STEVEN L. FARRIS

COMPLETELY TRANSITIVE

AND

CLASSIFYING

ALL

EDGE-TRANSITIVE

VERTEXPOLYHEDRA

Part I: Necessary Class Conditions

ABSTRACT. Recently A. Dress completed the classification of the regular polyhedra in E 3 by adding one class to the enumeration given by Gr/inbaum on this subject. This classification is the only systematic study of a collection of polyhedra possessing special symmetries which uses the generalized definition of a polygon allowing for skew polygons as well as planar polygons in E 3. This study gives necessary conditions for polyhedra to be vertex-transitive and.edgetransitive. These conditions are restrictive enough to make the task of completely enumerating such polyhedra realizable and efficient. Examples of this process are given, and an explanation of the basic process is discussed. These 'new' polyhedra are appearing more frequently in applications of geometry, and this examination is a beginning of the classifications of polyhedra having special symmetries even though there are many other such classes which lack this scrutiny.

INTRODUCTION

In [3] Griinbaum gave a systematic classification of the known regular (i.e. flag-transitive) polyhedra in E 3. In that paper he used the generalized notion of a polygonal face of a polyhedron allowing polygonal faces to be skew as well as planar. This more generalized definition of a polygon made the classification more natural and more complete. Just recently in F2] A. Dress completed the classification of all the regular polyhedra using this more general definition of a polygon. However, no examinations of the other, more or less symmetrical classes of polyhedra using the generalized notion of a polygon have appeared in the literature. This study began as a detailed look at the class of fully-transitive polyhedra. That is, the polyhedra which are simultaneously vertex-transitive, edgetransitive, and face-transitive. Although this class is the same as the class of regular polyhedra when it is restricted to finite, planar polyhedra, it is vastly larger than the class of regular polyhedra when these restrictions are removed. The results presented here will yield a complete classification of those polyhedra in E 3 which are simultaneously vertex-transitive and edgetransitive. This extensive enumeration will appear in Parts II-IV (or V) of this report. In Part II a large, complete classification of just the finite, fullytransitive polyhedra will appear. In Part III the complete classification of the finite, vertex-transitive and edge-transitive polyhedra which are not face-transitive will be given. Finally, in Part IV, and Part V if necessary, the corresponding infinite classifications will be given. Since this enumeration uses the generalized notion of a polygon, the less obvious results presented Geometriae Dedicata 26 (1988), 111-124. © 1988 by Kluwer Academic Publishers.

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STEVEN L. FARR!S

below will be proven in more detail than usual. The remarks of Branko Gr/inbaum and an independent reviewer are acknowledged. Following Gr/inbaum [3], a finite polygon (or n-gon) P = I-V1, V2,..., 1,1,] in a Euclidean space E k is the figure formed by the distinct points (vertices) V~, V2. . . . . V, of E k, together with the segments (edges) [V~,V~+1] for i = 1, 2 , . . . , n - 1, and [V,,V 1]. An infinite polygon P = [ . . . , V l, Vo, V1, V2 . . . . ] is defined in a similar way except the sequence of distinct points of E k is infinite and every compact subset of E k meets only finitely many edges. A polygon is regular if its group of symmetries acts transitively on itsflags, i.e. the ordered pairs of a vertex and edge which are mutually incident. If the vertices of a polygon P all lie in one plane, then P is planar, otherwise P is skew. A polyhedron P = {F~: iE 1} is a family of polygons F i (called the faces of P) in E k such that: (i) each compact subset of E k meets only finitely many faces of P; (ii) each edge E of one face Fj of P is an edge of precisely one other face of P; (iii) each vertex of one face F~ of P belongs to at least two other faces of P, and all the faces of P which contain V form one 'circuit' (that is, they can be labelled cyclically so that neighboring faces share an edge); (iv) the faces of P form a connected family (that is, any two faces can be connected by a finite sequence of faces such that two faces adjacent in the sequence share at least one edge). The vertices of the faces of P are called the vertices of P, and the edges of the faces of P are called the edges of P. If the number of vertices of P is finite we say P is finite, otherwise we say P is infinite. In 1-3] Griinbaum omits condition (iii) in the definition above. In order to ensure that the classification sought will be complete this condition is included here. Further, there is no restriction of the dimension k in the results that follow although the enumerations that follow this study will be carried out in E 3. The concepts of a flag, a symmetry, the symmetry group, and regularity of a polyhedron are similar to those for a polygon. The primary difference is that a flag is an ordered triple of a vertex, an edge, and a face of the polyhedron which are all mutually incident. The properties of vertextransitivity, edge-transitivity, face-transitivity, and full-transitivity are similar to flag-transitivity. Clearly, if a polyhedron P is vertex-transitive, then every vertex of P has the same number of faces and edzes at it as any other vertex of P, and we

