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Enumeration of platonic maps on the torus
Discrete Mathematics 61 (1986) 71-83 North-Holland
71
E N U M E R A T I O N OF PLATONIC MAPS ON THE TORUS Winfried KURTH Institut f-fir Mathematik, Technische Universitiit Clausthal, D-3392 Clausthal-Zellerfeld, Fed. Rep. Germany Received 28 September 1984 Revised 29 May 1985 Certain maps (graph embeddings) on the torus are counted, namely those with all faces triangles, respectively quadrilaterals, resp. hexagons, and all vertices having the same degree (which then must be 6, 4 or 3, resp.). These are the toroidal analogues of the spherical maps corresponding to the five Platonic solids. Techniques from combinatorics and number theory are applied to obtain the results.
1. I n t r o d u c t i o n
A map K on a torus T is an embedding of a finite graph G on T, such that each component of T \ G (each face of K) is homeomorphic to a disc. G may have loops and multiple edges. By the dual K* of the map K we mean the map which results if we take a vertex for each face of K and an edge for each edge of K, joining two vertices of K* whose corresponding faces are incident along an edge of K. We call a map K Platonic of type {p, q } if all vertices of K are of degree q and all vertices of K* are of degree p. The term "Platonic" is used since the maps with the above property are the natural generalizations of the spherical maps corresponding to the Platonic solids. In the literature the terminology varies between "regular" [1, 2, 10], "quasiregular" [17], "uniform" [27], "equivelar" [19] and "Platonic" [11]. Platonic maps on arbitrary closed surfaces are constructed and investigated in [10]. If a map K is Platonic of type {p, q}, then K* is Platonic of type {q, p}. From Euler's formula, together with a simple counting argument, it follows that the only types of Platonic maps on the torus are {3, 6}, {4, 4} and {6, 3}. Two maps on the torus are called isomorphic if there exists an autohomeomorphism of the torus which sends vertices to vertices, edges to edges and faces to faces and preserves incidence. In most instances maps are considered only up to isomorphism. Let Tp.q(V) denote the number of Platonic maps of type {p, q} with exactly v vertices on the torus. In this paper we develop simple formulae for the calculation of Tp,q(V). If a map of type {6, 3} on the torus has v vertices and f faces, then 0012-365X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
72
W. Kurth
v = 2f, and because of duality the number of maps of type {6, 3} with f = ½v faces equals the number of maps of type {3, 6} with ½v vertices. Hence T6,3(v) = ~ Ta,6(½v), t0,
if 2Iv, if2 ,~ v.
Therefore it is sufficient to determine T4,4(v) and T3,6(v). For this purpose we introduce a standard form of Platonic maps on the torus in Section 2. In Section 3 we use Burnside's Lemma to get first expressions for T4,4(v) and T3,6(v). These are simplified in Section 4 by some number theory yielding our main result (Theorem 4.3). In Section 5 we briefly discuss some stronger versions of regularity and the corresponding numbers of maps on the toms. Maps can also be defined in entirely combinatorial terms. This is carried out in [17] and [24], for example. However, the way a map is defined is not essential for our arguments. There is a considerable literature on polyhedral realizations of maps in Euclidean 3-space. We mention [3, 13, 19], where further references can be found. An especially remarkable fact is the existence of toroidal polyhedra which bear maps of type {3, 6} and are vertex-transitive under their symmetry groups. Such (necessarily everywhere fiat) tori have been found by U. Brehm [4] and, independently, by H. Leitzke [18]. A construction is given in [14]. We do not discuss polyhedral realizations in this paper.
2. c-maps
Let (]4 and G3 be tilings of the plane R 2 with squares or triangles, respectively, induced by the infinite graphs {(x, y) ER2:X E~' v y E~ '}
and {(x, y) e R 2 : X C:7/ v y E Z v x + y E~'_}.
We may interprete G3 as a tiling with regular triangles if we let the coordinate axes intersect at an angle of in. This enables us to represent all possible automorphisms of (93 by isometries, as in the case of (]4. In the following, the index j stands for 3 or 4. The symmetry group ~ of Gs is the group of all isometrics which map Gs onto itself. It is obvious that ~4 ~: ~ . A translation T e ~ acts o n R E in the way
(x, y)--, (x, y) + (tl, t2), with fixed integers tl, t2. We shall identify T with the vector (tl, t2). Let S, T e ~ be two linearly independent translations and (S, T) the group generated by them. "lifts group is dearly isomorphic to (Z 2, +). It acts on R 2 and
73
Enumeration o f Platonic maps on the torus
,
v
m
ff
,
, :
(a)
(b) Fig. 1
on the embedded tiling Gj, generating the torus R2/(S, T) as its orbit space. This torus can be represented by the parallelogram spanned by S and T (which is a fundamental region of (S, T)) if opposite sides are identified. (Fig. 1 shows two examples with j = 3.) Obviously, GJ(S, T) is a Platonic map of type {3, 6} or {4, 4}, respectively, embedded into this toms. We call a map given in this form a c-map ('c' from 'coordinates') and denote it by Mj(A), if A is the 2 x 2 matrix with columns S and T:
A = (Sl
S2
tl)
t2
for S = (SI, S2), T = (tl, t2).
Fig. l(a) illustrates the c-map M
2
1
, ,s ows
1
-2
The number v of vertices of a c-map Ms(A) is clearly given by v = Idet AI. We call two c-maps isomorphic if they are isomorphic as maps (see Section 1). Two c-maps are equal if the orbits of R 2 under the action of the corresponding groups are identical as sets. Note that the c-maps of Figs. l(a) and (b) are isomorphic, but not equal. The following conclusion is almost immediate (and can be found in [15, p. 27]): Lemma 2.1. Let A and B be 2 x 2 matrices with integral entries. The c-maps
Mj(A) and Mj(B) are equal if and only if there exists a unimodular matrix U with integral entries satisfying A U = B. The matrix representation of a c-map becomes unique if only matrices (~ m) with the restrictions 0 ~
74
W. Kurth
?
" " r ~'
¢ ~
i~
xJ
f'~
")
t"'%
Fig. 2
form, see [21].) Clearly, v = nr. We call the integers n, m and r the characteristic parameters of the c-map. If (x, y) - (v, w) means (x, y) - (v, w) ~ (S, T), S and T forming the columns of A, the combinatorial meaning of the characteristic parameters of the c-map Mj(A) is easily seen to be as follows: n is the smallest positive integer satisfying (0, 0) - (n, 0), r is the smallest positive integer for which there exists an integer x with (0, 0 ) - (x, r), m is the smallest non-negative integer satisfying (0, 0) (m, r). This is demonstrated in Fig. 2, where the points equivalent to (0, 0) are marked with circles. Lemma 2.2. Let
A=(~
t2tl)
and
A=detA.
