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ISOGONAL PRISMATOIDS.
1. Introduction.
According to Webster [18], a prismatoid is "a polyhedron having all of its vertices in two parallel planes", which we shall always take as distinct. Not every text agrees completely with this definition, but it is precisely the one we need. This is a very wide class of polyhedra it includes prisms, antiprisms, pyramids and many other polyhedra. Therefore we shall restrict attention to isogonal prismatoids, that is, prismatoids in which all vertices are equivalent under isometric symmetries of the polyhedron. We note that we shall understand the isogonality restriction in a strong sense: since prismatoids have two distinguished planes, only symmetries which map the pair of planes onto itself will be considered. Two classes of examples of isogonal prismatoids are known since antiquity: certain prisms and antiprisms. The prisms and antiprisms have two congruent bases (which are polygons contained in the parallel planes mentioned in the definition) together with a mantle connecting the two bases; the mantle consists of quadrangles in case of prisms, and triangles in case of antiprisms. In Figure 1 are shown examples of Archimedean prisms, in which the bases are regular convex polygons and the mantle faces are squares, as well as examples of Archimedean antiprisms, in which the bases are regular convex polygons and the mantle consists of equilateral triangles. It is well known that every convex isogonal prismatoid is isomorphic (or combinatorially equivalent) to an Archimedean prism or antiprism.
The aim of the present paper is to investigate more general isogonal prismatoids, in which faces may be selfintersecting polygons, and different faces may have intersections that are not common edges or vertices. As we shall see, this enlarges the family of isogonal prismatoids much beyond the convex prisms and antiprisms. In fact, it will turn out that this family contains orientable polyhedra of positive genus, as well as non-orientable polyhedra.
The only source of which we are aware, in which prismatoids other than Archimedean prisms and antiprisms are discussed in some detail, is Brckner [2, Sections 114 and 140]. Uniform prismatoids (that is, prisms and antiprisms with regular polygons as faces) have been discussed by several authors, as part of more inclusive investigations of uniform polyhedra in general. Coxeter et al. [3] give a list of uniform prisms and antiprisms, as well as references to the earlier literature. The enumeration of uniform polyhedra given by Coxeter et al. was shown to be complete by Sopov [16] and Skilling [15]. Har'El [9] gives skeletal illustrations of all non-prismatic uniform polyhedra, but illustrates only the pentagonal uniform prisms and antiprisms. In the more general case considered by Brckner one can get a glimpse of some of the unusual nonuniform isogonal prisms and antiprisms. However, Brckner's presentation is completely ad hoc, with no particular guiding ideas and no clear classification or description principles; moreover, it is very incomplete and misses some of the most interesting polyhedra of the types it purports to enumerate. We shall show that it is possible to describe many additional prismatoids in a satisfactory and natural manner, and illustrate how varied are the shapes that prismatoids can have.
In order to do this, we have to face another curious shortcoming regarding polyhedra namely the fact that the literature contains no satisfactory methods or ideas for a classification of polyhedra of the general kind we are concerned with. We shall attempt to give usable definitions of polyhedra, and of ways of deciding whether two polyhedra are "of the same types", and apply them to investigate isogonal prismatoids.
The present paper is organized as follows. In Section 2 we give precise definitions of the concepts we need. In Section 3 we present a complete classification of prisms and antiprisms (not necessarily convex or uniform), while Sections 4 and 5 discuss other isogonal prismatoids. Additional comments are collected in Section 6.
2. Polygons and polyhedra.
There is no generally accepted terminology for polyhedra in which faces may selfintersect, or intersect each other in various ways. Below we shall give the definitions which seem appropriate to the topic at hand; however, since the polyhedra are built up of polygons, we shall first supply and illustrate the corresponding definitions for planar polygons. The exposition here follows the one in Grnbaum [7], simplified as appropriate for the restricted classes of polygons and polyhedra under investigation. Specifically, we are concerned here only with polygons and polyhedra that are called unicursal in [7], and only with epipedal realizations of such polyhedra.
An abstract polygon is a fixed simple circuit c, that is, a system consisting of a finite, cyclically ordered set v of distinct elements, and the set e of distinct unordered pairs of adjacent elements of v. The elements of v are the "vertices" of the polygon, and the pairs represent the "edges" of the abstract polygon.
A geometric polygon (or polygon for short) P is the image of an abstract polygon c under a map f which associates with each element of v a point (vertex ofP) in a Euclidean plane E2, and with each pair from e the line-segment (edge of P) having as endpoints the images of the elements of v that constitute the pair. If P has n edges we shall call it an n-gon. We note that different vertices of v may be represented by the same point of the plane; this does not affect the incidences of the vertices of P with its edges, although it entails the possibility of edges that have coinciding vertices and are therefore represented by single-point line-segments. Also, edges may cross or overlapp in various ways, or even coincide.