VERTEX-TRANSITIVE

AND EDGE-TRANSITIVE

POLYHEDRA

ll3

call this number the valency of P. Further, if P is an edge-transitive polyhedron, then all the edges of P are congruent.

1.

ANGLE

CLASSES

A face angle (or angle) ~ of a polyhedron P is the planar angle ct between two consecutive edges of a face F of P with 0 < ~ ~< =. Two angles cq and ~2 of a polyhedron P will be called equivalent provided there is a symmetry of P mapping ~1 onto ~2. The equivalence classes will be called angle classes. Similar relations will be used for the vertices, edges, and faces of a polyhedron. The first two results are obvious and their proofs are therefore omitted.

LEMMA 1.1. I f a polyhedron P is edge-transitive, then P has at most two face classes. LEMMA 1.2. I f a polyhedron P is edge-transitive, then P has at most four angle classes. When an edge E is the side of an angle ~ we shall say E contains 7. T H E O R E M 1. I f a polyhedron P is vertex-transitive and edge-transitive, then P has at most three angle classes. Proof. By Lemma 1.2, P has at most four angle classes. Suppose P has exactly four angle classes denoted by9~1,~2, ~1, and ~2- Let arbitrary angles in these classes be denoted by ax, ct2, ill, and flz respectively. Let V be a vertex of P. Vertex-transitivity of P forces at least one angle from each class to be at V. With no loss of generality suppose an 71 angle is adjacent to a fll angle at V(see Figure 1). There must be a symmetry a of P which maps E 1 onto E 2 since P is edge-transitive. The map tr must fix Vand the fll angle pictured, and send the 71 angle onto the next angle adjacent to the fll angle making it an a r This is forced by the fact that the other two angles contained by E l a n d E 2 must belong to angle classes other than fll since P is edge-transitive, and every edge of P must contain one angle from each class. Repeating this argument, V has only ctl, and fll angles at it, contradicting vertex-transitivity. This completes the proof. Let P be a vertex-transitive and edge-transitive polyhedron. If P is not face-transitive, then P has exactly two face classes which will be denoted by and -~, and arbitrary faces from these classes will be denoted by F and H respectively. If P has exactly two angle classes, then they will be designated by U and ~, and arbitrary angles from these classes will be denoted by ct and fl respectively. Finally, if P has exactly three angle classes, then they will be designated by U, ~1, ~2, and arbitrary angles from these classes will be

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STEVEN L. FARRIS

•.

V

Fig. 1. Possible angle configuration at the vertex Vwhen P is vertex-transitive, edge-transitive, and has four distinct angle classes.

denoted by e, ill, and f12 respectively. The reasons for the choice of notation will become evident in the results that follow. 2. EDGE CONFIGURATIONS We now examine the possible configuration of angles at the edges of a vertex-transitive and edge-transitive polyhedron. The first result follows primarily from Theorem 1. COROLLARY 1. If a polyhedron P is vertex-transitive, edge-transitive, and not face-transitive, then one of the following holds: (i) P has exactly two angle classes, and the configuration of angles at every edge of P is as in Figure 2(a). (ii) P has exactly three angle classes, and the configuration of angles at every edge of P is as in Figure 2(b).

Proof. By Theorem 1, P has at most three angle classes, and since P is not face-transitive, then P has at least two angle classes. The result is immediate from the fact that P has exactly two face classes with faces from each class containing • angles and fl-type angles respectively forcing the edge configurations claimed. We now turn to fully-transitive polyhedra.

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115

(o.)

F H I

(L) Fig. 2. The possible angle arrangements at each edge of P when P is not face-transiti ve. In (a) P has two distinct angle classes; in (b) P has three distinct angle classes.