The characteristic parameters n, m, r o f the c-map Mj(A ) are the unique solutions o f the following system o f congruences under the restrictions nr = [A], 0 <~m < n,
0
(,)
nt2 =--0 (mod A),
mt2 ----rtl (mod A),
ns2-= 0 (mod A),
rsl = ms2 (mod A).
Proof. Taking B = ( g m) in Lemma 2.1, we see that there is a uniquely determined unimodular matrix U=(Uik), i, k = 1, 2, such that A U = B . By solving this matrix equation, we can express the numbers Uik in terms of the entries of A and B. The condition that these numbers are integers can be put into the congruences (.). The further condition Idet U[ = 1 is equivalent to nr = IAI. On the other hand, (*) together with the above restrictions implies that n, m and r are the (uniquely determined) characteristic parameters of Mj(A). []
Enumeration of Platonic maps on the torus
75
Until now, we have left open the question whether every Platonic map on a toms is isomorphic to a c-map. Indeed, this is the case. Theorem 2.3. Every Platonic map o f type {3, 6} or {4, 4} on the torus is isomorphic to some c-map Mj(A ) (j = 3 or j = 4, respectively).
We give only the outline of a proof. Let K be a Platonic map of type {3, 6} or {4, 4} on the toms. If the torus is represented by a parallelogram, we can cover the plane with copies of this parallelogram and get a tiling of the plane with some triangular or tetragonal tiles, respectively. The tiles need not to be regular, but there is always a homeomorphism mapping the tiling onto Gj. The translations along the sides of the original parallelogram are transformed into translations of Gj, and they generate a c-map which is isomorphic to the original map K. Altshuler [2] and Negami [20, pp. 168-171] give detailed proofs of this theorem for the case j = 3. They both specify the c-map M3(A) to be given by its characteristic parameters. (Their proofs follow a way slightly different from ours.) Cf. Hutchinson [16, p. 43] for an extension to Kepler graphs on the toms. A certain generalization to other surfaces can be found in [10] and [17]. Our problem of counting Platonic maps on the toms is now reduced to the enumeration of classes of isomorphic c-maps. Every isomorphism between two c-maps can be extended to the covering plane and induces an isometry belonging to ~ and having essentially the same effect on the c-map as the original isomorphism. E.g. the isomorphism illustrated in Fig. 1 corresponds to a rotation about the origin. Some symmetries of Gj are of minor interest, because they leave all c-maps unchanged. These are just the translations (x, y) and the half-turns with centres (½x, ½y), where x, y e Z. They form a normal subgroup of ~ which will be denoted by ~ . Symmetries lying in the same coset with respect to ~ have the same effect on c-maps. Therefore it is convenient to use the elements of the factor group 5j = ~ / ~ instead of the symmetries themselves. We will refer to the elements of 5j as isomorphisms (of c-maps), having in mind that they are indeed whole classes of isomorphisms. It is easily verified that 54 is the four-group (~2 X (~2, and ot~3 the symmetric group 5e3. The elements of 54 and 53, together with representatives from ~4 or ~3 and their effects on a point (x, y) ~ R 2, are listed in Table 1. The action of tP5 is illustrated in Fig. 1. (b) is the image of (a). Now we are able to determine the characteristic parameters of all isomorphic images of a given c-map by applying Lemma 2.2 to the transformed generating translations. In each of the 4 + 6 cases, there is a system of congruences relating the new parameters to the original ones. (In the case of ,-~3, this is the essence of Negami's Theorem 3.6 [20].) We shall use these relations to determine the c-maps which are invariant under a given isomorphism.
76
W. Kurth Table 1. Isomorphisms of c-maps ~4
Element
Representative from qd4
Image of (x, y)
I/)1 lp2
Identity Reflection in the line y = 0 Tetragonal rotation Reflection in the line x = y
(x, y) (x, - y ) ( - y , x) 0', x)
Representative from %
Image of (x, y)
Identity Reflection in the line y = 0 Reflection in the line x = 0 Reflection in the line - x = y Hexagonal rotation Trigonal rotation
(x, y) (x + y, - y ) ( - x , x + y) (-y, -x) ( - y , x + y) ( - x - y, x)
1/)3
'~4
~¢3 Element q01 q02 q% ~4 tp5 q~6
3. Enumeration of Platonic maps
To formulate the following results we need some number-theoretic functions. Let tr and fl be functions on the positive integers. The convolution product tr * fl is defined by
din
~' is called multiplicative if gcd(m, n) = 1 implies ol(mn) = o~(m)o~(n). If a' and fl are multiplicative then e * f l is multiplicative, too. (See [22; 23] for details on convolution.) The following functions are defined for every positive integer n (j and k are integers). l(n) = 1, :tk(n)=
t(n) = n, 1, 0,
ifkln i f k Jf n,
0, x 3 ( n ) ---- +1,
if31n if n - +1 (mod 3),
{ 0, +1,
if2ln if n--- +1 (mod 4),
x4(n) =
1, if n is a square y(n) = 0, otherwise, pj.k(n) = [{X e 7/: O<_x < n ^ x 2 + jx + k - 0 (mod n)}l, 3=1.1,
1;k = l * ~ k ,
o=l*t.
77
Enumeration of Platonic maps on the torus
pj, k(n) is the number of residues solving x 2 + jx + k =- 0 (mod n). We shall need only P0,1, P0,-1 and P1,1. 17k(n) is the number of divisors of n, including 1 and n, which are divisible by k. tr(n) is the sum of the divisors of n. Clearly, the functions 1, t, Z3, Z4, ]', 17and cr are multiplicative. So is Pj, k. (See e.g. [15, p. 96].) We need this fact in Section 4. [r] denotes the greatest integer not exceeding r. The following lemma will be the basis for our main theorem. Lemma 3.1. (a) The number of Platonic maps of type {4, 4} on the torus with
exactly v vertices is T4,4(13 ) "-" 1(or(v) + 17(!/) -Jr- 172(v) -J¢-( ~ * P0,1)(v) --t- (~:'g p0,_l)(v)). (b) The number o f Platonic maps of type {3, 6} on the torus with exactly v
vertices is T3,6(u ) : l(o'(o) q- 3(17(v) - 172(o) + 2174(v)) at- 2 ( r * Pl,1)(v)). Proof. Let us consider the set
Sj(v)=
0
"n,m, reT/AO<~m
~ = 3 , 4).
