The classification of quasiregular polyhedra of genus 2
The Classification of QuasiRegular Polyhedra of Genus 2
Reinhard Franz 0
Daniel H u s o n 0
0 Fakult/it fiir Mathematik, Universit/it Bielefeld,Universit~itsstraBe , W4800 Bielefeld 1, Federal Republic of Germany
The method of chamber systems is used to provide a complete list of all possible tessellations of the closed, orientable surface of genus 2 by (topological) ngons and mgons (n, m > 2) satisfying a certain local symmetry condition. Using a computer program it is shown that (up to homeomorphism) there are precisely 379 such quasiregular polyhedra. S. Bilinski constructed the first one for each of the 17 possible combinations of m and ngons using geometrical methods. It is the intention of the authors to demonstrate the usefulness and suitability of chamber systems in dealing with problems of the above type.

1. QuasiRegularPolyhedra
A (topological) polyhedron ~ (of genus p) is a compact, orientable 2manifold (in
N:a) of genus p divided into simply connected open regions by a finite n u m b e r of
arcs (and simple closed curves) called edges. Such a region, together with its
boundary, is called a face. Edges meet only at their endpoints called vertices and
each vertex is incident to at least three edges (where loops are counted twice) (see
[
16
]).
As in [
2
], a (topological) polyhedron :~ is called (locally) quasiregular if each
of its vertexcycles has the form
(1)
with n > m > 2 and s > 1. In other words, when "going a r o u n d " a vertex, we
alternately encounter medged and nedged faces (exactly s of each). We call g*
globally quasiregular if its a u t o m o r p h i s m group acts transitively on its set of edges.
There is a simple construction of globally quasiregular polyhedra which we
would like to mention: Start with a regular m a p J / o f type {n, m}. Truncate ~t';
that is, " c u t off" the vertices of ~t' up to the midpoints of the edges; or equivalently,
join up the midpoints of the edges of J/¢. This gives a globally quasiregular
p o l y h e d r o n with parameters n and m and s = 2. For genus y = 2 this gives several
p o l y h e d r a for free (see p. 140 of [
4
]).
F o r a quasiregular p o l y h e d r o n ~ of genus p, with ct0 vertices, c~t edges, and
~2 faces (qm o f which have m edges and q, o f which have n edges), the following
system o f D i o p h a n t i n e equations holds:
SCto= m q m = n q , = ~1,
0~2 = qm + q,,
(2)
and
2(1  p) = cto  cq + ~2
(Euler formula),
with m, qm, n, q,, a n d s as above. N o t e that the n u m b e r s cto, ~l, and ~2 are
completely determined by the parameters m, q,,, n, q,, a n d s.
F o r p = 2 there are exactly 17 choices of positive integers m, qm, n, q,, a n d s
that satisfy (2). In [
2
] and [
3
] Bilinski shows that, for each o f these 17 solutions,
a c o r r e s p o n d i n g p o l y h e d r o n ~ exists. In [
9
] quasiregular p o l y h e d r a are
discussed in terms of their associated " c h a m b e r systems." There it is shown, in the
case qm = 4, m = 3, q. = 3, n = 4, s = 4, h o w c h a m b e r systems can be used to
determine all the c o r r e s p o n d i n g h o m e o m o r p h i s m classes of (oriented)
quasiregular polyhedra. 1
This paper describes an algorithm based on the chambersystem approach. By
p r o d u c i n g all a p p r o p r i a t e c h a m b e r systems, we obtain the new result that there
exist precisely 379 classes of quasiregular p o l y h e d r a of genus 2. Exactly ten of
these classes consist of globally quasiregular polyhedra. Representatives for the
latter are displayed in Figure 4.
2. The Chamber System of a Polyhedron
T o define the c h a m b e r system of a p o l y h e d r o n ,# of genus p, let us consider a
b a r y c e n t r i c s u b d i v i s i o n o f # ' (see Fig. 1). Such a triangulation ~ ' o f ~ can be
obtained b y first a d d i n g to the interior of each edge a n d each face of ~ exactly
one new vertex a n d then, for every face, connecting, via new edges, the vertex
placed in that face to all the vertices contained in the b o u n d a r y of that face. T h u s
~Though not explicitly mentioned there, the ten systems (~;~,fl) listed in [
9
] represent all
homeomorphism classes of oriented quasiregular polyhedra of the type considered there. Six of the
ten oriented polyhedra are homeomorphic to their "mirror image" (by an orientationpreserving
homeomorphism!), the polyhedra 2a2b2c  I, 2a2b2c  2 and the polyhedra 26262d  2, 26262d  3
are mirror images of each other. Hence, there are precisely eight homeomorphism classes of such
potyhdedra if orientation is neglected.
