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ARS MATHEMATICA CONTEMPORANEA 5 (2012) 333–354
Developments and open problems on
chiral polytopes
Daniel Pellicer
National University of Mexico, Morelia, Michoacan, Mexico
Received 30 December 2010, accepted 28 January 2012, published online 30 May 2012
Abstract
This paper presents the current state of the theory of abstract chiral polytopes together
with 39 open problems on this topic.
Keywords: Chiral polytope, regular and chiral.
Math. Subj. Class.: 05C25, 51M20
1 Introduction
Abstract polytopes are combinatorial structures that generalize the concept of convex poly-
topes. Although some ideas can be found earlier [32], the work of Coxeter and Grünbaum
in the 1970’s inspired later work such as [37] and it mainly deals with those polytopes with
a high degree of symmetry.
Regular abstract polytopes are those admitting all symmetries by reflections, which
generate all other possible symmetries. Most results regarding regular abstract polytopes
are summarized in [37].
Generally, an object is said to be chiral whenever it lacks mirror symmetries. In prin-
ciple it may have no symmetries at all. In the 1980’s, in [57] the term “chiral polytopes”
was reserved for those abstract polytopes admitting all possible symmetries by abstract ro-
tations but no mirror symmetries. In some way, “chiral polytopes” became a short name for
“the most symmetric chiral polytopes”. In this paper we shall be consistent with the work
done so far and use the term “chiral” exclusively for polytopes admitting all rotational sym-
metries, although we acknowledge the existence of other interesting classes of polytopes
with no mirror symmetries.
After regular polytopes, chiral polytopes are regarded as the most symmetric ones.
However, there is a considerable difference in the knowledge we have about these two
classes of polytopes. This is due in part to the distinct properties of regular and chiral poly-
topes that make chirality harder to study. For instance, in contrast with regular polytopes,
E-mail address: pellicer@matmor.unam.mx (Daniel Pellicer)
Copyright c© 2012 DMFA Slovenije
334 Ars Math. Contemp. 5 (2012) 333–354
there are no convex chiral polytopes, and we do not know of a natural family of chiral
polytopes providing an example in each rank. Consequently, several questions that have
been answered for regular polytopes remain open for chiral polytopes. Part of the goal of
this paper is to provide a number of properties with respect to which regular and chiral
polytopes behave in different ways. We also state a number of open problems dealing with
chiral polytopes.
In Section 2 we state the main definitions and give a brief exposition of properties of
regular and chiral polytopes. From Section 3 to Section 8 we discuss group actions, Petrie
paths, topological considerations, maps and polyhedra, higher ranks and realizations, all in
relation with regular and chiral polytopes.
2 Definitions
We start by introducing abstract regular and chiral polytopes, referring to [37], [57] for
details.
An abstract d-polytope (or abstract polytope of rank d) K is a partially ordered set
satisfying the following properties. The elements of K are called faces, and its maximal
totally ordered subsets are called flags. There is a minimum face F−1 and a maximum face
Fd, and every flag of K contains precisely d + 2 elements (including F−1 and Fd). This
induces a rank function from K to the set {−1, 0, . . . , d} such that rank(F−1) = −1 and
rank(Fd) = d. The faces of rank i are called i-faces, the 0-faces are called vertices, the
1-faces are called edges and the (d− 1)-faces are called facets. When convenient we shall
abuse notation and identify the section G/F−1 := {H |H ≤ G} with the face G itself in
K. Given a vertex v, the section Fd/v := {H |H ≥ v} is called the vertex-figure of K
at G. For every incident faces F and G such that rank(F ) − rank(G) = 2, there exist
precisely two faces H1 and H2 such that G < H1, H2 < F . This property is referred to
as the diamond condition. The diamond condition implies that for any flag Φ and any i ∈
{0, . . . , d− 1} there exists a unique flag Φi that differs from Φ only in the i-face. This flag
is called the i-adjacent flag of Φ. Finally, K must be strongly flag-connected, in the sense
that for any two flags Φ, Φ′ there exists a sequence of flags Ψ0 = Φ,Ψ1, . . . ,Ψm = Φ′
such that Φ ∩ Φ′ ⊆ Ψk and Ψk−1 is adjacent to Ψk, for k = 1, . . . ,m.
Whenever the (polygonal) sectionG/F := {H |F ≤ H ≤ G} between an (i−2)-face
F and an incident (i+1)-faceG of an abstract polytope depends only on i and not on F and
G, we say that K is equivelar. If F and G are faces, with ranks i−2 and i+ 1 respectively,
of an equivelar polytope K, the section G/F is isomorphic to an abstract pi-gon where
pi ≤ ∞ is fixed. We define the Schläfli type of an equivelar polytope K as {p1, . . . , pd−1}.
Throughout this paper we will encounter only polytopes with pi ≥ 3 for i = 1, . . . , d− 1.
Regular and chiral polytopes defined below are examples of equivelar polytopes.
An automorphism of a polytope K is an order-preserving permutation of its faces. The
automorphisms of K form a group denoted by Γ(K).
We say that a d-polytope K is regular whenever Γ(K) acts transitively on the flags
of K. In this case, Γ(K) is generated by involutions ρ0, . . . , ρd−1 where ρi is the unique
automorphism of K mapping a fixed base flag Φ to its i-adjacent flag Φi. The generating
set {ρ0, . . . , ρd−1} satisfies the relations
ρ2i = ε for all i, (2.1)
(ρiρj)
2 = ε whenever |i− j| ≥ 2, (2.2)
D. Pellicer: Developments and open problems on chiral polytopes 335
where ε denotes the identity element. These generators also satisfy the intersection condi-
tions given by
〈ρi | i ∈ I〉 ∩ 〈ρi | i ∈ J〉 = 〈ρi | i ∈ I ∩ J〉, (2.3)
for all I, J ⊆ {0, . . . , d − 1}. The order of the element ρi−1ρi in Γ(K) is the Schläfli
number pi. In this paper pi may be either finite or∞.
A group together with a generating set {ρ0, . . . , ρd−1} is called a string C-group if it
satisfies (2.1) and (2.2), and the intersection condition (2.3). String C-groups are in a one-
to-one correspondence with automorphism groups of regular polytopes; in particular, every
regular polytope can be reconstructed from its automorphism group and its distinguished
generators.
The rotation subgroup of the automorphism group of a regular d-polytope K is defined
as the subgroup of Γ(K) consisting of words of even length on the generators ρ0, . . . , ρd−1
and is denoted by Γ+(K). For convenience, the rotation subgroup of Γ(K) is simply re-
ferred to as the rotation subgroup ofK. For i = 1, . . . , d−1 we define the abstract rotation
σi to be ρi−1ρi, that is, the automorphism of K mapping the base flag Φ to (Φi)i−1. It is
not hard to see that Γ+(K) = 〈σ1, . . . , σd−1〉 and that those generators satisfy the relations
(σiσi+1 · · ·σj)2 = ε (2.4)
for i < j. In particular, σiσi+1 · · ·σj = σ−1j · · ·σi+1σ
−1
i for j ≥ i+ 1. The order of σi is
precisely the entry pi in the Schläfli symbol.
We define abstract half-turns as the involutions τi,j := σi · · ·σj for i < j. For consis-
tency we also define τ0,i := τi,d := ε and denote σi by τi,i. Then the abstract rotations and
half-turns satisfy the intersection condition given by
〈τi,j | i ≤ j; i−1, j ∈ I〉∩〈τi,j | i ≤ j; i−1, j ∈ J〉 = 〈τi,j | i ≤ j; i−1, j ∈ I∩J〉 (2.5)
for I, J ⊆ {−1, . . . , d}.