It contains all c-maps with v vertices of type {3, 6} or {4, 4}, respectively. For each divisor n of v there are n possible values of m. Hence St(v ) has a(v) elements in both cases, j = 3 and j = 4. Some of these elements are isomorphic by some isomorphism belonging to 5~. Enumerating the Platonic maps with v vertices, we have to take these isomorphisms into account. Each element of 5j induces a permutation on St(v ). The orbits of St(v ) with respect to the group action correspond to the isomorphism classes of Platonic maps with v vertices and are to be counted. According to Burnside's Lemma (see e.g. [5]), the number of orbits equals the average number of invariant c-maps of the group elements. Thus we have to determine the number,.of invariant c-maps for each ~i (i = 1 , . . . , 4) and ~i (i = 1 , . . . , 6). Let us denote this number by c(V2i, v), resp. c(q~i, v). We begin with the elements of & . The identity element leaves each c-map invariant. Hence c(V/1, v) = a(v). The image of the c-map M4(g mr) under aP2 is M4(g mr). Left unchanged by this transformation, its characteristic parameters must be n, r and m again. Inserting the entries of A = (~ _mr)into the congruence mt2 =--rtl (mod A) from Lemma 2.2, we get - m = m (mod n). The remaining three congruences in (*) give no information in this case. Because of 0 ~
c( 2, v)= 17(v) + 172(v).
78
W. Kurth
The image of M4(~ '~) under ~3 is M4( ° 7,r). Lemma 2.2 gives the following conditions for invariance under ~P3:
nm =- 0 (rood nr),
m 2 --- - r 2 (rood nr),
n 2 - 0 (rood nr),
0 =- mn (rood nr).
It follows that r I m, r I n, and thus r 2 [ v. So we have to choose r to be a quadratic divisor of v. Clearly, n/r = v/r 2. Furthermore, we get
For given r, there are po, l(vlr 2) possible values for m l r satisfying this congruence and 0 ~ m l r < nlr. Hence
0)=
r2]l~
o0,1
=(r*po,1)(o).
The isomorphism W4 is treated like ~3. To avoid repetitions, we only state the result: c(~4, v) = (~,* p0.-1)(v). By the way, a c-map of type {4, 4} is invariant under ~3 or ~4 if and only if its automorphism group acts transitively on its edges. Therefore the corresponding conditions for invariance were derived by Gethner and Hutchinson [11] as conditions for edge-transitivity. As a non-commutative group, 5~3 has non-trivial inner automorphisms. Conjugate elements have the same number of invariant c-maps. (See [12] for a detailed discussion of related questions.) Thus c(tp2, v ) = c ( t p 3 , v)=c(q~4, v),
and
c( w, 0)= c( w, v). Trivially, c(cpl, v) = o(v). The image of M3(~) '~) under q% is M3(~ m+~). The condition for invariance is taken from (.) again (we omit redundant congruences): 2m -= - r (rood n). It is easily verified that the number of possible values for m varies between 0 (if n is even and r odd), 1 (if n is odd), and 2 (if n and r are both even). Hence
¢(tjO2, O)---- 17(O) -- 172(O) + 2174(O). Finally, q% yields M3( ° -,+r), and the conditions for invariance are
n(m + r)-=O(modnr),
m ( m + r ) - - r 2 (rood nr),
n 2 ~ 0 (rood nr),
0 - m n (rood nr).
We conclude that r ] m , t i n ,
thus r2Jv (as in the case of ~3), and (re~r)2+
Enumeration of Platonic maps on the torus
79
m/r + 1 ~ 0 (mod v/r2). These are at the same time the conditions for a c-map of type {3, 6} to be edge-transitive [11]. They yield c(tp5, v) = (7 * pl,1)(v). By Burnside's Lemma, 4 T4,4(1/) ._ 1 ~ i=1
c(~i ' 13) and
6 T3,6(13)= 1 ~ i=1
Inserting the above results proves our lemma.
c(qgi, 13).
[]
We can get an additional result from the identity c(q03, v ) = c((p2, v). If we determine c(tp3, v) directly by the congruences (*), we find C(~3, 13) -- (Y* P0,-1)(13)"
Thus the following identity holds. Lemma 3.2. 7 * Po,-1 = v - 1"2+
2T4.
4. Simplifying the results
Lemma 3.2 suggests that there are similar identities for 7"Po,1 and Y*PL~, too. In order to obtain such results, we make use of the multiplicativity of 7 * P0,1 and 7*Pl,x. Both functions are completely determined by their values for prime-power arguments. I f p is a prime and a / > 0 , the values PO, l(p a) and pl,l(p a) are the numbers of solutions of x 2 + 1 = 0 ( m o d p " ) and x 2 + x + 1 - 0 (modp"), respectively. Of course, congruences of this type have been thoroughly investigated. (See e.g. Hardy and Wright [15, Chapter VIII].) Having calculated the values of Po,1 and P1,1, it needs a simple summation to get the values of 7"Po,1 and ~*PL~- We only state the results here. Lemma 4.1. Let p be a prime and a > O. Furthermore, let the modulus j be 4, if i = 0 , and 3, i f i = 1. Then the following equations hold for i = 0 , 1:
p~,,(1) = Pi, l(i + 2 ) = 1,
pi,,(pa) = {20,
pi.,((i + 2) "+~) = 0 ,
/fp-=X(modj), /f p -- - 1 (mod j),
( 7 * PL,)(1) = ( 7 " PLy)(( i + 2) a)
(7 * P~.I)(P ~) =
a+l, 1, O,
--
1,
/ f p -- 1 (modj), if p -- - 1 (mod j) and 21a, if p - - 1 (mod j) and 2 ~ a.
Now the announced identities can be established.