In this paper we shall be interested mainly in triangles, quadrangles and isogonal polygons, the latter including, in particular, the regular polygons. Since the literature on isogonal polygons is meager, and that on regular polygons contains a considerable amount of misleading statements, we present some details concerning these concepts. In analogy to the definition for polyhedra, a polygon is called isogonal provided its isometric symmetries act transitively on its vertices. A polygon P is called regular if its isometric symmetries act transitively on the flags of P, where a flag is the pair consisting of an edge and one of its endpoints. (Many other definitions of regular polygons, equivalent to the one given here, are possible.) If k=[n/2] then there are k different regular n-gons, denoted by {n/d}, where d = 1, 2, , k. (Throughout, we shall consider as equal all polygons which can be mapped onto each other by similarity transformations.) We note that, contrary to frequently encountered assertions, a regular ngon exists, and is a well-defined geometric object, even if the integers n and d are not relatively prime; see Figure 2 for examples. (Concerning this topic and its history see Grnbaum [5], [7].)
Isogonal polygons seem to have been first investigated by Hess [10]; however, due to the lack of a consistent point of view and to disregard of the deeper insights of Meister [11] and Wiener [19], this long work is unusable for any further investigations. A systematic approach to isogonal polygons, including a full description and classification, appears in Grnbaum [6]; here we only briefly recall the results. In order to avoid trivialities, we exclude from further consideration isogonal polygons all vertices of which coincide.
For odd n=2k+1, the only isogonal polygons are the regular polygons {n/d} where d = 1, 2, ..., k. In Figure 2 we show all the types of isogonal n-gons for n = 5, 7, 9 and 15; these examples are typical and should be sufficient to show what happens if n and d are not relatively prime.
For even n several cases need to be distinguished. A schematic illustration of the somewhat special case n = 4 is shown in Figure3. For even n e" 6, the isogonal ngons form [(n+2)/4] families, of which [n/4] are continuous and can be parametrized by one real parameter each; an exceptional family occurs for n = 4k + 2, and consists of the single (regular) polygon {n/(2k+1)}. If n = 4k + 2 with k e" 1 then, for each d = 1, 2, ... , k, there is a continuous family denoted n/d of isogonal n-gons that starts with the regular polygon {n/d} and ends with the regular polygon {n/e}, where e = 2k + 1 d. For n = 6 and n= 10 this is illustrated in Figures 4 and 5; the three continuous families that occur for n = 14 are shown in Grnbaum [6]. Similarly, for n = 4k with k e" 2 there are k continuous families n/d, where d = 1, 2, ... , k ; each family n/d starts at the regular polygon {n/d}, and except for the family n/k ends at the regular polygon {n/e}, where e= 2k - d. The family n/k reaches from {n/k} to {n/(2k)}. The case n = 8 is illustrated in Figure 6.
Before starting the detailed discussion of isogonal prismatoids we have to make explicit what we understand as polyhedra, and how we distinguish between "types" of polyhedra. This definition will turn out to be applicable to polygons as well, and to be an extension of the classification of polygons proposed long ago by Steinitz [17]. We begin by defining abstract polyhedra.
A finite family of abstract polygons is an abstract polyhedron provided:
(i) each "edge" of each of the polygons (which are the "faces" of the abstract polyhedron) is an "edge" of precisely one other "face";
(ii) all "faces" that contain a "vertex" form a single simple circuit; and
(iii) the family of "faces" is connected in the sense that any two belong to a chain of "faces" in which adjacent "faces" share an "edge".
Since each "face" of an abstract polyhedron can be considered as the boundary of a topological disk, an abstract polyhedron can be interpreted as a cell complex decomposition of a 2-manifold. Two abstract polyhedra are isomorphic if there is an incidence preserving bijection between their "vertices", their "edges", and their "faces". An abstract prismatoid has two disjoint "faces" which together comprise all "vertices"; it is isogonal if the group of automorphisms acts transitively on the set of "vertices". For any abstract isogonal polyhedron one can define its vertex-symbol, the cyclic list of sizes of the polygons at one (hence every) vertex of the polyhedron; of the different possible symbols, the one lexicographically first is usually chosen. Abstract prisms and antiprisms have vertex-symbols (4.4.n) resp. (3.3.3.n), where n indicates the number of sides of the basis.
A geometric polyhedron, or polyhedron P for short, is a representation of an abstract polyhedron (said to be the underlying abstract polyhedron of P) in the Euclidean 3space E3, such that "vertices" are represented by points, "edges" by segments, and "faces" by (planar) polygons, giving the vertices, edges and faces of the polyhedron.