For any regular polytope K, the index of Γ+(K) in Γ(K) is at most 2. Whenever
Γ+(K) has index 2 in Γ(K) we say that K is orientably regular; otherwise K is said to be
non-orientably regular.
We say that the d-polytope K is chiral whenever its automorphism group induces two
orbits on flags with the property that adjacent flags belong to different orbits. This elimi-
nates all possible abstract reflections.
Note that every polygon is isomorphic (as a poset) to a regular convex polygon. There-
fore there are no chiral 2-polytopes.
Although the facets and vertex-figures of a chiral d-polytope K can be either regular or
chiral, the 2-faces and (d− 2)-faces of K must necessarily be regular (see [57, Proposition
9]).
Some quotients of the Euclidean tessellations {3, 6}, {4, 4} and {6, 3} of the Euclidean
plane by triangles, squares and hexagons respectively are examples of orientably regular
and chiral polyhedra (rank 3 polytopes). Let T denote the group of translations of the plane
which fix a given tessellation. We denote by {p, q}ū the quotient of the regular tessellation
{p, q} of the plane by the subgroup Tū of T generated by the translations with respect to
the linearly independent vectors ū and ūR, where R stands for the rotation by 2π/q. For
{p, q} = {4, 4} and {p, q} = {3, 6} it is well known that the quotient {p, q}ū is regular
if and only if ū = (a, 0) or ū = (a, a) with the vectors taken with respect to the basis
{e, eR} where e denotes a vector with the size and direction of an edge of the tessellation.
336 Ars Math. Contemp. 5 (2012) 333–354
Moreover, the quotient {p, q}ū is chiral whenever it is not regular (see [19], [37, Section
1D]). The same is true for {p, q} = {6, 3} if we choose e as the vector between the centres
of two hexagons sharing an edge. Note that for a = 0, 1 the maps {4, 4}(1,a), {3, 6}(1,0)
and {6, 3}(1,0) fail to be strongly flag-connected, and thus, they are not abstract polyhedra.
The automorphism group Γ(K) of a chiral polytope is generated by elements σ1, . . . ,
σd−1, where σi maps a base flag Φ to (Φi)i−1, in which case, σi cyclically permutes the i-
and (i− 1)-faces of K incident with the (i− 2)- and (i+ 1)-faces of Φ. Furthermore, the
generators σi also satisfy (2.4) as well as the intersection conditions in (2.5). Because of the
obvious similarities between the automorphism groups of chiral polytopes and the rotation
subgroups of regular polytopes we shall also refer to the generators σi of the automorphism
groups of chiral polytopes as abstract rotations. The products τi,j := σiσi+1 · · ·σj (i < j)
will be referred to as abstract half-turns. Furthermore, we say that the rotation subgroup
of a chiral polytope is its automorphism group, and when convenient, we shall denote the
automorphism group Γ(K) = 〈σ1, . . . , σd−1〉 of the chiral polytope K by Γ+(K).
It was proved in [57] that any group Γ together with a generating set {σ1, . . . , σd−1}
satisfying (2.4) and the intersection conditions in (2.5) is the rotation subgroup of either an
orientably regular polytope, or a chiral polytope. Furthermore, Γ is the rotation subgroup
of a regular polytope if and only if there is a group automorphism η of Γ such that
σiη =
 σ
−1
i for i = 1,
σ21σi for i = 2,
σi for i > 2.
(2.6)
The following lemmas provide some properties of the generators of the automorphism
groups of chiral polytopes.
Lemma 2.1. Let Γ(K) = 〈σ1, . . . , σd−1〉 be the automorphism group of a chiral d-poly-
tope, then σiσj = σjσi whenever |i− j| ≥ 3.
Proof. Without loss of generality assume that i < j. We use (2.4) to prove that σi com-
mutes with σ−1j .
σiσ
−1
j = σi(σi+1σi+2 · · ·σjσi+1σi+2 · · ·σj−1)
= (σiσi+1 · · ·σj)σi+1σi+2 · · ·σj−1
= (σ−1j σ
−1
j−1 · · ·σ
−1
i )σi+1σi+2 · · ·σj−1
= σ−1j (σ
−1
j−1σ
−1
j−2 · · ·σ
−1
i )σi+1σi+2 · · ·σj−1
= σ−1j (σiσi+1 · · ·σj−1)σi+1σi+2 · · ·σj−1
= σ−1j σi(σi+1σi+2 · · ·σj−1)
2 = σ−1j σi.
Lemma 2.2. Let Γ(K) = 〈σ1, . . . , σd−1〉 be the automorphism group of a chiral d-poly-
tope and τ the product σ1σ2 · · ·σd−1. Then
1. τσiτ =
{
σ−1i for i = 1, d− 1,
σi for i = 3, 4, . . . , d− 3,
D. Pellicer: Developments and open problems on chiral polytopes 337
2. σ−12i σ2j+1σ2i =
{
σ−22i σ
−1
2j+1 if j = i− 1, i
σ2j+1 if j 6= i− 1, i.
Proof. We use (2.4) and Lemma 2.1.
τσ1τ = σ1σ2 · · ·σd−1(σ21)σ2σ3 · · ·σd−1 = σ1σ2σ3(σ21)σ4σ5 · · ·σd−1σ2σ3 · · ·σd−1
= σ1σ2σ3σ
2
1(σ4σ5 · · ·σd−1σ2σ3 · · ·σd−1) = (σ1σ2σ3)σ21(σ−13 σ
−1
2 )
= (σ−13 σ
−1
2 σ
−1
1 )σ
2
1σ
−1
3 σ
−1
2 = (σ
−1
3 σ
−1
2 )σ1(σ
−1
3 σ
−1
2 ) = (σ2σ3)σ1(σ2σ3)
= σ−11 .
The proof of τσd−1τ = σ−1d−1 is analogous.
On the other hand, for 4 ≤ i ≤ d− 4,
τσiτ = σ1σ2 · · ·σd−1(σi)σ1σ2 · · ·σd−1 = σ1σ2 · · ·σd−1σ1σ2 · · ·
· · ·σi−3(σi)σi−2σi−1 · · ·σd−1
= (σ1σ2 · · ·σd−1)σ1σ2 · · ·σi−3σiσi−2σi−1 · · ·σd−1
= (σ−1d−1σ
−1
d−2 · · ·σ
−1
1 )σ1 · · ·σi−3σi(σi−2σi−1 · · ·σd−1)
= σ−1d−1σ
−1
d−2 · · ·σ
−1
i−2σi(σ
−1
d−1σ
−1
d−2 · · ·σ
−1
i−2)
= (σ−1d−1σ
−1
d−2 · · ·σ
−1
i−2)(σi)σ
−1
d−1σ
−1
d−2 · · ·σ
−1
i−2
= (σi−2 · · ·σd−1)σ−1d−1σ
−1
d−2 · · ·σ
−1
i+3(σi)σ
−1
i+2σ
−1
i+1 · · ·σ
−1
i−2
= σi−2σi−1(σiσi+1σi+2σi)σ
−1
i+2σ
−1
i+1(σ
−1
i σ
−1
i−1σ
−1
i−2)
= σi−2σi−1(σ
−1
i+2σ
−1
i+1)σ
−1
i+2σ
−1
i+1(σi−2σi−1σi)
= σi−2σi−1[(σi+1σi+2)
2]−1σi−2σi−1σi = σi−2σi−1σi−2σi−1σi = σi.
The same procedure, but with less steps, can be used for i ∈ {3, d − 3}. This proves the
first part.
The second part is a direct consequence of Lemma 2.1 and (2.4).