80
W. Kurth
Lemma 4.2. y*po, l=l*x4,
y*pl,j
= 1"X3 •
Proof. Because all involved functions are multiplicative, it suffices to prove the identities for prime-power arguments. According to the definition of X4, the value (1 * X4)(v) is the difference between the number of divisors - 1 and the number of divisors ---3 (rood 4). If v = 2", a I> 0, this difference is one. Let p be an odd prime, a > 0, v = p". The divisors of v are 1, p, p 2 , . . . , p,,. If p - 1 (rood 4), all give the same residue, and we get (1 * X4)(v) = a + 1. In the case p -- 3 (rood 4), the divisors have alternating residue 1 or 3 (rood 4), according as the exponent is even or odd. Thus ( l * x 4 ) ( v ) = 1, if 21a, and 0, if 2 ~ a. Comparison with Lemma 4.1 (i = 0, ] = 4) proves y * Po.1 = 1 * X4. The other identity is obtained analogously. [] In order to simplify the enumerative formula in Lemma 3.1(a), we substitute the results from Lemma 3.2 and Lemma 4.2: T4,4(`0 ) = 1(0" dr 17 -Jr- 172 "31-(1 * •4) + "[" - 172.31_ 2174)(,0)
=
1((1 • L) + 2(1 • 1) + 2(1 • :r4) +
= -I(1 * (t + 2 +
21"r4
({4,
1
+ X4))(,0)
3,
2,
})
if 4ld ~ + x4(d))
=Zd~l,,(d+2+{02, ' if4~
=I o a+
(1 * X4))(,0)
dJ
if d -
1,
(rnod4) 3
= ~ ([id] + 1). d[v
The analogous calculation for T3,6(,0) yields T3,6(,0) "-- ~(O + 3(17 -- 172+ 2174) + 2(1 * X3))(,0) = -~(1 *(t + 3 - 3:rZ + 6:r4 + 2X3))(,0)
({6' i }
=!6 d~l~, a+
3, o,
if d =
(mod4)
3,
+
{o, o }) 2, -2,
if d --- 1 (rood 3) 2
.
The last two terms give together 6, 5, - 2 , 3, !~, 1, 0, 5, 4, 3, 2, or 1, depending on
Enumeration o f Platonic maps on the torus
81
the residue of d (mod 12). For a smoother version of this result we add an auxiliary value in the cases d -= 2, 4, 6 (mod 12) which are underlined above. This leads to part (b) of the following theorem.
Theorem 4.3. (a) The number o f Platonic maps of type {4, 4} on the torus with exactly v vertices is
T4,4(v) = ~ ([ld]+ 1). air
(b) The number o f Platonic maps o f type {3, 6} on the torus with exactly v vertices is
T3,6(v) = ~ ([16d] + 1 + to(d)), air
where to(d) is defined by
to(d) =
- 1, 1, O,
if d - 2 or d -= 6 (mod 12), if d-=4 (moO 12), otherwise.
5. Maps with higher degree of symmetry There are some kinds of special Platonic maps on the torus which are characterized by a certain richness of their automorphism group. We have already mentioned edge-transitivity. Here we discuss two other symmetry conditions. A 'regular' map in the sense of [6, 8, 9, 17] admits two automorphisms, the first of which cyclically permutes the edges bounding a face, while the other cyclically permutes the edges which meet at one vertex of this face [9, Chapter 8]. Applied to Platonic maps on the torus, this turns out to be equivalent to the condition that the corresponding c-maps are invariant under all rotations, that means, under ~3 or under q95 and q96, respectively. Following Wilson [27, 28], we will call such a map rotary. The number of rotary c-maps of type {4, 4} with v vertices is obviously c0P3, v) = (1 * X4)(v). Considering c-maps of type {3, 6}, we observe that ¢6 = ¢2 and ¢5 = ¢2. Hence the invariant c-maps of ¢5 and ¢6 coincide, and the corresponding number is here c(¢5, v) = (1 * Z3)(v). A greater degree of symmetry is reached when the automorphism group of the map acts transitively on the flags. (A flag is a triple consisting of a vertex, an edge incident with that vertex, and a face incident with that edge.) The term for flag-transitivity is 'reflexible' in [9] and 'regular' in [7, 25, 26, 27, 28]. In our theory, the corresponding c-map of such a map is distinguished by the property that all elements of 5j (j"= 3, 4) leave it invariant. It suffices to require invariance with respect to g'2 and IP3 (for type {4, 4}) or with respect to ¢2 and ¢s (for type {3, 6}), since these are generators of 5j.
82
W. Kurth
7 I
L .
[ .
I
I
!
I I
I I
I I
I :
' .
.
.
.
.
/
I
.
' Z _ Z ' .
.
(~<)
.
.
.
(o)
.
.
.
"
(c)
Fig. 3
Applying the method from Section 3 once more, we get as the number c(Sj, v) of totally invariant c-maps with v vertices of type {4, 4}: C(,J~4, V) =
1, 0,
1, of type {3, 6}: c(~3, v) = 0,
if v is a square or twice a square otherwise, if v is a square or three times a square, otherwise.
This is at the same time the number of flag-transitive maps, since each such map admits only one representation as a c-map. When counting rotary maps, however, we have to take isomorphisms into account again. Theorem 5.1. The number of rotary maps of type {4, 4}, respectively {3, 6}, on the torus with exactly v vertices is
½((1 * Xj)(v) + c(~j, v)), where c(3rj, v) is defined above and j = 4, resp. 3. Proof. It suffices to consider the action of the group {~Pl, ~'12), resp. ((Pl, I]02), on the set of all rotary c-maps with v vertices. The corresponding numbers of fixed c-maps are (1 ,%s)(V) and c(~ s, v). Burnside's Lemma yields the above numbers of orbits. [] Fig. 3 illustrates the discussed degrees of symmetry with three Platonic maps of type {4, 4} on the torus. The map (a) is flag-transitive, (b) is only rotary, and (c) is not even rotary.
Acknowledgment The author wishes to thank Professor W. Klotz and Professor L. Lucht for their suggestions and encouragement.