.
t" /
o, s l"s
~ ' has three types of vertices and we call a vertex of ~ ' either a 0, 1, or 2vertex
depending on whether it lies in (the interior of) a vertex, an edge, or a face of P.
Note that every face of ~ ' possesses exactly one vertex of each of the three types.
The intersection of two different faces of ~ ' is either empty, a vertex, or it
consists of exactly one edge connecting two vertices of different types. By the color
of an edge e of ~ ' we mean the number k with the property that e contains a
/vertex and a jvertex but not a kvertex, where {i,j, k} = {0, 1, 2}.
Now the chamber system G(~) = (~r, o//)of ~ is defined as the dual of the graph
given by the 1skeleton of ~". That is, the vertex set ~ of G consists of the faces
of ~ ' called chambers and two different vertices x, y e ~ are connected in G by an
edge of color k (a kedge) in q / i f and only if, as faces of ~ ' , they intersect in an
edge of color k. Note that different barycentric subdivisions of ~ yield isomorphic
chamber systems.
It follows from our construction that each vertex of the chamber system of a
polyhedron ~ is incident to exactly one/edge (for all i ~ {0, 1, 2}). Furthermore
the graph does not contain any loops (since, as a 2manifold, ~ has no boundary),
is bipartite (since ~ is orientable), and has genus p. Removing all the edges of
color k (k e {0, 1, 2}) yields a spanning (planar) subgraph whose connected
components (called ijcomponents) are cycles consisting of alternating i and jedges
({i,j, k} = {0, 1, 2}), called ijcycles. Note that there is a canonical onetoone
correspondence between the 0.1, 02, and l2cycles and the faces, edges, and
vertices of ~ , respectively. Furthermore, we have the following, theorem which
allows a classification of quasiregular polyhedra in terms of their associated
chamber systems (see [
9
]):
Theorem A. There exists a onetoone correspondence between the homeomorphism
classes of quasiregular polyhedra of genus p as defined above and the isomorphism
classes of finite, connected, edgecolored, bipartite graphs
G = (X, q / = {{a,b}la, b ~ X } × {0, 1,2})
satisfying the following conditions:
(P1) There exist natural numbers m, n with 2 < m < n such that G consists of
exactly q,, Olcycles of cardinality 2m and q, Olcycles of cardinality 2n.
(P2) Every O2cycle is of cardinality 4.
(P3) All 12cycles have the same eardinality, say 4s, which is larger than 4.
(P4) Every O2cycte intersects exactly one Olcycle of cardinality 2n and one
of cardinality 2m.
(P5) The parameters q,,, m, q,, n, and s satisfy the equation
2(1  p) = mq___~_ mqm + q,, + q,.
s
Remark. Since the numerical invariants m, q,,, n, q,, and s of the quasiregular
polyhedron ~ coincide with the numerical invariants of the associated chamber
system G(~) defined in (P1), (P2), and (P3) they are denoted by the same symbols.
3. Algorithm POLYHEDRA
Theorem A shows that the two following problems can be translated into each
other:
(a) Producing all homeomorphism classes of quasiregular polyhedra of genus
p.
(b) Enumerating all isomorphism classes of connected arccolored bipartite
graphs G = (X, q / c {{a, b} [a, b e X) × {0, 1, 2}) satisfying conditions
(P1)(P5).
In this paper we formulate a simple algorithm which generates "standard
representatives" for all isomorphism classes of chamber systems corresponding to
a given set of parameters m, qm, n, q,, S (n > m > 2, s > I), and p satisfying (2).
We start with a simple version of the algorithm based on the "bruteforce" method
of generating permutations (see [
19
]), which is too slow to produce complete
results. Then we discuss how to speed up the algorithm.
Picture Algorithm P O L Y H E D R A operating on a single graph G = (X, ~),
successively adding and deleting edges in that graph, and, whenever appropriate,
producing output by "printing" a copy of the graph in its current state.
Before the algorithm is applied, graph G is initialized as follows. Give graph
G exactly N = 4q,nm = 4q,,n vertices which, for the sake of simplicity, are identified
with the numbers 1. . . . , N, thus X = {1. . . . . N}. Set ~¢:= {1,..., N/2} and g~:=
{N/2 + 1. . . . . N}. Next we define the 0 and 1edges in q/(note that our algorithm
will leave these edges fixed): Any two vertices i and i + 1 are to be connected by
a 0edge if i is odd. Then 1edges are used to create q,~ consecutive 01cycles of
cardinality 2m, and q, such cycles of cardinality 2n in G (property (P1)!). Figure
( D ........( D  . . . . ® ........® . . . . ( b ........®
d .......® . . . . ® ........, 9 . . . . ® ........@
2 indicates exactly how this is to be done. Note that the vertex set of d (or ~ )
induces a graph consisting of q,, (or qn) consecutive 01cycles of cardinality m (or
n, respectively).
