the faces of type F{1,2}

2are

regular, skew quadrilaterals. The vertex symbol is (4s.6c.4s.6c) so the vertex ﬁgure is a rectangle. This is a

uniform apeirohedron.

The Wythoﬃan P2is {4,6|4}, the dual of {6,4|4}. The faces are convex squares of type F{1,2}

2and there

are six circling each vertex with vertex symbol (46

c). The vertex ﬁgure is a regular, antiprismatic hexagon.

In the apeirohedron P01 the faces of type F{0,1}

2are convex dodecagons (truncated hexagons) and the faces

of type F{1,2}

2are regular, skew quadrilaterals. The vertex symbol is (4s.122

c) yielding an isosceles triangle

as a vertex ﬁgure. For a carefully chosen initial vertex the dodecagons are regular and this Wythoﬃan is

uniform.

For the apeirohedron P02 , we get a ﬁgure which is congruent to P02 of {4,6|4}. The faces of type F{0,1}

2

are convex, regular hexagons; the faces of type F{1,2}

2are convex squares; and the faces of type F{0,2}

2are

convex rectangles. At each vertex there is a rectangle, a square, a rectangle, and a hexagon, in cyclic order,

yielding a vertex symbol of (4c.4c.6c.4c). The vertex ﬁgure is then a convex quadrilateral. For certain choices

of initial vertex the faces are all regular and the apeirohedron is uniform.

In the apeirohedron P12 the faces of type F{0,1}

2are convex, regular hexagons and the faces of type F{1,2}

2

are skew octagons (truncated skew quadrilaterals). The vertex symbol is (6c.82

s) resulting in an isosceles

triangle as a vertex ﬁgure. For an initial vertex choice outside the convex hull of the base face of {6,4|4}

the skew octagons will sometimes become convex octagons (possibly regular) and the convex hexagons will

sometimes become antiprismatic, regular hexagons. In this case the Wythoﬃan would be uniform.

Finally, examine P012. In this apeirohedron the faces of type F{0,1}

2are convex dodecagons (truncated

hexagons), the faces of type F{1,2}

2are skew octagons (truncated skew quadrilaterals), and the faces of type

F{0,2}

2are convex rectangles. The vertex symbol is (4c.8s.12c) corresponding to a triangular vertex ﬁgure.

Wythoﬃan Skeletal Polyhedra in Ordinary Space, I 27

P0P1P2P01

P02 P12 P012

Fig. 11 The Wythoﬃans derived from {6,4|4}.

The ﬁnal Petrie-Coxeter polyhedron is {6,6|3}with symmetry group G({6,6|3}) = hr0, r1, r2i, where

r0:= (t0t1)2t2(t0t1)2,r1:= t1t3, and r2:= t2, and t0, . . . , t3are as before (see [41, p. 224]). In particular,

r0and r2are plane reﬂections and r1is a half-turn. Again the initial vertices are points of the base face of

{6,6|3}that lie in the fundamental region. As before we place this restriction on the initial vertex choices to

make the geometry of the Wythoﬃan similar to the geometry of {6,6|3}. Note that {6,6|3}is geometrically

self-dual, and so the collections of Wythoﬃans P2and P12 are just the same as those of P0and P01,

respectively. For pictures of the Wythoﬃans see Figure 12.

The ﬁrst Wythoﬃan, P0, is the regular apeirohedron {6,6|3}itself. It has regular, convex hexagons for

faces. Six such hexagons meet at each vertex yielding a regular, antiprismatic hexagon for the vertex ﬁgure.

In P1the faces of type F{0,1}

2are regular, convex hexagons while the faces of type F{1,2}

2are regular,

antiprismatic hexagons. Alternating about each vertex are two skew hexagons and two convex hexagons so

the vertex symbol is (6c.6s.6c.6s). The vertex ﬁgure is then a convex rectangle. All faces are regular so this

Wythoﬃan is uniform.

In P2the resulting ﬁgure is again the regular apeirohedron {6,6|3}, thanks to the self-duality of {6,6|3}.

In the apeirohedron P01 the faces of type F{0,1}

2are convex dodecagons (truncated hexagons) and the faces

of type F{1,2}

2are regular, antiprismatic hexagons. There are two dodecagons and one hexagon meeting at

each vertex, yielding a vertex symbol (6s.122

c) and an isosceles triangle as the vertex ﬁgure. For a speciﬁc

choice of initial vertex the dodecagons are regular and the Wythoﬃan is uniform.

In the apeirohedron P02 the faces of type F{0,1}

2are regular, convex hexagons; the faces of type F{1,2}

2

are regular convex hexagons; and the faces of type F{0,2}

2are convex rectangles. At each vertex there is a

hexagon of the ﬁrst type, a rectangle, a hexagon of the second type, and another rectangle, giving a vertex

symbol of (6c.4c.6c.4c). The resulting vertex ﬁgure is a skew quadrilateral. If a certain initial vertex is chosen

the faces of type F{0,2}

2are squares and the Wythoﬃan is uniform.

In the apeirohedron P12 the faces of type F{0,1}

2are regular, convex hexagons. The faces of type F{1,2}

2

are skew dodecagons which appear as the truncations of regular, antiprismatic hexagons. There are two

dodecagons and one hexagon at each vertex yielding an isosceles triangle as the vertex ﬁgure corresponding

to the vertex symbol (6c.122

s). The dodecagons are not regular so the Wythoﬃan is not uniform.

28 Schulte and Williams

Finally consider P012. In this apeirohedron the faces of type F{0,1}

2are convex dodecagons (truncated

hexagons), the faces of type F{1,2}

2are skew dodecagons (truncated, anstiprismatic hexagons), and the faces

of type F{0,2}

2are convex rectangles. The vertex symbol is (4c.12c.12s) yielding a triangular vertex ﬁgure.

As before, the skew dodecagons are not regular so the Wythoﬃan is not uniform.

P0P1P2P01

P02 P12 P012

Fig. 12 The Wythoﬃans derived from {6,6|3}.

Acknowledgment. We are grateful to the anonymous referees for their careful reading of our original

manuscript and their helpful suggestions that have improved our paper.

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