Lemma 2.3. Let Γ(K) = 〈σ1, . . . , σd−1〉 be the automorphism group of a chiral d-poly-
tope. Then
σ1σ2 · · ·σkσk+2 = σk+1σ1σ2 · · ·σk+2
for all k ≥ 2.
Proof. Note that (2.4) implies that σk+1σk−1σkσk+1 = σ−1k σ
−1
k−1 = σk−1σk. The lemma
then follows from the commutativity of σk+1 with σi for i < k − 1.
Lemma 2.4. Let Γ(K) = 〈σ1, . . . , σd−1〉 be the automorphism group of a chiral d-poly-
tope. Then
1. σ−11 σ3σ5 · · ·σ2k+1 = σ2σ4 · · ·σ2kσ1σ2 · · ·σ2k+1,
2. σ1σ2σ4σ6 · · ·σ2k = σ3σ5 · · ·σ2k−1σ1σ2 · · ·σ2k.
Proof. We proceed by induction over k. Since (σ1σ2)2 = ε it follows that σ−11 σ3 =
σ2σ1σ2σ3. On the other hand, σ1σ2σ4 = σ3σ1σ2σ3σ4 is a particular instance of Lemma
2.3.
The inductive step follows directly from Lemma 2.3.
338 Ars Math. Contemp. 5 (2012) 333–354
Each chiral d-polytopeK occurs in two isomorphic enantiomorphic forms, in a sense in
a right and left handed version which can be thought of as mirror images of each other. The
two enantiomorphic forms correspond to the two possible choices of flag orbit for the base
flag. The polyhedra {4, 4}(2,1) and {4, 4}(1,2) constitute an example of the two enantiomor-
phic forms of a chiral polytope. The group Γ(K) together with the distinguished generators
σ1, . . . , σd−1 completely determines the chiral polytope K as well as the enantiomorphic
form in consideration. The group of the other enantiomorphic form is also Γ(K), but with
respect to the generators σ−11 , σ
2
1σ2, σ3, . . . , σd−1. We refer to [58] for details.
The dualK∗ of a d-polytopeK is the d-polytope consisting of the set of faces ofK with
the order reversed. Note that the dual of a regular (resp. chiral) polytope is again regular
(resp. chiral).
3 Group structures
There are several groups associated with chiral abstract polytopes. The most obvious one
is the automorphism group of the polytope itself.
Not every group is the automorphism group of a chiral polytope. For instance, abelian
groups cannot be generated by two or more elements σi of order at least 3 satisfying (2.4).
In recent years there have been attempts to describe all finite simple groups in terms
of their actions on particular sets (see for example [31] and [30]). This gives interest to
the problem of determining which simple groups are automorphism groups of chiral d-
polytopes. Alternating groups and projective special linear groups were first investigated
in [46] and [58] respectively.
Problem 1 Determine which finite simple groups are automorphism groups of chiral
d-polytopes.
Of particular interest are the sporadic simple groups.
Problem 2 Determine which sporadic finite simple groups are automorphism groups of
chiral d-polytopes.
For any d-polytope K (not necessarily regular or chiral) and i = 1, . . . , d − 1 we
define the (involutory) permutation ri on the flags of K by Φri = Φi. The subgroup
〈r0, . . . , rd−1〉 of the symmetric group on the set of flags of K is often referred to as the
monodromy group of K (see for example [27], [29]).
Whenever the polytopeK is regular, its monodromy group is isomorphic to its automor-
phism group. For less symmetric polytopes this is no longer the case. If K is chiral then
there is a contravariant group isomorphism between its automorphism group with respect
to the base flag Φ and the permutation group determined by the action on the flag orbit
containing Φ of the set of even words on the generators r0, . . . , rd−1 of the monodromy
group (see [49, Proposition 2.7]).
LetK be a d-polytope andP be a regular d-polytope with automorphism group Γ(P) =
〈ρ0, . . . , ρd−1〉. We say that K admits a flag action from P if there is a group homomor-
phism from Γ(P) to the monodromy group of K mapping ρi to ri for all i. The polytope K
is isomorphic to the quotient P/N where N is the stabilizer in Γ(P) of a flag of K. Here
P/N can be thought of as the combinatorial structure resulting from identifying the image
Φw of the base flag Φ with all flags of the form Φhw where h ∈ N , for every w in the
monodromy group. We refer to [22] and [37, Section 2D] for more information on flag
actions and quotients.
D. Pellicer: Developments and open problems on chiral polytopes 339
We say that the regular polytope P is a regular cover of the polytope K whenever K
admits a flag action from P . Moreover, P is the minimal regular cover of K if it admits
a flag action from all other regular covers of K. It is not hard to see that whenever the
generators r0, . . . , rd−1 of the monodromy group of K satisfy the intersection condition
(2.3) then Mon(K) is the automorphism group of the minimal regular cover of K. Further
details can be found in [26] and [41].
It is known that the monodromy group of all polyhedra are string C-groups [41], and
therefore every polyhedron has a minimal regular cover. This is not the case for higher
rank polytopes. In [40] an example of a 4-polytope whose monodromy group is not a
string C-group is given.
The mix of two regular polytopes is defined in [37, Section 7A]. This definition is
extended to the mix of orientably regular or chiral polytopes in [5] and [49] as follows.
Let Γ+(K) and Γ+(P) be the rotation subgroups of orientably regular or chiral d-
polytopes with abstract rotations given by σ1, . . . σd−1 and δ1, . . . , δd−1 respectively. The
mix Γ+(K)♦Γ+(P) of Γ+(K) and Γ+(P) is the subgroup 〈(σ1, δ1), . . . , (σd−1, δd−1)〉 of
the direct product Γ+(K) × Γ+(P). Whenever Γ+(K)♦Γ+(P) satisfies the intersection
condition with respect to the generators γi := (σi, δi), the group Γ+(K)♦Γ+(P) is the
rotation subgroup of an orientably regular or chiral polytope K♦P called the mix of K and
P .
The monodromy group of a chiral polytope K is the mix of K with its enantiomorphic
form. The resulting group is a string C-group whenever the facets or vertex-figures of K
are regular. The corresponding minimal regular cover is called mixed regular cover in [49,
Section 4]. However it is not known whether all chiral polytopes with chiral facets and
chiral vertex-figures have a minimal regular cover.
Problem 3 Is there a chiral polytope whose monodromy group is not a string C-group?
Even if the previous problem has a negative answer, it is still interesting to determine
whether this implies that some chiral polytopes have no minimal regular cover.
Problem 4 Does every chiral polytope have a minimal regular cover?
It is easy to see that whenever K and P are regular and K♦P is a polytope, the latter
is the minimal regular polytope which covers simultaneously K and P . An equivalent
statement for the case when either K or P is chiral must hold for an appropriate definition
of a quotient of a chiral polytope.
Problem 5 Develop a theory of chiral covers of polytopes.
Any d-polytope K is a quotient of the universal polytope {∞d−1} := {∞, . . . ,∞}
with the Coxeter group
[∞d−1] = 〈ρ0, . . . , ρd−1 | (ρiρj)2 = id for |i− j| ≥ 2〉
as automorphism group. Let M be the subgroup of [∞d−1] such that K = {∞k−1}/M .
The subgroup M is normal in [∞d−1] if and only if K is regular. If K is chiral, then
M is normal in the subgroup [∞k−1]+ consisting of the elements of [∞k−1] that can be
expressed as a word of even length in the generators. The chirality group of a chiral poly-
tope K is defined in [5] as X(K) := M/(M ∩ ρ0Mρ0), or equivalently as X(K) =
(M · ρ0Mρ0)/M . Intuitively, the order of X(K) is a measure of how far from regular
340 Ars Math. Contemp. 5 (2012) 333–354
(of how chiral) is K. This concept preserves the idea of the chirality group of hypermaps
defined in [4].