G r a p h G is now ready to have the algorithm applied to it. Before proceeding
to define the algorithm, we briefly introduce some more notation. To 2connect
two vertices i and j means to connect them by a 2edge. A vertex i that is not
2connected to any other vertex is called free. Note that to help ensure property
(P4) the algorithm may only 2connect two vertices i and j, i < j, if i • d a n d j • ~ .
Because we are only interested in bipartite graphs, we can always assume that
two 2connected vertices i and j are either both even or both odd. Furthermore,
if, while running the algorithm, we want to 2connect an odd vertex i • d to an
odd vertex j • :~, then to ensure property (P2) the vertices i + 1 and j + 1 must
also be 2connected. An ordered pair (i, i + 1) of vertices (in d or ~ ) is called a
Opair (in ~d or ~ , respectively) if they are joined by a 0edge, which is the case
whenever i is odd. We 2connect two pairs (i, i + 1) and ( j , j + 1) by 2connecting
i to j and i + 1 to j + 1. A 0pair of free vertices is called free. Note that initially
all 0pairs in G are free. Finally, a 0pair (i, i + 1) is called a head if i is the smallest
vertex for some 01cycle.
Algorithm P O L Y H E D R A
Input: Initially (i.e., at recursion level 0) graph G as defined above, all vertices free.
At recursion level t: G r a p h G with 2t new 2edges added.
Output: One representative for each isomorphism class of connected, bipartite
graphs that satisfy (P1)(P5).
{
(,)
}
if d contains a free 0pair then {
let A be the smallest free 0pair in d
for each free 0pair B in ~ (in ascending order) do {
call C O N N E C T (A, B)
}
else if the graph G = (~, ~ ) is "OK" then print it
}
}
Procedure C O N N E C T (A, B)
(
2connect A and B in 0//
call P O L Y H E D R A
disconnect A and B in ~//
In the else if line graph G is " O K " if it has the following properties:
(OK1) Every 0pair has been 2connected.
(OK2) Every 12cycle is of cardinality 4s.
(OK3) The graph is not isomorphic to any graph previously printed out by
the algorithm.
4. Speedingup the Algorithm
While " r u n n i n g " the recursive algorithm P O L Y H E D R A let us indicate the current
"level of recursion" by a subscript t. So initially Algorithm P O L Y H E D R A o is
called and for the tth nested call of the algorithm we write P O L Y H E D R A t.
N o w assume that Algorithm P O L Y H E D R A is running and that it has already
2connected the first t < N/4 of the 0pairs in d and that it has just entered
P O L Y H E D R A t (see Fig. 3). Algorithm P O L Y H E D R A , takes the smallest free
0pair A in ~¢ and 2connects it to the smallest free 0pair B in ~ . It then goes
on to call P O L Y H E D R A t + 1 (which m a y or m a y not lead to some graphs being
printed out). Once this call has been completed, the 0pairs A and B are
disconnected. Algorithm P O L Y H E D R A t would now go on to select the next free
0pair B' > B in ~ and then to reenter the main loop to 2connect A and B', etc.
However, if A is a head, then repeating the main loop will not produce any graphs
not isomorphic to ones already printed during the first execution of the main loop.
The reason for this is that, with respect to isomorphism, the 0pair A and all
following 0pairs A' in d "play the same role": each is a free 0pair contained in
Fig. 3. An example of the typiea.lstate of a graph G as the algorithm enters POLYHEDRA, In this
ease t = 4 and the algorithm is just about to 2connect the 0pair (9, 10). Vertices are indicated by
smallcirclesand 0 (1or 2)edgesare drawn as linesthat are dotted (dashedor unbroken,respectively).
The Classificationof QuasiRegular Polyhedraof Genus 2
a 01cycle of cardinality 2m, which consists entirely of free 0pairs. Hence, if we
were to 2connect A to B', then in P O L Y H E D R A t ÷ 1 some A ' > A would be
2connected to B and we would not obtain a new isomorphism type of graph.
N o w consider Algorithm P O L Y H E D R A t under the assumption that A is not a
head. If at some point the main loop chooses a free 0pair B' in ~ that is a head,
then the above argument applies similarly to B' and again, after performing
P O L Y H E D R A t + 1 with A and B' 2connected, Algorithm P O L Y H E D R A t can be
aborted.
So we have seen that it suffices if o u r algorithm considers only graphs G that
have the following two properties for any pair of 0pairs A and A' in d 2connected
to 0pairs called B and B' in ~ :
(a) If A is a head and A < A', then B < B'.