Enumeration o f Platonic maps on the torus
83
References [1] A. Altshuler, Construction and enumeration of regular maps on the toms, Discrete Math. 4 (1973) 201-217. [2] A. Altshuler, Hamiltonian circuits in some maps on the toms, Discrete Math. 1 (1972) 299-314. [3] A. Altshuler, Polyhedral realization in R 3 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes, Discrete Math. 1 (1971) 211-238. [4] U. Brehm, Polyeder gegebenen Geschlechts mit minimaler Eckenzahl, Tagungsbericht Oberwolfach 19 (1978) 5. [5] D. Cohen, Basic Techniques of Combinatorial Theory (Wiley, New York, 1978). [6] H.S.M. Coxeter, Configurations and maps, Rep. Math. Colloq. (2) 8 (1948) 18-38. [7] H.S.M. Coxeter, Regular Complex Polytopes (Cambridge University Press, Cambridge, 1974). [8] H.S.M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950) 413-455. [9] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, 4th ed. (Springer, Berlin, 1980). [10] A.L. Edmonds, J.H. Ewing and R.S. Kulkarni, Regular tessellations of surfaces and (p, q, 2)triangle groups, Ann. Math. 116 (1982) 113-132. [11] E. Gethner and J.P. Hutchinson, Platonic graphs on the toms, to appear. [12] S.W. Golomb and A.W. Hales, On enumerative equivalence of group elements, J. Combin. Theory 5 (1968) 308-312. [13] P. Gritzrnann, The toroidal analogue to Eberhard's theorem, Mathematika 30 (1983) 274-290. [14] B. Griinbaum and G.C. Shephard, Polyhedra with transitivity properties, C.R. Math. Acad. Sci., Soc. R. Canad. 6 (1984) 61-66. [15] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th ed. (At the Clarendon Press, Oxford, 1979). [16] J.P. Hutchinson, Automorphism properties of embedded graphs, J. Graph Theory 8 (1984) 35-49. [17] G.A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978) 273-307. [18] H. Leitzke, 0ber ringf6rmige Polyeder, unpublished manuscript. [19] P. McMullen, Ch. Schulz and J.M. Wills, Equivelar polyhedral manifolds in E 3, Israel J. Math. 41 (1982) 331-346. [20] S. Negami, Uniqueness and faithfulness of embedding of toroidal graphs, Discrete Math. 44 (1983) 161-180. [21] M. Newman, Integral Matrices (Academic Press, New York, 1972). [22] I. Niven and H.S. Zuckerman, An Introduction to the Theory of Numbers, 4th ed. (Wiley, New York, 1980). [23] H.N. Shapiro, Introduction to the Theory of Numbers (Wiley, New York, 1983). [24] A. Vince, Combinatorial maps, J. Combin. Theory Ser. B 34 (1983) 1-21. [25] A. Vince, Regular combinatorial maps, J. Combin. Theory Ser. B 35 (1983) 256-277. [26] S.E. Wilson, Non-orientable regular maps, Ars Combin. 5 (1978) 213-218. [27] S.E. Wilson, Operators over regular maps, Pacific J. Math. 81 (1979) 559-568. [28] S.E. Wilson, Riemann surfaces over regular maps, Canad. J. Math. 30 (1978) 763-782.
71
E N U M E R A T I O N OF PLATONIC MAPS ON THE TORUS Winfried KURTH Institut f-fir Mathematik, Technische Universitiit Clausthal, D-3392 Clausthal-Zellerfeld, Fed. Rep. Germany Received 28 September 1984 Revised 29 May 1985 Certain maps (graph embeddings) on the torus are counted, namely those with all faces triangles, respectively quadrilaterals, resp. hexagons, and all vertices having the same degree (which then must be 6, 4 or 3, resp.). These are the toroidal analogues of the spherical maps corresponding to the five Platonic solids. Techniques from combinatorics and number theory are applied to obtain the results.
1. I n t r o d u c t i o n
A map K on a torus T is an embedding of a finite graph G on T, such that each component of T \ G (each face of K) is homeomorphic to a disc. G may have loops and multiple edges. By the dual K* of the map K we mean the map which results if we take a vertex for each face of K and an edge for each edge of K, joining two vertices of K* whose corresponding faces are incident along an edge of K. We call a map K Platonic of type {p, q } if all vertices of K are of degree q and all vertices of K* are of degree p. The term "Platonic" is used since the maps with the above property are the natural generalizations of the spherical maps corresponding to the Platonic solids. In the literature the terminology varies between "regular" [1, 2, 10], "quasiregular" [17], "uniform" [27], "equivelar" [19] and "Platonic" [11]. Platonic maps on arbitrary closed surfaces are constructed and investigated in [10]. If a map K is Platonic of type {p, q}, then K* is Platonic of type {q, p}. From Euler's formula, together with a simple counting argument, it follows that the only types of Platonic maps on the torus are {3, 6}, {4, 4} and {6, 3}. Two maps on the torus are called isomorphic if there exists an autohomeomorphism of the torus which sends vertices to vertices, edges to edges and faces to faces and preserves incidence. In most instances maps are considered only up to isomorphism. Let Tp.q(V) denote the number of Platonic maps of type {p, q} with exactly v vertices on the torus. In this paper we develop simple formulae for the calculation of Tp,q(V). If a map of type {6, 3} on the torus has v vertices and f faces, then 0012-365X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
72
W. Kurth
v = 2f, and because of duality the number of maps of type {6, 3} with f = ½v faces equals the number of maps of type {3, 6} with ½v vertices. Hence T6,3(v) = ~ Ta,6(½v), t0,
if 2Iv, if2 ,~ v.
Therefore it is sufficient to determine T4,4(v) and T3,6(v). For this purpose we introduce a standard form of Platonic maps on the torus in Section 2. In Section 3 we use Burnside's Lemma to get first expressions for T4,4(v) and T3,6(v). These are simplified in Section 4 by some number theory yielding our main result (Theorem 4.3). In Section 5 we briefly discuss some stronger versions of regularity and the corresponding numbers of maps on the toms. Maps can also be defined in entirely combinatorial terms. This is carried out in [17] and [24], for example. However, the way a map is defined is not essential for our arguments. There is a considerable literature on polyhedral realizations of maps in Euclidean 3-space. We mention [3, 13, 19], where further references can be found. An especially remarkable fact is the existence of toroidal polyhedra which bear maps of type {3, 6} and are vertex-transitive under their symmetry groups. Such (necessarily everywhere fiat) tori have been found by U. Brehm [4] and, independently, by H. Leitzke [18]. A construction is given in [14]. We do not discuss polyhedral realizations in this paper.
2. c-maps
Let (]4 and G3 be tilings of the plane R 2 with squares or triangles, respectively, induced by the infinite graphs {(x, y) ER2:X E~' v y E~ '}
and {(x, y) e R 2 : X C:7/ v y E Z v x + y E~'_}.
We may interprete G3 as a tiling with regular triangles if we let the coordinate axes intersect at an angle of in. This enables us to represent all possible automorphisms of (93 by isometries, as in the case of (]4. In the following, the index j stands for 3 or 4. The symmetry group ~ of Gs is the group of all isometrics which map Gs onto itself. It is obvious that ~4 ~: ~ . A translation T e ~ acts o n R E in the way
(x, y)--, (x, y) + (tl, t2), with fixed integers tl, t2. We shall identify T with the vector (tl, t2). Let S, T e ~ be two linearly independent translations and (S, T) the group generated by them. "lifts group is dearly isomorphic to (Z 2, +). It acts on R 2 and
73
Enumeration o f Platonic maps on the torus
,
v
m
ff
,
, :
(a)
(b) Fig. 1
on the embedded tiling Gj, generating the torus R2/(S, T) as its orbit space. This torus can be represented by the parallelogram spanned by S and T (which is a fundamental region of (S, T)) if opposite sides are identified. (Fig. 1 shows two examples with j = 3.) Obviously, GJ(S, T) is a Platonic map of type {3, 6} or {4, 4}, respectively, embedded into this toms. We call a map given in this form a c-map ('c' from 'coordinates') and denote it by Mj(A), if A is the 2 x 2 matrix with columns S and T:
A = (Sl
S2
tl)
t2
for S = (SI, S2), T = (tl, t2).