The chirality group X(K♦P) of the mix of a chiral polytope K and a directly regular
polytope P is shown in [5] to be a subgroup of X(K). These two groups sometimes
coincide.
Problem 6 Can we characterize all pairs (K,P) consisting of a chiral polytope K and
a regular polytope P such that X(K♦P) = X(K)?
4 Petrie paths
A Petrie path π of a d-polytope K with rank at least 3 is a closed or infinite walk on the
underlying graph of K (the set of vertices and edges) such that for 2 ≤ k ≤ d− 1, every k
consecutive edges of π belong to the same k-face, but the (k+ 1)-th does not belong to the
common k-face of the previous k edges. An equivalent recursive definition of Petrie paths
was given in [18, Chapter 12.4].
According to [65], a Petrie map Π(ξ) of a d-polytopeK is the (left) action of the product
rξ(0)rξ(1) · · · rξ(d−1) of the generators r0, . . . , rd−1 of the monodromy group of K, for a
given permutation ξ of {0, 1, . . . , d − 1}. A Petrie scheme of K is a sequence of flags
(Φ,ΠΦ, . . . ,Πm−1Φ), where Φ is a flag of K, Π = Π(ξ) is a Petrie map, ΠmΦ = Φ and
ΠkΦ 6= Φ for 1 ≤ k ≤ m − 1; or the sequence (. . . ,Π−1Φ,Φ,ΠΦ,Π2Φ, . . . ) whenever
ΠkΦ 6= Φ for all k 6= 0.
A Petrie scheme P of a polytope K is said to be acoptic whenever each proper face of
K is contained in at most one flag of P ([26, Section 6]). Several families of polytopes
were proved to be acoptic in [65].
Problem 7 Determine conditions under which the Petrie scheme of a chiral polytope is
acoptic.
The following theorem is equivalent to Theorem 2.5 in [65].
Theorem 4.1. Any Petrie map of a d-polytope with monodromy group 〈r0, . . . , rd−1〉 is a
conjugate of the Petrie map Π(ι) where ι is the identity permutation of {0, . . . , d− 1}.
The Petrie schemes associated to the Petrie map Π(ξ) of a regular d-polytopeK are pre-
cisely the sequences (Φ, ψΦ, . . . , ψm−1Φ) or (. . . , ψ−1Φ,Φ, ψΦ, ψ2Φ, . . . ) where ψ :=
ρξ(d−1)ρξ(d−2) · · · ρξ(0) ∈ Γ(K), since the actions of Π(ξ) and ψ on any flag coincide. It
follows from Theorem 4.1 that all Petrie paths are equivalent under Γ(K).
Whenever K is a chiral d-polytope with d even, the Petrie map Π(ξ) acts on the base
flag like an automorphism of K. In particular, rd−1rd−2 · · · r0 acts like σ1σ3 · · ·σd−1.
Theorem 4.1 and the contravariant isomorphism of Γ(K) to a permutation group on the
orbit of the base flag (see Section 3) imply that there are precisely two classes of Petrie
schemes under Γ(K), namely the one induced by the product rd−1rd−2 · · · r0 (or by the
automorphism σ1σ3 · · ·σd−1) and the one induced by the product rd−1rd−2 · · · r2r0r1 (or
by the automorphism σ−11 σ3 · · ·σd−1).
If K is a chiral d-polytope with d odd, then the numbers of factors of the Petrie maps
are odd, and therefore all Petrie schemes must involve an even number of Petrie maps,
when there are finitely many. That is, for any odd number k and any Petrie map Π =
Π(ξ), ΠkΦ 6= Φ. The square of a Petrie map acts on the base flag like an automor-
phism of K. For instance, if Π is the action of the product ri−1ri−2 ri−3 · · · r0 then Π2
D. Pellicer: Developments and open problems on chiral polytopes 341
acts like σ1σ3 · · ·σd−2σ1σ2 · · ·σd−1σ2σ4 · · ·σd−1. As above we can conclude that there
are precisely two classes of Petrie schemes under Γ(K), the one induced by the product
rd−1rd−2 · · · r0 (or by the automorphism σ1σ3 · · ·σd−2σ1σ2 · · ·σd−1σ2σ4 · · ·σd−1) and
the one induced by the product rd−1rd−2 · · · r2r0r1 (or by the automorphism σ−11 σ3 · · ·
σd−2σ1σ2 · · ·σd−1σ21σ2σ4 · · ·σd−1).
An inductive argument shows that for every d − 1 ≥ k > j ≥ 2, the j-face of
rk−1rk−2 · · · r0(rd−1rd−2 · · · r0)d−k−1Φ is contained in the k-face of Φ. It follows that
the set of vertices and edges of the flags of a Petrie scheme determine a Petrie path of the
polytope. Moreover, the Petrie paths of a polytope coincide with the set of vertices and
edges of its Petrie schemes.
The following problem is, thus, a weaker version of Problem 7.
Problem 8 Determine conditions under which the Petrie path of a chiral polytope is a
polygon.
We shall refer to the Petrie paths of a chiral polytope K associated to the Petrie map
rd−1rd−2 · · · r0 by left Petrie paths, and those associated to the Petrie map rd−1rd−2 · · · r2
r0r1 by right Petrie paths. Note that the length (number of edges) of the right Petrie paths
of K correspond to the length of the left Petrie paths of its enantiomorphic form, and vice
versa. Clearly, either all left (right) Petrie paths are polygons, or none of them is.
Problem 9 Is there a chiral polytope whose left Petrie paths are polygons, but whose
right Petrie paths are not?
It was pointed out in [60, Section 3] that the lengths of the left and right Petrie paths
of chiral polyhedra have the same length. This is true for d-polytopes whenever d is not a
multiple of 4.
Proposition 4.2. The right and left Petrie paths of any chiral d-polytope have the same
length for d odd.
Proof. We claim that the group elements defining two steps on the right and left Petrie paths
are conjugates. In fact, if d = 3 this is straightforward; on the other hand, if d = 2k + 1
then by Lemma 2.4 it follows that
σ−11 σ3 · · ·σ2k−1σ1σ2 · · ·σ2kσ21σ2σ4 · · ·σ2k
= σ2σ4 · · ·σ2k−2(σ1σ2 · · ·σ2k−1)2σ2kσ1σ3 · · ·σ2k−1σ1σ2 · · ·σ2k
= σ2σ4 · · ·σ2kσ1σ3 · · ·σ2k−1σ1σ2 · · ·σ2k.
Conjugating the last expression by (σ2σ4 · · ·σ2k)−1, we obtain the desired result.
Proposition 4.3. The right and left Petrie paths of any chiral d-polytope have the same
length for d ≡ 2 modulo 4.
Proof. We prove that the group element defining a step on a right Petrie path is conjugate
to the inverse of the element defining a step on the left Petrie path.
First note that commutativity of σi and σj if |i − j| ≥ 3 (see Lemma 2.1) implies that
σ−11 σ3σ5 · · ·σd−1 is a conjugate of
σd−1σd−3 · · ·σ3σ−11 . (4.1)
342 Ars Math. Contemp. 5 (2012) 333–354
We conjugate the element in (4.1) by σ(d/2)−1σ(d/2)−3 · · ·σ2 to obtain the element
σd−1σd−3 · · ·σ(d/2)+2σ−1d/2σ(d/2)−2 · · ·σ1 (4.2)
(see Lemma 2.2 (2)).
By conjugating the element in (4.2) by the element τ = σ1 · · ·σd−1 (see Lemma 2.2
(1)) we obtain
σ−1d−1σd−3σd−5 · · ·σ(d/2)+2σ
−1
d/2σ(d/2)−2σ(d/2)−4 · · ·σ3σ
−1
1 .