(b) If B is a head and B < B', then A < A'.
We can ensure that only such graphs are considered by adding the following
line to the bottom of the main loop:
if A is a head or B is a head then end
where the statement end means end the "current incarnation" of P O L Y H E D R A
(but not the whole algorithm).
We take a further step to speed up the algorithm. It consists of making the
procedure C O N N E C T ensure that, from the beginning, the algorithm only
considers and produces graphs whose l2cycles have the correct cardinality. F o r
any intermediate graph produced by the algorithm, let ~(i) denote the
12component that contains the vertex i. Any such set ~(i) is either a 12cycle, or a
chain using alternating 1 and 2edges (a 12chain). Let i e d and j e ~ be two
vertices not incident to a 2edge. Then ~f(i)and ~(j) are 12chains. If ~(i) n ~(j)
~ , then oK(i)= c~(j) and 2connecting i and j produces a 12cycle of cardinality
#~(i). Otherwise, 2connecting i and j produces a 12chain of cardinality
:~ ~(i) + ~ ~(j).
Replace the entire procedure C O N N E C T by this new version:
Procedure C O N N E C T ' (A, B)
{
assume A = (i, i + 1) and B = (j, j + 1)
if ~(/) = c~(j) and ~ c~(/) = 4s
or ~(/) ~ c~(j) and ~ c¢(/) + ~ c~(j) < 4s then {
2connect i and j in ~'
i f ~ ( i + 1) = c~(j + 1) and 4~c~(i + 1) = 4s
or c~(i + 1) ~ T(j + 1) and ~ ( i + 1) + # T ( j + 1) < 4s then {
2connect i + 1 and j + 1 in q/
call P O L Y H E D R A
disconnect i + 1 and j + 1
}
disconnect i and j
Finally, it follows from properties (a) and (b) that adding the lines
if A is not the smallest 0pair in ~¢ then
if A is a head and the smallest free 0pair in ~ is a head then end
to P O L Y H E D R A directly under the line marked (,) ensures that the graphs
considered by the algorithm are connected.
After making all the the above modifications to the algorithm, in the else if
line only the property (OK3) has to be checked, (OK1) and (OK2) are necessarily
satisfied. Here is the modified algorithm:
Algorithm P O L Y H E D R A '
Input: Initially (i.e., at recursion level 0) graph G as defined above, all vertices free.
At recursion level t: graph G with 2t new 2edges added.
Output: One representative for each isomorphism class of connected, bipartite
graphs that satisfy (P1)(P5).
if d contains a free 0pair then {
let A be the smallest free 0pair in d
if A is not the smallest 0pair in d then
if A is a head and the smallest free 0pair in ~ is a head then
end
for each free 0pair B in ~ (in ascending order) do {
call C O N N E C T ' (A, B)
if A is a head or B is a head then end
else if the graph G = (~, ~/) is not isomorphic to one already printed
then print it
5. Results and Comments
Implementing the final version of Algorithm P O L Y H E D R A on a computer yields
the following results:
Theorem B. There are exactly 379 homeomorphism classes of quasiregular
polyhedra Of genus 2, ten of which are globally quasiregular and 225 of which have
orientationreversinxd automorphisms (see Table 1).
Figure 4 displays a representative of each of the ten classes of globally
quasiregular polyhedra of genus 2.
A remark on the computation time: our implementation of Algorithm
POLYHEDRA' needed approximately 4000 minutes for the computation of the 90 classes
12
162
161
17
Fig. 4. Representatives of all ten classes of globally quasiregular polyhedra of genus 2.
* Columns 26 contain the 17 solutions of the system of Diophantine
equations (2) for p = 2, N is the cardinality of the vertex sets, Q is the
n u m b e r of h o m e o m o r p h i s m classes of the corresponding quasiregular
polyhedra, Q+ is the n u m b e r of h o m e o m o r p h i s m classes of the corresponding
oriented quasiregular polyhedra, and G is the number of globally
quasiregular polyhedra.
of quasiregular polyhedra in Case 1, where the associated chamber systems have
N = 336 vertices.
Modified versions of Algorithm POLYHEDRA are presently being used to
solve classification problems concerning more general polyhedra. Similar
algorithms based on the theory of "Delaney symbols" developed by Dress and others
(see, e.g., [
6
], [
8
], [101, and [
13
]) have been used successfully to enumerate
periodic tilings of the plane (see, e.g., [111, [121, and [171). Work is presently being
done on developing computer graphics programs that automatically draw the
associated structure to a given Delaney symbol or chamber system. Following
Tutte (see [
20
]) recursion formulas counting homeomorphism classes of various
types of"flagged" regular polyhedra can also be developed (see [11, [
7
], [
14
], and
[
15
]).
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