Fig. l(a) illustrates the c-map M
2
1
, ,s ows
1
-2
The number v of vertices of a c-map Ms(A) is clearly given by v = Idet AI. We call two c-maps isomorphic if they are isomorphic as maps (see Section 1). Two c-maps are equal if the orbits of R 2 under the action of the corresponding groups are identical as sets. Note that the c-maps of Figs. l(a) and (b) are isomorphic, but not equal. The following conclusion is almost immediate (and can be found in [15, p. 27]): Lemma 2.1. Let A and B be 2 x 2 matrices with integral entries. The c-maps
Mj(A) and Mj(B) are equal if and only if there exists a unimodular matrix U with integral entries satisfying A U = B. The matrix representation of a c-map becomes unique if only matrices (~ m) with the restrictions 0 ~
74
W. Kurth
?
" " r ~'
¢ ~
i~
xJ
f'~
")
t"'%
Fig. 2
form, see [21].) Clearly, v = nr. We call the integers n, m and r the characteristic parameters of the c-map. If (x, y) - (v, w) means (x, y) - (v, w) ~ (S, T), S and T forming the columns of A, the combinatorial meaning of the characteristic parameters of the c-map Mj(A) is easily seen to be as follows: n is the smallest positive integer satisfying (0, 0) - (n, 0), r is the smallest positive integer for which there exists an integer x with (0, 0 ) - (x, r), m is the smallest non-negative integer satisfying (0, 0) (m, r). This is demonstrated in Fig. 2, where the points equivalent to (0, 0) are marked with circles. Lemma 2.2. Let
A=(~
t2tl)
and
A=detA.
The characteristic parameters n, m, r o f the c-map Mj(A ) are the unique solutions o f the following system o f congruences under the restrictions nr = [A], 0 <~m < n,
0
(,)
nt2 =--0 (mod A),
mt2 ----rtl (mod A),
ns2-= 0 (mod A),
rsl = ms2 (mod A).
Proof. Taking B = ( g m) in Lemma 2.1, we see that there is a uniquely determined unimodular matrix U=(Uik), i, k = 1, 2, such that A U = B . By solving this matrix equation, we can express the numbers Uik in terms of the entries of A and B. The condition that these numbers are integers can be put into the congruences (.). The further condition Idet U[ = 1 is equivalent to nr = IAI. On the other hand, (*) together with the above restrictions implies that n, m and r are the (uniquely determined) characteristic parameters of Mj(A). []
Enumeration of Platonic maps on the torus
75
Until now, we have left open the question whether every Platonic map on a toms is isomorphic to a c-map. Indeed, this is the case. Theorem 2.3. Every Platonic map o f type {3, 6} or {4, 4} on the torus is isomorphic to some c-map Mj(A ) (j = 3 or j = 4, respectively).
We give only the outline of a proof. Let K be a Platonic map of type {3, 6} or {4, 4} on the toms. If the torus is represented by a parallelogram, we can cover the plane with copies of this parallelogram and get a tiling of the plane with some triangular or tetragonal tiles, respectively. The tiles need not to be regular, but there is always a homeomorphism mapping the tiling onto Gj. The translations along the sides of the original parallelogram are transformed into translations of Gj, and they generate a c-map which is isomorphic to the original map K. Altshuler [2] and Negami [20, pp. 168-171] give detailed proofs of this theorem for the case j = 3. They both specify the c-map M3(A) to be given by its characteristic parameters. (Their proofs follow a way slightly different from ours.) Cf. Hutchinson [16, p. 43] for an extension to Kepler graphs on the toms. A certain generalization to other surfaces can be found in [10] and [17]. Our problem of counting Platonic maps on the toms is now reduced to the enumeration of classes of isomorphic c-maps. Every isomorphism between two c-maps can be extended to the covering plane and induces an isometry belonging to ~ and having essentially the same effect on the c-map as the original isomorphism. E.g. the isomorphism illustrated in Fig. 1 corresponds to a rotation about the origin. Some symmetries of Gj are of minor interest, because they leave all c-maps unchanged. These are just the translations (x, y) and the half-turns with centres (½x, ½y), where x, y e Z. They form a normal subgroup of ~ which will be denoted by ~ . Symmetries lying in the same coset with respect to ~ have the same effect on c-maps. Therefore it is convenient to use the elements of the factor group 5j = ~ / ~ instead of the symmetries themselves. We will refer to the elements of 5j as isomorphisms (of c-maps), having in mind that they are indeed whole classes of isomorphisms. It is easily verified that 54 is the four-group (~2 X (~2, and ot~3 the symmetric group 5e3. The elements of 54 and 53, together with representatives from ~4 or ~3 and their effects on a point (x, y) ~ R 2, are listed in Table 1. The action of tP5 is illustrated in Fig. 1. (b) is the image of (a). Now we are able to determine the characteristic parameters of all isomorphic images of a given c-map by applying Lemma 2.2 to the transformed generating translations. In each of the 4 + 6 cases, there is a system of congruences relating the new parameters to the original ones. (In the case of ,-~3, this is the essence of Negami's Theorem 3.6 [20].) We shall use these relations to determine the c-maps which are invariant under a given isomorphism.
76
W. Kurth Table 1. Isomorphisms of c-maps ~4
Element
Representative from qd4
Image of (x, y)
I/)1 lp2
Identity Reflection in the line y = 0 Tetragonal rotation Reflection in the line x = y
(x, y) (x, - y ) ( - y , x) 0', x)
Representative from %
Image of (x, y)
Identity Reflection in the line y = 0 Reflection in the line x = 0 Reflection in the line - x = y Hexagonal rotation Trigonal rotation
(x, y) (x + y, - y ) ( - x , x + y) (-y, -x) ( - y , x + y) ( - x - y, x)
1/)3
'~4
~¢3 Element q01 q02 q% ~4 tp5 q~6
3. Enumeration of Platonic maps
To formulate the following results we need some number-theoretic functions. Let tr and fl be functions on the positive integers. The convolution product tr * fl is defined by
din
~' is called multiplicative if gcd(m, n) = 1 implies ol(mn) = o~(m)o~(n). If a' and fl are multiplicative then e * f l is multiplicative, too. (See [22; 23] for details on convolution.) The following functions are defined for every positive integer n (j and k are integers). l(n) = 1, :tk(n)=
t(n) = n, 1, 0,
ifkln i f k Jf n,
0, x 3 ( n ) ---- +1,
if31n if n - +1 (mod 3),
{ 0, +1,
if2ln if n--- +1 (mod 4),
x4(n) =
1, if n is a square y(n) = 0, otherwise, pj.k(n) = [{X e 7/: O<_x < n ^ x 2 + jx + k - 0 (mod n)}l, 3=1.1,
1;k = l * ~ k ,
o=l*t.