By a suitable modification of the previous two steps it follows that σ−11 σ3σ5 · · ·σd−1
is a conjugate of the expression obtained by replacing each σi and σd−i by σ−1i and σ
−1
d−i
respectively. As a consequence, this expression is a conjugate of σ−12d−1σ
−1
2d−3 · · ·σ
−1
1 ,
which is the inverse of the relation defining the left Petrie path.
There are examples of chiral 4-polytopes with the property that the left and right Petrie
paths have different lengths (see [17], [51] and [58]). No such example is known for 4k-
polytopes with k ≥ 2.
Problem 10 Find examples of chiral 4k-polytopes (where k ≥ 2) with the property that
the length of the right Petrie paths does not coincide with the length of the left Petrie paths,
or prove their non-existence.
The examples of chiral polytope-like structures of rank 4 with right Petrie paths of
length m and left Petrie paths of length n < m given in [17] yield the pairs
(m,n) ∈ {(3, 2), (4, 2), (5, 3), (6, 3), (5, 4), (9, 5), (18, 6), (30, 8)}.
However, the structures corresponding to the first 5 pairs are not polytopes. In [58] the
authors prove the existence of infinite families of chiral polytopes using projective linear
groups. The length of the right and left Petrie paths is the order of a matrix, which can be
found by a recursive procedure. The infinite family of chiral 4-polytopes described in [51]
contains polytopes with right Petrie paths of length 4 and left Petrie paths of length 4k for
each k ≥ 2.
Problem 11 Determine the pairs (m,n) for which there exists a chiral 4-polytope with
right Petrie paths of length m and left Petrie paths of length n.
It is also interesting to determine the existence and properties of chiral 4-polytopes with
the property that the lengths of the right and left Petrie paths coincide.
Problem 12 Determine the integers n for which there exists a chiral 4-polytope with
right and left Petrie paths of length n.
5 Topological considerations
We say that a regular d-polytope K is spherical whenever it is isomorphic to a regular
tessellation of the (d − 1)-sphere, or equivalently, it is isomorphic to the face lattice of a
convex regular polytope. There are spherical regular d-polytopes for all rank d ≥ 2. We
say that K is Euclidean (resp. hyperbolic) whenever it is isomorphic to the face lattice
of a geometrically regular tessellation of an Euclidean (resp. hyperbolic) (d − 1)-space.
D. Pellicer: Developments and open problems on chiral polytopes 343
Euclidean regular d-polytopes exist for each d ≥ 2, whereas hyperbolic regular d-polytopes
exist only for d = 3, 4, 5. By a tessellation we understand a locally finite decomposition
of the space into compact tiles, each of them isomorphic to a convex polytope (see [37,
Chapter 6]). In this paper the terms “Euclidean polytope” and “hyperbolic polytope” are
used exclusively to describe tessellations of the full Euclidean or hyperbolic space, not
quotients of the latter.
Following this idea, we can define a spherical chiral polytope as the face lattice of a
convex chiral polytope, and an Euclidean (resp. hyperbolic) chiral polytope as a chiral
tessellation of an Euclidean (resp. hyperbolic) space. However, there exist no chiral, Eu-
clidean or hyperbolic chiral polytopes (see for example [33, Theorem 11.2]).
A space-form is a quotient of the (spherical, Euclidean or hyperbolic) d-dimensional
space E by a discrete group of isometries which acts freely on E. The projective d-space
and the d-torus are examples of spherical and Euclidean space forms respectively.
A polytope is locally spherical whenever its facets and vertex-figures are spherical. In
[37, Chapter 6B] a general theory on locally spherical regular polytopes is provided, and
every such regular polytope is shown to be a tessellation of a spherical, Euclidean or hy-
perbolic space-form arising as a quotient (from a combinatorial, geometric and topological
point of view) of a regular tessellation of the spherical, Euclidean or hyperbolic space by
a normal sparse subgroup. Similar arguments can be applied to prove an analogous result
for chiral polytopes (see also [60, Section 5]).
In [37, Chapter 6C] it is shown that besides the spherical regular polytopes, the only
regular tessellations of spherical space-forms are the projective regular (n+1)-polytopes in
the projective n-space Pn. The author does not know whether the existence or nonexistence
of chiral tessellations of spherical space-forms has been established.
Problem 13 Determine whether there exist chiral tessellations of spherical space-forms.
For each n ≥ 2 there exist infinite families of regular tessellations of the n-torus (see
[37, Sections 6D, 6E]). Chiral tessellations of the 2-torus are classified in [19]. Later it was
proved in [25, Section 4] that no other Euclidean space-form admits chiral tessellations
(see also [37, Section 6H]). In particular, all toroidal chiral n-polytopes (tessellation of the
n-torus) have rank 3.
The only locally finite tessellations of the hyperbolic n-space with convex regular poly-
topes are the universal polytopes with Schläfli type {p, q}, with 1/p+1/q < 1/2 for n = 2;
{4, 3, 5}, {5, 3, 4}, {5, 3, 5} and {3, 5, 3} for n = 3; or {3, 3, 3, 5}, {5, 3, 3, 3}, {4, 3, 3, 5},
{5, 3, 3, 4} and {5, 3, 3, 5} for n = 4.
Little is known about regular or chiral tessellations on hyperbolic space-forms. Chiral
tessellations of quotients of the hyperbolic plane are discussed in Section 6. The first
examples of chiral tessellations of quotients of the hyperbolic 3-space are described in [17,
Table 3], and later also in [64]. Recently some examples of chiral tessellations of quotients
of the hyperbolic 4- and 5-spaces were found [12].
Problem 14 Examine locally spherical chiral 5-polytopes of hyperbolic type.
Among all polytopes which are not locally spherical, special attention has been given
to locally toroidal polytopes.
A polytope is locally toroidal whenever its facets and vertex-figures are spherical or
toroidal, but not both spherical.
344 Ars Math. Contemp. 5 (2012) 333–354
Locally toroidal regular polytopes exist only for ranks d = 4, 5, 6 (see [37, Chapter 10].
Whenever there exist chiral polytopes with facets isomorphic to a regular or chiral
(d − 1)-polytope K1 and vertex-figures isomorphic to a regular or chiral polytope K2, but
not both K1 and K2 regular, there exists a universal polytope {K1,K2}ch with the property
that every polytope with facets isomorphic to K1 and vertex-figures isomorphic to K2 is
a quotient of {K1,K2}ch (see [60, Theorem 2]). This will be discussed in more detail in
Section 7. Abusing notation, we may assume that Γ+(K1) = 〈σ1, . . . , σd−2〉 subject to a
certain set of relations R1 and Γ+(K2) = 〈σ2, . . . , σd−1〉 subject to a set of relations R2.
Then Γ({K1,K2}ch) = 〈σ1, . . . , σd−1〉 subject to the set of relations R1 ∪ R2 together
with (σ1σ2 · · ·σd−1)2 = ε (see also [54] and [37, Section 4A]). In particular, this holds
whenever K1 and K2 are spherical or toroidal.
The central two problems on locally toroidal polytopes are to determine whether {K1,
K2}ch or {K1,K2} (the regular equivalent of {K1,K2}ch) exists, and if it does, whether
it is finite. In the regular case, the classification of all spherical or toroidal K1 and K2 for
which there exist polytopes with facets of type K1 and vertex-figures of type K2 is almost
complete. In case of existence, in most of the cases it is known whether {K1,K2} is finite
or infinite, however certain cases remain open (see [37, Chapters 10, 11, 12], [42], [43],
[44], [45]).
The same problem can be stated for locally toroidal chiral polytopes.