L E M M A 2. If a fully-transitive polyhedron P has exactly two angle classes, then every edge oJ P contains two ~ angles and two fl angles. Proof. By edge transitivity of P every edge must contain at least one angle and one /~ angle. Suppose the other two angles at every edge of P belong to the same class, say 9j. Let E be an edge of P, and let V be the endpoint of E containing two ~ angles of E. Let F and F' be the two faces of P containing E. With no loss of generality let F' be the face containing two c~ angles of E. Further, let E' be the other edge of F' at V, and V' the other endpoint of E' (see Figure 3). If the angle of F' at V' is an c¢ angle, then all the angles of F' must be angles. Otherwise, an ~ angle of F' would be adjacent to a/~ angle of F!, and this ~ angle of F' could not be mapped onto the ~ angle of F' at Vby any symmetry of P. This, however, contradicts the face-transitivity of P since no symmetry of P can map F onto F'. On the other hand, if the angle of F' at V' is a /~ angle, then the other angles of E' would be 7 angles. By a similar argument this would contradict vertex-transitivity. This completes the proof. The following is an analogous result to Corollary 1 for fully-transitive polyhedra. T H E O R E M 2. If a polyhedron P is fully-transitive, then one of the following

holds:

116

STEVEN

L.

FARRIS

/

"~

f

.

°

~.°

•

E/

'5'

\',

°°\ ,1~

F

Fig. 3. The angle configurationof any edge of a fully-transitivepolyhedronhaving exactlytwo distinct angle classes with three from one class and one from the other at each edge, and surrounding edges.

(i) The angles of P belong to just one angle class. (ii) The angles of P belong to exactly two angle classes, and the configuration of angles at every edge of P is as in Figure 4(a). (iii) The angles of P belong to exactly three angle classes, and the configuration of angles at every edge of P is as in Figure 4(b).

Proof. By Theorem 1, P has at most three angle classes. (i) If P has just one angle class, then there is nothing to prove. (ii) Suppose P has exactly two angle classes. By Lemma 2 each edge of P contains two c~ angles and two fl angles. The possible configurations of these angles other than that of Figure 4(a) would contradict either face-transitivity or vertex-transitivity by arguments similar to those used in Lemma 2. The result in this case follows. (iii) If P has exactly three angle classes, then every edge of P must contain two angles from one of the angle classes, and one angle from each of the other two angle classes. With no loss of generality, say every edge contains two c~ angles, one fi~ angle, and one f12 angle. The configuration of Figure 4(b) is also forced by arguments similar to those used in Lemma 2. This establishes the theorem. This knowledge of the possible edge configurations will give information

VERTEX-TRANSITIVE

AND EDGE-TRANSITIVE

POLYHEDRA

117

(al

I

(~ Fig. 4. The possible arrangements at each edge of a fully-transitive polyhedron P. In (a) P has exactly two distinct angle classes, and in (b) P has three distinct angle classes.

about the configuration of angles at the vertices and faces of vertextransitive and edge-transitive polyhedra.

3. ANOLE CIRCUITS

Following the definition of a circuit of faces at a vertex, a circuit of angles at a vertex V of a polyhedron P shall be a sequence of angles a 1 . . . . , ~, at V such that ai and ai+l have an edge in c o m m o n for i = 1 , . . . , n - 1, and ~1 and ~, have an edge in c o m m o n . Similarly one can define a circuit of angles of a face F of P whether or not F is finite. The following gives the condition on circuits of angles at every vertex of a vertex-transitive and edge-transitive polyhedron. THEOREM

3.1. If a polyhedron P is vertex-transitive and edge-transitive

and V is a vertex of P, then P satisfies the following: (i) If P has exactly two angle classes, then a circuit of angles at V is ~, fi, :~,

fl,..., ft. (ii) If P has exactly three angle classes, then a circuit of angles at V is :c,

Proof. By Corollary 1 and T h e o r e m 2 we k n o w the angles at V must alternate between :~- and fl-type angles. The only thing to prove is that if case (ii) holds, then the fi-type angles must alternate between B~ and f12

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L. F A R R I S

V

Fig. 5. The arrangement of angles at a vertex V of a vertex-transitive and edge-transitive polyhedron P having three distinct angle classes and two fll angles adjacent to an • angle at V.