77
Enumeration of Platonic maps on the torus
pj, k(n) is the number of residues solving x 2 + jx + k =- 0 (mod n). We shall need only P0,1, P0,-1 and P1,1. 17k(n) is the number of divisors of n, including 1 and n, which are divisible by k. tr(n) is the sum of the divisors of n. Clearly, the functions 1, t, Z3, Z4, ]', 17and cr are multiplicative. So is Pj, k. (See e.g. [15, p. 96].) We need this fact in Section 4. [r] denotes the greatest integer not exceeding r. The following lemma will be the basis for our main theorem. Lemma 3.1. (a) The number of Platonic maps of type {4, 4} on the torus with
exactly v vertices is T4,4(13 ) "-" 1(or(v) + 17(!/) -Jr- 172(v) -J¢-( ~ * P0,1)(v) --t- (~:'g p0,_l)(v)). (b) The number o f Platonic maps of type {3, 6} on the torus with exactly v
vertices is T3,6(u ) : l(o'(o) q- 3(17(v) - 172(o) + 2174(v)) at- 2 ( r * Pl,1)(v)). Proof. Let us consider the set
Sj(v)=
0
"n,m, reT/AO<~m
~ = 3 , 4).
It contains all c-maps with v vertices of type {3, 6} or {4, 4}, respectively. For each divisor n of v there are n possible values of m. Hence St(v ) has a(v) elements in both cases, j = 3 and j = 4. Some of these elements are isomorphic by some isomorphism belonging to 5~. Enumerating the Platonic maps with v vertices, we have to take these isomorphisms into account. Each element of 5j induces a permutation on St(v ). The orbits of St(v ) with respect to the group action correspond to the isomorphism classes of Platonic maps with v vertices and are to be counted. According to Burnside's Lemma (see e.g. [5]), the number of orbits equals the average number of invariant c-maps of the group elements. Thus we have to determine the number,.of invariant c-maps for each ~i (i = 1 , . . . , 4) and ~i (i = 1 , . . . , 6). Let us denote this number by c(V2i, v), resp. c(q~i, v). We begin with the elements of & . The identity element leaves each c-map invariant. Hence c(V/1, v) = a(v). The image of the c-map M4(g mr) under aP2 is M4(g mr). Left unchanged by this transformation, its characteristic parameters must be n, r and m again. Inserting the entries of A = (~ _mr)into the congruence mt2 =--rtl (mod A) from Lemma 2.2, we get - m = m (mod n). The remaining three congruences in (*) give no information in this case. Because of 0 ~
c( 2, v)= 17(v) + 172(v).
78
W. Kurth
The image of M4(~ '~) under ~3 is M4( ° 7,r). Lemma 2.2 gives the following conditions for invariance under ~P3:
nm =- 0 (rood nr),
m 2 --- - r 2 (rood nr),
n 2 - 0 (rood nr),
0 =- mn (rood nr).
It follows that r I m, r I n, and thus r 2 [ v. So we have to choose r to be a quadratic divisor of v. Clearly, n/r = v/r 2. Furthermore, we get
For given r, there are po, l(vlr 2) possible values for m l r satisfying this congruence and 0 ~ m l r < nlr. Hence
0)=
r2]l~
o0,1
=(r*po,1)(o).
The isomorphism W4 is treated like ~3. To avoid repetitions, we only state the result: c(~4, v) = (~,* p0.-1)(v). By the way, a c-map of type {4, 4} is invariant under ~3 or ~4 if and only if its automorphism group acts transitively on its edges. Therefore the corresponding conditions for invariance were derived by Gethner and Hutchinson [11] as conditions for edge-transitivity. As a non-commutative group, 5~3 has non-trivial inner automorphisms. Conjugate elements have the same number of invariant c-maps. (See [12] for a detailed discussion of related questions.) Thus c(tp2, v ) = c ( t p 3 , v)=c(q~4, v),
and
c( w, 0)= c( w, v). Trivially, c(cpl, v) = o(v). The image of M3(~) '~) under q% is M3(~ m+~). The condition for invariance is taken from (.) again (we omit redundant congruences): 2m -= - r (rood n). It is easily verified that the number of possible values for m varies between 0 (if n is even and r odd), 1 (if n is odd), and 2 (if n and r are both even). Hence
¢(tjO2, O)---- 17(O) -- 172(O) + 2174(O). Finally, q% yields M3( ° -,+r), and the conditions for invariance are
n(m + r)-=O(modnr),
m ( m + r ) - - r 2 (rood nr),
n 2 ~ 0 (rood nr),
0 - m n (rood nr).
We conclude that r ] m , t i n ,
thus r2Jv (as in the case of ~3), and (re~r)2+
Enumeration of Platonic maps on the torus
79
m/r + 1 ~ 0 (mod v/r2). These are at the same time the conditions for a c-map of type {3, 6} to be edge-transitive [11]. They yield c(tp5, v) = (7 * pl,1)(v). By Burnside's Lemma, 4 T4,4(1/) ._ 1 ~ i=1
c(~i ' 13) and
6 T3,6(13)= 1 ~ i=1
Inserting the above results proves our lemma.
c(qgi, 13).
[]
We can get an additional result from the identity c(q03, v ) = c((p2, v). If we determine c(tp3, v) directly by the congruences (*), we find C(~3, 13) -- (Y* P0,-1)(13)"
Thus the following identity holds. Lemma 3.2. 7 * Po,-1 = v - 1"2+
2T4.
4. Simplifying the results
Lemma 3.2 suggests that there are similar identities for 7"Po,1 and Y*PL~, too. In order to obtain such results, we make use of the multiplicativity of 7 * P0,1 and 7*Pl,x. Both functions are completely determined by their values for prime-power arguments. I f p is a prime and a / > 0 , the values PO, l(p a) and pl,l(p a) are the numbers of solutions of x 2 + 1 = 0 ( m o d p " ) and x 2 + x + 1 - 0 (modp"), respectively. Of course, congruences of this type have been thoroughly investigated. (See e.g. Hardy and Wright [15, Chapter VIII].) Having calculated the values of Po,1 and P1,1, it needs a simple summation to get the values of 7"Po,1 and ~*PL~- We only state the results here. Lemma 4.1. Let p be a prime and a > O. Furthermore, let the modulus j be 4, if i = 0 , and 3, i f i = 1. Then the following equations hold for i = 0 , 1:
p~,,(1) = Pi, l(i + 2 ) = 1,
pi,,(pa) = {20,
pi.,((i + 2) "+~) = 0 ,
/fp-=X(modj), /f p -- - 1 (mod j),
( 7 * PL,)(1) = ( 7 " PLy)(( i + 2) a)
(7 * P~.I)(P ~) =
a+l, 1, O,
--
1,
/ f p -- 1 (modj), if p -- - 1 (mod j) and 21a, if p - - 1 (mod j) and 2 ~ a.