Since there are no spherical chiral polytopes and there are toroidal chiral polytopes only
of rank 3, then any locally toroidal polytope with chiral facets and/or vertex-figures must
have rank 4. Some progress in this case can be found in [8].
The following two problems were stated as Problem 19 in [60].
Problem 15 Determine all spherical or toroidal (d− 1)-polytopes K1 and K2 such that
at least one ofK1 orK2 is chiral and there exists a chiral d-polytope with facets isomorphic
to K1 and vertex-figures isomorphic to K2.
Problem 16 Determine all finite universal locally toroidal polytopes.
We can still consider the case where both spherical or toroidal (d−1)-polytopes are reg-
ular. In this case, whenever it exists, the corresponding universal locally toroidal polytope
must be regular.
Problem 17 Determine all spherical (d−1)-polytopesK1 andK2 for which there exists
a chiral d-polytope with facets isomorphic to K1 and vertex-figures isomorphic to K2.
6 Rank 3
All polyhedra (rank 3 polytopes) are locally spherical (see Section 5) since both the facets
and vertex-figures are isomorphic to convex polygons. It follows that each finite polyhedron
can be represented as a tessellation on a compact surface without boundary.
A map is a 2-cell embedding of a graph on a surface. Maps have been broadly stud-
ied (see for example [39]). Note that not every map satisfies the diamond condition, and
therefore not all maps are 3-polytopes.
The definitions of flags, automorphisms, regularity and chirality extend naturally from
polytopes to maps. Regular and chiral maps have been extensively studied since the early
20th century ([3]). All chiral maps on the torus were described by Coxeter in [19, Section
3.2]. They are classified in three families, each of which contains maps with the same
D. Pellicer: Developments and open problems on chiral polytopes 345
Schläfli type {3, 6}, {4, 4} or {6, 3}. Several years later chiral maps on surfaces with
higher genus were discovered. Edmonds found a map with type {7, 7} on an orientable
surface with genus 7. This map is described in [16, Chapter 21.3]. Sherk constructed a
family of chiral maps with type {6, 6} ([61]), and Garve proved that there is no chiral map
on surfaces with genus 2, 3, 4, 5 or 6 ([21]). Since then, several chiral maps on surfaces
with higher genus have been found (see [66], [13], [46] and [11]).
Regular and chiral maps have been considered from the viewpoints of the surfaces (see
for example [6], [11] and [13]), of the automorphism groups (see [46]) and of the graphs
(see [7], [28] and[62]). However little has been said about which regular and chiral maps
are polyhedra. In each orientable surface with genus 2 or greater there exists a regular map
with one vertex. Clearly, this is not a polyhedron. Similarly, several chiral maps are not
polyhedra, such as the map C11.4 in [11].
It would be convenient to know whether a result for polytopes extends for maps, and
also which results for maps translate directly to polytopes. This motivates the following
problem.
Problem 18 Characterize chiral maps which are not polyhedra.
Residual finiteness was used in [63, Theorem 6.3] to prove that given any Schläfli
type {p, q} with 1p +
1
q ≤
1
2 , there are infinitely many regular maps whose faces are p-
gons, q of them around each vertex. Similar techniques can be used to prove that there are
infinitely many finite chiral maps (and even chiral polytopes) with Schläfli type {p, q} with
1
p +
1
q ≤
1
2 . However, given a surface S with negative Euler characteristic there exist at
most finitely many regular and chiral maps on S (see for example [13]).
No chiral map lies on a non-orientable surface (see for example [13]). Also, as noted
above, there are no chiral maps on orientable surfaces from genus 2 to genus 6. Further-
more, it was proved in [15, Lemma 6.2] that orientable surfaces with genera p + 1, with p
prime and p−1 not divisible by 3, 5 or 8, do not admit chiral maps. This result implies that
there are infinitely many genera g such that there are no chiral maps on orientable surfaces
with genus g.
Problem 19 Determine all positive integers g for which there are chiral maps (polyhe-
dra) on orientable surfaces with genus g.
In orientable surfaces with small genera, regular maps appear to be more numerous than
chiral maps. For instance, on surfaces of genera 2 ≤ g ≤ 5 there are 220 regular maps and
only 16 chiral maps (see [13]). However, as we increase the genus, apparently the number
of chiral polytopes grows faster than the number of regular polytopes. For instance, on
orientable surfaces of genera 2 ≤ g ≤ 101 there are 3378 regular maps and 594 chiral
maps (see [11]).
Problem 20 Let nr(g) and nch(g) denote respectively the number of regular and chiral
polytopes on orientable surfaces of genus at most g. Determine
lim
g→∞
nr(g)
nch(g)
if the limit exists.
Some infinite regular polyhedra arise naturally as tessellations of the Euclidean or hy-
perbolic plane. In fact, we can construct one of them for each Schläfli type {p, q} with
1
p +
1
p ≤
1
2 .
346 Ars Math. Contemp. 5 (2012) 333–354
Very little is known about infinite chiral polyhedra. Permutation groups were used in
[51] to find chiral polyhedra with Schläfli symbols {8, 4}, {∞, 6} and {∞,∞}. Chiral
polyhedra with Schläfli types {6, 6}, {6, 4} and {4, 6} were found in [55] as geometric
objects in E3, and proved to be combinatorially chiral in [50] (see Section 7).
Problem 21 Determine all pairs {p, q} with p, q ∈ {3, 4, . . . } ∪ {∞} for which there
exist infinite chiral maps (polyhedra) with Schläfli type {p, q}.
The classification of noncompact surfaces was achieved in [52]. Whenever an infinite
polyhedron has finite faces and vertex-figures, it can be understood like a map on a non-
compact surface where the faces are considered to be discs, two of them identified in a line
segment in the boundary whenever they share an edge. The identifications must be made
in such a way that the cyclic order of the identifications around a face and around a vertex
are the same as the order of the corresponding edges in the polyhedron. The next problem
follows the spirit of Problem 19.
Problem 22 Determine which noncompact surfaces without boundary admit embed-
dings of chiral maps.
7 Higher ranks
Convex regular polytopes, regular tessellations of the Euclidean spaces and the regular
toroidal quotients of the latter are natural examples of regular polytopes of each rank d ≥ 3.
In contrast, only sporadic examples of chiral d-polytopes are known to exist for d ≥ 4. That
is, there is no “natural” construction known of chiral polytopes of each rank. Moreover, for
some time it was not even clear that there exist chiral polytopes of each rank d ≥ 3.
The first examples of chiral structures in the sense of abstract polytopes of rank greater
than 3 are due to Coxeter, and were obtained as quotients of regular tessellations of the
hyperbolic 3-space, by forcing the right and left Petrie walks to have distinct lengths (see
[17] and Sections 4, 5). More examples of this type were found in [47], [64] and [58].
In the latter, instead of tessellations with compact tiles, finitely many of which meet at
every vertex, the honeycombs {4, 4, 3}, {4, 4, 4}, {6, 3, 3}, {3, 6, 3} and {6, 3, 6} are con-
sidered. In each case, the faces are infinite (non-convex) polyhedra, whose vertices lie on
a horosphere. The vertex-figures may be either convex regular polyhedra, or again infinite
polyhedra with vertices on a horosphere.
Besides the locally finite tessellations of the hyperbolic n-space with convex regular
tiles (see Section 5) and the five honeycombs mentioned in the previous paragraph together
with their duals, we may consider the honeycombs {6, 3, 4}, {6, 3, 5} and their duals in hy-
perbolic 3-space, the honeycombs {3, 4, 3, 4} and {4, 3, 4, 3} in hyperbolic 4-space, and the
honeycombs {3, 3, 4, 3, 3}, {3, 3, 3, 4, 3}, {3, 4, 3, 3, 3}, {4, 3, 3, 4, 3} and {3, 4, 3, 3, 4} in
hyperbolic 5-space. Recently some examples of chiral quotients of these honeycombs were
found [10].