angles. Suppose this fails at V (see Figure 5). Clearly this configuration contradicts the fact that the two ~ angles pictured are equivalent. This completes the proof. Unlike the vertex angle circuits, the results for face angle circuits must be separated into two cases. The first case is for polyhedra that are vertextransitive, edge-transitive, and not face-transitive. This result follows directly from Corollary 1. The second case is for fully-transitive polyhedra, and this result is proven in a manner similar to Theorem 3.1. Thus, the proofs will be omitted. COROLLARY 2. If a polyhedron P is vertex-transitive, edge-transitive , and not face-transit!re, then P has two face classes ~ and -¢'A, and one of the following holds: (i) P has exactly two angle classes, and both classes of faces consist of

VERTEX-TRANSITIVE AND EDGE-TRANSITIVE POLYHEDRA

119

regular faces with the faces of ~ having ~ face angles and the faces of f9 having fl face angles. (ii) P has exactly three angle classes, and the faces from one class of faces are regular having ct angles as face angles, and the faces from the other class offaces have angle circuits ill, ~62, ill, f12 .... , fi2 if they are finite and ... ill, f12, ill, f12,.., if they are infinite. By the above result one class of faces is comprised of regular faces. With no loss of generality assume the faces of ~ are always regular with c~ angles as face angles, and the faces of .~ contain fl angles in (i) and fll and flz angles in (ii). T H E O R E M 3.1. Let P be a fully-transitive polyhedron, and let F be any face

of P. One of the following holds: (i) P has exactly one angle class. (ii) P has exactly two angle classes, and a circuit of angles of F is c~, fl, :~,

fl,..., fl if F is finite and .... ct, fl, ~, ~ . . . . if F is infinite. (iii) P has exactly three angle classes, and a circuit of angles o f f is ~, ill,

~, f12,.. ., f12 if F is finite and ..., ct, ill, ~, f12, c~, E l , . . . if F is infinite. 4.

PETRIE

POLYHEDRA

Finally, we establish a relationship between polyhedra that are vertextransitive and edge-transitive and polyhedra that are fully-transitive using the concept of a Petrie polyhedron. Recall from Grfinbaum [3] or Coxeter [1], for example, that a Petrie polygon II of a polyhedron P is a polygon with vertices and edges chosen from the vertices and edges of P so that each two successive edges of YI are on the same face of P, but no three successive edges of YI are incident with the same face of P. When the collection of all Petrie polygons of P form a polyhedron, the new polyhedron is called the Petrie polyhedron of P, and it is denoted by n(P). It is easy to show that P = n(~(P)). From this the following result is immediate, and its proof is omitted. L E M M A 4. I f a polyhedron P admits a Petrie polyhedron, n(P), then P and n (P) have the same symmetry group. From this result the main result of this section is obtained. T H E O R E M 4. I f a vertex-transitive and edge-transitive polyhedron P admits

a Petrie polyhedron it(P), then either P or n(P) is fully-transitive. Proof Assume P is not fully-transitive. Referring to Figure 2, it is immediate that any two Petrie polygons H 1 and 172 both contain an :~

120

STEVEN

L. F A R R I S

angle. The symmetry a that maps the first ~ angle to the second a angle maps I-I1 to I-I2 . The result follows immediately.

5.

