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Abstract and Figures

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 48 regular skeletal polyhedra (Grünbaum–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.
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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
the date of receipt and acceptance should be inserted later
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 (...,v2, v1, 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 rG\ {1}and En=SrGr(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 uEnbe a
point that is not held invariant under any non-identity transformation in G. For rGdefine 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:= TrGH[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 F1(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
F1and F3, exactly one vertex, one edge and one facet. In practice, when listing the elements of a flag we
often suppress F1and 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 Φi1and Φiare adjacent (differ by one
face), and ΦΨΦifor each i. Finally, if Fand Gare a (j1)-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/F1. 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:F0Eof 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:F1V1, with V12V0consisting of the elements
β1(F) := {β0(G)|G∈ F0and G6F}
for F∈ F1; and a surjection β2:F2V2, with V22V1consisting 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 vR1R2and v0R0
1R0
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
of Cis the set of types of the vertices of . 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 (a simplex
or the empty set) of C, there exists a sequence Φ=Φ0, Φ1, . . . , Φk1, Φk=Ψof 2-simplices of C, all containing
, such that Φi1and Φiare adjacent for i= 1, . . . , k.
Recall that the star of a face in a simplicial complex is the subcomplex consisting of all the simplices
which contain , and all their faces. The link of is the subcomplex consisting of all the simplices in the
star of which do not intersect . 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/F1of 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 iIbut 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 iI, 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 iIand
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 iIwe 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 Pof 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={2i|iI}then PI= (P)I. Thus the dual
Phas the same set of seven Wythoffians as the original polyhedron P. Moreover, if Pis self-dual then the
Wythoffians PIand PIare 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 uE3we 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(iI), 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:= \
iI
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 iIthere 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 vc
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 t2, 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.t2.t2) 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.t2). 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.t2.t2). 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.t2) 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.t2.t2) (where here we use t2to 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.t2).
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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