Problem 23 Determine the existence of infinite families of chiral quotients of all regular
honeycombs in hyperbolic 3-, 4- and 5-space.
One procedure of constructing polytopes of rank d is to determine whether it is possible
to assemble a d-polytope from a set of (d − 1)-polytopes as facets of a prescribed type.
Whenever the facets are all isomorphic, the d-polytope is known as an extension of the
(d− 1)-polytope.
D. Pellicer: Developments and open problems on chiral polytopes 347
In [53] it was proved that for every regular d-polytopeK there exists a universal regular
extension U(K) with the following properties.
(a) Every other regular extension of K is a quotient of the U(K);
(b) U(K) is an infinite polytope, and the last entry of its Schläfli type is∞;
(c) the automorphism group of U(K) is obtained from Γ(K) by adding an extra involu-
tory generator ρd satisfying relations (2.2).
The equivalent result for chiral polytopes was established in [59]. Specifically, it was
established that every chiral d-polytope K with regular facets admits a universal chiral ex-
tensionU ch(K) satisfying the equivalent to (a) and (b) as for the universal regular extension
of a regular polytope, with the property that the automorphism group of U ch(K) is obtained
from Γ(K) by adding an extra generator σd with the extra relations given by (2.4).
If K is orientably regular, the polytope obtained from Γ+(K) by adding a generator σd
and relations (2.4) is its universal regular cover U(K). In this sense, U ch(K) does not exist
for regular K. However, it is not known yet whether there exists a chiral polytope covering
all other chiral extensions of a regular polytope K.
Problem 24 Determine whether for every (any) regular polytope K there exists a chiral
extension which covers every extension of K.
The requirement of regularity for the facets of K is necessary for the existence of
U ch(K) since the facets of the extension are isomorphic to K. Therefore, this construc-
tion cannot be applied twice. However, if K is a chiral d-polytope with regular facets and
vertex-figures, we may obtain the infinite chiral (d + 2)-polytope (U(U(K)∗))∗, the dual
of the universal extension of the dual of the universal extension of K. This result can be
applied to chiral 4-polytopes with regular facets and vertex-figures (for example, those in
[47]) to obtain infinite chiral 5- and 6-polytopes.
The first finite chiral 5-polytopes were found in [14] with the aid of a computer. Later
similar methods were used to find finite chiral d-polytopes for d = 6, 7, 8 [12]. (A computer
based method was also used to construct the atlas in [24].) Then a proof of existence of
finite d-polytopes for every rank d ≥ 3 was given in [49], providing a positive answer to
Problem 14 in [60].
There are regular extensions of each regular polytope K. Furthermore, if K is finite, it
admits finite regular extensions (see for example [20] and [53]). The Schläfli type of any
equivelar extension of an equivelar polytope K will coincide on the first d− 1 entries with
the Schläfli type of K. It was determined in [48] that every regular polytope admits regular
extensions where the last entry of the Schlälfi type is an arbitrary even number pd. On
the other hand, Hartley proved in [23] that the hemicube does not admit regular extensions
with odd numbers as last entries on their Schläfli symbols.
Little is known about chiral extensions of regular or chiral polytopes. The main result
in this direction is the construction of U ch(K) given in [59]. It is natural then to ask the
following questions.
Problem 25 Does every chiral polytope K with regular facets admit a chiral extension
with prescribed last entry of the Schläfli symbol?
Problem 26 Does every chiral finite polytopeK with regular facets admit a finite chiral
extension with prescribed last entry of the Schläfli symbol?
348 Ars Math. Contemp. 5 (2012) 333–354
A necessary condition for a regular polytope to admit chiral extensions is orientability.
The only known general result in this direction is given in [49], where chiral extensions
were found for minimal regular covers of the so-called scattered chiral polytopes. This
was used to provide a recursive construction of chiral polytopes of all ranks higher than 3.
However, very few polytopes are regular covers of scattered chiral polytopes.
Problem 27 Does every orientably regular d-polytope admit a chiral extension (d ≥ 2)?
Problem 28 Does every orientably regular finite d-polytope admit a finite chiral exten-
sion (d ≥ 2)?
We recall that every chiral polytope K with regular facets admits an infinite chiral ex-
tension, namely U ch(K). It is interesting to determine whether this is true for orientably
regular polytopes.
Problem 29 Does every orientably regular polytope admit an infinite chiral extension?
Problems 27, 28 and 29 seem hard to answer, and some partial results would be of
interest. Chiral polytopes with a d-simplex as facet were found by Conder and Devillers
for d = 6, 7, 8 [12]. The next problem is a general version of the dual of Problem 13 of
[60].
Problem 30 Does every convex regular polytope admit a chiral extensions?
Given two (d−1)-polytopesK1 andK2, we say that a d-polytopeP is an amalgamation
of K1 and K2 whenever all of its facets are isomorphic to K1 and all of its vertex-figures
are isomorphic to K2 (see [37, Chapter 4]). In other words, an amalgamation of K1 and K2
is an extension of K1 where we are additionally prescribing the type of vertex-figure to be
K2.
In general it is hard to determine whether there exist amalgamations of two given poly-
topes K1 and K2. A necessary condition is that the vertex-figures of K1 are isomorphic to
the facets of K2, but this condition is not sufficient. Whenever there exists a chiral amal-
gamation of two polytopes K1 and K2 such that at least one K1 and K2 is chiral and the
other one is regular or chiral, there exists a universal amalgamation {K1,K2}ch with the
property that any other amalgamation of K1 and K2 is a quotient of it (see [57, Section 6]).
We include here the chiral part of Problem 8 and Problem 11 of [60].
Problem 31 Given regular (or chiral) (d − 1)-polytopes K1 and K2, does there exist a
chiral d-polytope P with faces isomorphic to K1 and vertex-figures isomorphic to K2?
Problem 32 Determine the group of the universal amalgamation {K1,K2}ch and give
conditions under which this is finite, whereK1 andK2 are regular or enantiomorphic forms
of chiral polytopes, not both regular.
A polytope K is said to be (combinatorially) flat whenever every facet contains ev-
ery vertex. The hemi-cube and hemi-octahedron are examples of flat polytopes. More
generally, a polytope is said to be (k,m)-flat whenever every m-face contains every k-
face (see [37, Section 4E]). A (k,m)-flat polytope is also a (l, n)-flat polytope for every
0 ≤ l ≤ k < m ≤ n ≤ d− 1.
Chiral flat 3-, 4- and 5-polytopes are known to exist. The polyhedron C7.1 in [11]
D. Pellicer: Developments and open problems on chiral polytopes 349
with Schläfli type {6, 9} is flat. In [14, Section 4] a dual pair or chiral flat 4-polytopes
with Schläfli types {3, 4, 4} and {4, 4, 3} are described. Furthermore, in [51, Section 6.1]
a chiral flat 4-polytope with Schläfli type {4, 3, 6q} is constructed for each q ≥ 2. An
example of a chiral (0, 2)-flat 5-polytope is given in [14, Section 5]. Its Schläfli type is
{3, 6, 3, 6}.
Problem 33 Does there exist a flat d-polytopes for every d ≥ 3?
Problem 34 Does there exist a (k,m)-flat d-polytopes for each triple (k,m, d) with
0 < k + 1 < m ≤ d− 1?
To conclude this section we discuss regular and chiral d-polytopes with the fewest
possible flags.