REMARKS

(1) From the complete list of symmetry groups of E 3 o n e can classify all of the candidate sets of vertices of vertex-transitive polyhedra since they are just the orbits of points under the group action in E 3. This was carried out for the finite symmetry groups in E 3by Rober tson and Carter in I-4]. The candidate sets of edges from each set of vertices can be obtained for vertex-transitive and edgetransitive polyhedra using the fact that all edges of such a polyhedron are congruent. Using the angles created by such a candidate set of edges at any of the vertices, candidate sets of faces can be chosen, and tested to see if they form a vertex-transitive and edge-transitive polyhedron. For each polyhedron that is face-transitive only one type of face needs to be chosen, and if P is not facetransitive only two types of faces need to be chosen. Further, the necessary conditions established earlier will reduce the possible choices of faces making the enumeration task fairly efficient. As an example of this examine the vertices of the cuboctahedron. Carrying out the procedure with these vertices there are two candidate sets of edges, and twelve candidate sets of faces all of which produce vertextransitive and edge-transitive polyhedra. Six of these are fully-transitive, and they are depicted in Figure 6. In this figure just one of the congruent faces is shown in bold lines for each fully-transitive polyhedron within the cuboctahedron. The other six are Petrie polyhedra of the six fully-transitive polyhedra, and they are not face-transitive. The first three have planar faces and appear in Wenninger [5-1. These are the cuboctahedron itself, the cubohemioctahedron, and the octahemioctahedron. These are pictured in illustrations 11, 78, and 68 respectively in [5]. The other three are depicted in Figure 7. Each of these is depicted with two representative face types, one from each face class. Notice that all of these polyhedra use two of the three angle classes available at any one time. It is an easy task to check that using one or all three of these classes to form faces of a vertex-transitive and edge-transitive polyhedron is not possible. Further examples of fully-transitive polyhedra are given in Figure 8. Each of these has a Petrie polyhedron which is vertex-transitive and edgetransitive, but not face-transitive. Further examples of vertex-transitive and edge-transitive polyhedra which are not face-transitive appear in [5]. The

VERTEX-TRANSITIVE

6

8

4

AND E D G E - T R A N S I T I V E

8-GON FACES

8-OON FACES

12-GON FACES

6

6

4

POLYHEDRA

121

8-GON FACES

~5-~)N rACES

J2-GON FACES

Fig. 6. The six fully-transitive polyhedra having vertices equivalent to the vertices of the cuboctahedron.

classification of the finite, fully-transitive polyhedra alone contains 184 classes of polyhedra. The enumeration of the finite, vertex-transitive and edge-transitive polyhedra which are not face-transitive contains about the same number of classes as the finite, fully-transitive polyhedra due to Theorem 4. Finally, the corresponding infinite classifications, which must still be carried out, are expected to be even larger due to the increase in the number of symmetry groups when moving from the finite to the infinite.

122

STEVEN

L. F A R R I S

6 QUADRANGLES AND

4 HEXAOONS

,,} 8 TRIANGLES

AND

6

QUADRANGLES

8 TRIANGLES

AND

4 HEXAGONS

Fig. 7. Three of the six vertex-transitive and edge-transitive polyhedra that are not facetransitive, and whose vertices are equivalent to those of the cuboctahedron. These are the Petrie polyhedra of the three polyhedra pictured on the right in Figure 6.

(2) It appears as if similar results can be established for polyhedra which are simultaneously edge-transitive and face-transitive. This study would provide a nice dual to the one presented here, and it is worthy of further investigation. (3) The classifications obtained from these results will be useful for studying physical phenomena. For example, it is hoped that these classifications will yield a method of enumerating the periodic minimal surfaces in E 3 which appear in many objects of nature.

VERTEX-TRANSITIVE

4

AND EDGE-TRANSITIVE POLYHEDRA

I2-GON FACES HULL: ,366

12_ IO-GON FACES HULL: 555

x

I?_

123

IO-GON FACES HULL: 5555

X

INFINITE PLANAR

~" *,b

i8_

20-GON FACE8 HULL: 566

12

20-GON FACES HULL: 5454

Fig. 8. Further examples of fully-transitive polyhedra having Petrie. polyhedra that are vertextransitive, edge-transitive, and not face-transitive.

REFERENCES, 1. Coxeter, H. S. M., Regular Polytopes (3rd edn), Dover, New York, 1973, 2. Dress, A. W. M., 'A Combinatorial Theory-of Gr/inbaum's New Regular Polyhedra, Part lI: Complete Enumeration', Aequationes Math, 29 (1985), 222-243. 3. Grfinbaum, B., 'Regular Polyhedra- Old and New', Aequationes Math. 16 (1977), 1-20. 4. Robertson, S~ A. and Carter, S~, 'On the Platonic and Archimedean Solids', J. London Math. Soc. (2), 2 (1970), 125-132.

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STEVEN L. FARR1S

5, Wenninger, M. J., Polyhedron Models, Cambridge Univ. Press, 1971.

Author's address: Steven L. Farris, Department of Mathematical Sciences, Ball State University,

Muncie, IN 4 7306, U.S.A. (Received, March 11, 1986; revised version, December 8, 1986)