Now the announced identities can be established.
80
W. Kurth
Lemma 4.2. y*po, l=l*x4,
y*pl,j
= 1"X3 •
Proof. Because all involved functions are multiplicative, it suffices to prove the identities for prime-power arguments. According to the definition of X4, the value (1 * X4)(v) is the difference between the number of divisors - 1 and the number of divisors ---3 (rood 4). If v = 2", a I> 0, this difference is one. Let p be an odd prime, a > 0, v = p". The divisors of v are 1, p, p 2 , . . . , p,,. If p - 1 (rood 4), all give the same residue, and we get (1 * X4)(v) = a + 1. In the case p -- 3 (rood 4), the divisors have alternating residue 1 or 3 (rood 4), according as the exponent is even or odd. Thus ( l * x 4 ) ( v ) = 1, if 21a, and 0, if 2 ~ a. Comparison with Lemma 4.1 (i = 0, ] = 4) proves y * Po.1 = 1 * X4. The other identity is obtained analogously. [] In order to simplify the enumerative formula in Lemma 3.1(a), we substitute the results from Lemma 3.2 and Lemma 4.2: T4,4(`0 ) = 1(0" dr 17 -Jr- 172 "31-(1 * •4) + "[" - 172.31_ 2174)(,0)
=
1((1 • L) + 2(1 • 1) + 2(1 • :r4) +
= -I(1 * (t + 2 +
21"r4
({4,
1
+ X4))(,0)
3,
2,
})
if 4ld ~ + x4(d))
=Zd~l,,(d+2+{02, ' if4~
=I o a+
(1 * X4))(,0)
dJ
if d -
1,
(rnod4) 3
= ~ ([id] + 1). d[v
The analogous calculation for T3,6(,0) yields T3,6(,0) "-- ~(O + 3(17 -- 172+ 2174) + 2(1 * X3))(,0) = -~(1 *(t + 3 - 3:rZ + 6:r4 + 2X3))(,0)
({6' i }
=!6 d~l~, a+
3, o,
if d =
(mod4)
3,
+
{o, o }) 2, -2,
if d --- 1 (rood 3) 2
.
The last two terms give together 6, 5, - 2 , 3, !~, 1, 0, 5, 4, 3, 2, or 1, depending on
Enumeration o f Platonic maps on the torus
81
the residue of d (mod 12). For a smoother version of this result we add an auxiliary value in the cases d -= 2, 4, 6 (mod 12) which are underlined above. This leads to part (b) of the following theorem.
Theorem 4.3. (a) The number o f Platonic maps of type {4, 4} on the torus with exactly v vertices is
T4,4(v) = ~ ([ld]+ 1). air
(b) The number o f Platonic maps o f type {3, 6} on the torus with exactly v vertices is
T3,6(v) = ~ ([16d] + 1 + to(d)), air
where to(d) is defined by
to(d) =
- 1, 1, O,
if d - 2 or d -= 6 (mod 12), if d-=4 (moO 12), otherwise.
5. Maps with higher degree of symmetry There are some kinds of special Platonic maps on the torus which are characterized by a certain richness of their automorphism group. We have already mentioned edge-transitivity. Here we discuss two other symmetry conditions. A 'regular' map in the sense of [6, 8, 9, 17] admits two automorphisms, the first of which cyclically permutes the edges bounding a face, while the other cyclically permutes the edges which meet at one vertex of this face [9, Chapter 8]. Applied to Platonic maps on the torus, this turns out to be equivalent to the condition that the corresponding c-maps are invariant under all rotations, that means, under ~3 or under q95 and q96, respectively. Following Wilson [27, 28], we will call such a map rotary. The number of rotary c-maps of type {4, 4} with v vertices is obviously c0P3, v) = (1 * X4)(v). Considering c-maps of type {3, 6}, we observe that ¢6 = ¢2 and ¢5 = ¢2. Hence the invariant c-maps of ¢5 and ¢6 coincide, and the corresponding number is here c(¢5, v) = (1 * Z3)(v). A greater degree of symmetry is reached when the automorphism group of the map acts transitively on the flags. (A flag is a triple consisting of a vertex, an edge incident with that vertex, and a face incident with that edge.) The term for flag-transitivity is 'reflexible' in [9] and 'regular' in [7, 25, 26, 27, 28]. In our theory, the corresponding c-map of such a map is distinguished by the property that all elements of 5j (j"= 3, 4) leave it invariant. It suffices to require invariance with respect to g'2 and IP3 (for type {4, 4}) or with respect to ¢2 and ¢s (for type {3, 6}), since these are generators of 5j.
82
W. Kurth
7 I
L .
[ .
I
I
!
I I
I I
I I
I :
' .
.
.
.
.
/
I
.
' Z _ Z ' .
.
(~<)
.
.
.
(o)
.
.
.
"
(c)
Fig. 3
Applying the method from Section 3 once more, we get as the number c(Sj, v) of totally invariant c-maps with v vertices of type {4, 4}: C(,J~4, V) =
1, 0,
1, of type {3, 6}: c(~3, v) = 0,
if v is a square or twice a square otherwise, if v is a square or three times a square, otherwise.
This is at the same time the number of flag-transitive maps, since each such map admits only one representation as a c-map. When counting rotary maps, however, we have to take isomorphisms into account again. Theorem 5.1. The number of rotary maps of type {4, 4}, respectively {3, 6}, on the torus with exactly v vertices is
½((1 * Xj)(v) + c(~j, v)), where c(3rj, v) is defined above and j = 4, resp. 3. Proof. It suffices to consider the action of the group {~Pl, ~'12), resp. ((Pl, I]02), on the set of all rotary c-maps with v vertices. The corresponding numbers of fixed c-maps are (1 ,%s)(V) and c(~ s, v). Burnside's Lemma yields the above numbers of orbits. [] Fig. 3 illustrates the discussed degrees of symmetry with three Platonic maps of type {4, 4} on the torus. The map (a) is flag-transitive, (b) is only rotary, and (c) is not even rotary.
Acknowledgment The author wishes to thank Professor W. Klotz and Professor L. Lucht for their suggestions and encouragement.
Enumeration o f Platonic maps on the torus
83
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