Let mrd and m
ch
d denote the minimum of the numbers of flags of all regular and chiral
d-polytopes respectively. Then clearly mrd ≤ (n + 1)! since the d-simplex has precisely
this number of flags. Recently mrd was established for all d, with the value of (4
d)/2
for d ≥ 9 [9]. The number mchd seems to grow much faster than mrd, however little is
known about it. The construction of chiral polytopes of ranks d ≥ 4 in [49] is recursive,
and therefore it does not provide the function itself. An upper bound provided by this
construction is given by the recursive function fch where fch(3) = 84 and fch(n + 1) =
[(2fch(n) + 1)(fch(n)!)
2]!. However this bound may be very far away from the value of
mchd .
Problem 35 Find the value of mchd as a function of d.
8 Realizations
Following [37, Section 5A], a realization of an abstract polytope K in a geometric space
S is a mapping β = β0 from the vertex set of K into S. This indicates which points of S
correspond to the vertices of the realization of K. The image of the remaining faces of K
are defined recursively in the following way. For i = 1, . . . , d − 1 define βi from the sets
of i-faces of K to the power set of the set of images in S of the (i − 1)-faces of K, such
that for each i-face F of K,
βiF = {βi−1G |G ⊆ F}.
We say that a realization is faithful if βi is one-to-one for all i. The image of K under a
realization is called a geometric polytope.
A symmetry of the geometric polytope P is an isometry of S that preserves P . A
realization P of a polytope K is said to be symmetric if each automorphism of K induces
a symmetry of P . Note that if P is not symmetric, then the symmetry group of P is
isomorphic to a proper subgroup of Γ(K).
A geometric polytope P is said to be geometrically regular if its symmetry group is
transitive on its flags. Whenever P is a faithful realization of the abstract polytope K, the
symmetry group of P is isomorphic to the automorphism group of K, implying that K is
also regular as an abstract polytope.
A geometric polytope P is said to be geometrically chiral if its symmetry group has
two orbits on its flags in such a way that adjacent flags are in distinct orbits. The symmetry
group of a faithful geometrically chiral realization P of an abstract polytope K is isomor-
phic to a subgroup of the automorphism group ofK of index at most 2. If the index is 2 then
350 Ars Math. Contemp. 5 (2012) 333–354
K is regular as an abstract polytope, and the automorphisms represented by the symmetries
are precisely the elements of Γ+(K). We also say that K admits a chiral realization. On
the other hand, if the index is 1 then K is chiral as an abstract polytope. Consequently, if a
polytope admits a chiral realization it must be either chiral or orientably regular.
The study of realizations of polytopes involves two main branches: the description of
all realizations of a given polytope, and the determination of all polytopes realizable on a
given space.
For regular realizations, the first branch was thoroughly studied and is described in [37,
Chapter 5]. In the finite case, the set of all regular realizations of a given regular polytope
P has the structure of a convex cone. The structure of the cone is determined by the family
of irreducible orthogonal representations of the automorphism group of P . No such theory
has been developed for chiral realizations of regular or chiral polytopes. We include here
Problem 21 of [60].
Problem 36 Develop the details of the realization theory for chiral polytopes.
In [36] McMullen and Schulte provide a survey of all the geometrically regular discrete
polyhedra in E3 (see also [37, Section 7E]). There are 18 finite regular polyhedra, 6 infinite
regular polyhedra in the plane, and 24 infinite geometrically regular polyhedra such that the
affine hull of their vertices is E3. The latter are divided in two families: 12 blended poly-
hedra having reducible symmetry groups (the groups permutes a linear subspace of E3 and
its translates), and 12 pure polyhedra having irreducible symmetry groups. Furthermore,
they also classify the 8 regular 4-polytopes in E3.
A geometric realizationP of an abstract d-polytopeK is said to be of full rank whenever
either P is finite and its ambient space is the Euclidean d-space Ed, or P is infinite and its
ambient space is Ed−1. Full rank regular polytopes were classified in [33].
Discrete (finite) regular polytopes in the projective d-space Pd are in correspondence
with finite polytopes in the Eudlidean (d + 1)-space. All discrete polyhedra with planar
faces in the projective space P3 were found in [1] and [2]. The enumeration was completed
in [34], where the author determined all regular finite polyhedra in the Euclidean 4-space
E4. The classification of discrete regular 4-apeirotopes in E4 was achieved in [35].
Much less has been achieved for geometrically chiral polytopes.
It was proved in [33] that there are no geometric chiral polytopes of full rank. In
particular, there are no finite chiral polyhedra in E3.
The complete classification of discrete chiral geometric polyhedra in E3 was achieved
in [55] and [56]. Every chiral polyhedron in E3 belongs to one of the 6 infinite families
{P (a, b)}, {Q(c, d)}, {Q(c, d)∗}, {P1(a, b)}, {P2(c, d)} and {P3(c, d)}, each indexed by
two parameters.
The families P (a, b), Q(c, d) and Q(c, d)∗ contain polyhedra with finite faces and
Schläfli types {6, 6}, {4, 6} and {6, 4} respectively. Any of the parameters a, b, c and d
may be 0; otherwise a and c are rational multiples of the real parameters b and d respec-
tively. Furthermore, P (a, b), Q(c, d) and Q(c, d)∗ are respectively similar to P (sa, sb),
Q(sc, sd) and Q(sc, sd)∗ for every real s 6= 0. Therefore the parameters a, b, c and d can
be thought as integers. For pairs (a, b) and (a′, b′) of relatively prime integers with a ≥ 0
and (a, b) 6= ±(a′, b′), the polyhedron P (a, b) is not isomorphic (as an abstract polytope)
to P (a′, b′). Similarly, for (c, d) 6= ±(c′, d′) the polyhedra Q(c, d) and Q(c, d)∗ are not
isomorphic to Q(c′, d′) and Q(c′, d′)∗ respectively.
The familiesP1(a, b), P2(c, d) andP3(c, d) contain polyhedra with infinite helical faces
D. Pellicer: Developments and open problems on chiral polytopes 351
and Schläfli type {∞, 3}, {∞, 3} and {∞, 4} respectively. The parameters a, b, c and d are
any real numbers, and hence the set of parameters is not discrete (like it is for the families
containing polyhedra with finite faces).
In [50] it was proved that all chiral polyhedra P (a, b), Q(c, d) and Q(c, d)∗ are chiral
as abstract polytopes. This provides infinitely many examples of infinite chiral polyhedra
with the same Schläfli type.
On the other hand, all chiral polyhedra Pi(a, b), i = 1, 2, 3 are regular as abstract poly-
topes. Furthermore, all geometrically chiral instances of P1(a, b), P2(c, d) and P3(c, d) are
respectively isomorphic to the regular polyhedra {∞, 3}(a), {∞, 3}(b) and the orientable
double cover of {∞, 4}·,∗3 (in the notation of [37, Section 7E]), where a continuous change
on the parameters corresponds to a continuous movement of the vertices and edges of the
polyhedron. This provided the first examples of faithful discrete chiral realizations of reg-
ular polytopes.
The only known classifications of chiral polytopes in a given Euclidean space arise
from the results just mentioned. This suggests the following problem.
Problem 37 Classify all finite and infinite chiral k-polytopes in Rd (where k < d for
finite polytopes, and k ≤ d for infinite ones).
Of special interest is the case d = 4 for both, finite rank 3 polytopes and infinite rank
4-polytopes. These two instances are not known for chiral, but have been settled for the
regular case.
Problem 38 Classify all finite chiral polyhedra in E4.
Problem 39 Classify all infinite chiral 4-polytopes in E4.
9 Acknowledgements
The author is very grateful to Marston Conder, Egon Schulte, and Asia Weiss for some
helpful ideas and discussion. He also wishes to thank the anonymous referees for a number
of helpful suggestions.
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