We say that a polyhedron is acoptic (from the Greek koptw, to cut) if:
(i) all its faces are simple polygons, so that they can be unambiguously represented (or replaced) by simply-connected polygonal regions, and
(ii) the intersection of any two such regions consists of a union (possibly empty) of vertices and edges of each.
Clearly, convex polyhedra are acoptic, but so are many nonconvex polyhedra. Acoptic polyhedra are the ones for which cardboard models give a faithful representation; such a model is, in fact, an embedding in 3-space of the 2manifold determined by the abstract polyhedron.
A polyhedron is called aploic (from the Greek aploos, onefold, simple) if, whenever X and Y are two distinct "vertices", or distinct "edges", or distinct "faces" of the underlying polyhedron, then the affine hulls aff X and aff Y of X and Y are distinct. The analogous definition applies to polygons as well. Aploic polyhedra can be considered as that generalization of acoptic polyhedra to polyhedra with selfintersections which is closest to the "naive" understanding. However, it should be pointed out that the tradition which attempts to present aploic polyhedra (such as the Kepler-Poinsot regular polyhedra) by cardboard models is in many cases misguided and misleading; the most instructive models of non-acoptic aploic polyhedra are the skeletal ones, or those cardboard models in which the "hidden" parts are included and made visible through appropriate openings.
Each of the two planes that appear in the definition of a prismatoid may contain one or more faces, or may not contain any faces; for isogonal prismatoids the figures in the two planes must be congruent, and in each plane the symmetries of the polyhedron must act transitively on the vertices contained in that plane. The polygons contained in each of the two planes are said to form the basis of the prismatoid; if no polygon is a basis, we say that the prismatoid is basis-free. An obvious consequence of the isogonality condition is that the bases of any prismatoid which is not basis-free must be congruent isogonal polygons or isogonal compounds of polygons. (An "isogonal compound" of polygons is a collection of polygons such that the isometric symmetries of the plane act transitively on the vertices of the collection.)
Since all vertices of any isogonal polyhedron are cospherical, and all vertices of any prismatoid lie on two parallel planes, it is immediate that any aploic isogonal prismatoid can have only triangles and quadrangles as faces of its mantle, and that it is either basis-free or each basis consists of a single isogonal polygon. The prisms and antiprisms (that is, realizations of abstract prisms or antiprisms) are examples of the latter possibility, and we shall present their classification in the next section.
Before we can proceed with the classification, we need to define what constitutes a "type". While this is a staple in the theory of convex polyhedra, and even for more general acoptic polyhedra the problem of definition is not severe, in the present situation there seems to be no reasonable definition to be found in the literature. Here by "reasonable" is meant a definition that would be applicable to polyhedra of some degree of generality, and which reflects some natural properties we may wish to see preserved among polyhedra we assign to the same type but which, at the same time, is finer than the combinatorial classification which is given by the abstract polyhedra. (We note that we find it appropriate not to distinguish between two isogonal prismatoids or more general polyhedra if one can be mapped onto the other by a nonsingular affinity that is compatible with all their isometric symmetries.) It is not clear whether our definition is reasonable for very general polyhedra, but its application to aploic isogonal prismatoids appears to be both convenient and reasonable. The definition is analogous to the classification of polygons that goes back to Steinitz [17], and is similar to those in Robertson [12] and Grnbaum [7]; it is illustrated in Figures 7, 8 and 9.
Two polyhedra P0 and P1 are of the same geometric type provided there is a continuous family P(t), 0 d" t d" 1, of polyhedra, where P(0) = P0 and P(1) is P1 or a mirror image of P1, and such that:
(i) all members of the family have the same underlying abstract polyhedron P;
(ii) all members of the family have same group of isometric symmetries; (iii) for any two distinct faces F1 and F2 of the underlying abstract polyhedron P, the affine hull of the union of the images of F1 and F2 has the same dimension for every member P(t) of the family.
In particular, two aploic isogonal prismatoids are of the same geometric type provided they have the same underlying abstract polyhedron and the same symmetries, and there is a continuous family of polyhedra connecting them in such a way that each polyhedron in the family is an aploic isogonal prismatoid with the same properties. Since here we shall be interested only in the geometric types of the prismatoids under discussion, we shall usually simplify the language and speak of their type.
A finer classification would result if condition (iii) were expanded to include any two faces in the wider sense (that is, the set that includes the faces, edges and vertices) of the polyhedron P.
The definition of geometric type can obviously be applied to polygons as well, the only change being the replacement of "faces" in condition (iii) by "edges". In order to illustrate the concept of geometric type, and of the finer classification mentioned above, we show in Figures 7 and 8 the different geometric types of quadrangles and of acoptic pentagons, and in Figure 9 the different types of acoptic polyhedra with five vertices.
In the next section we shall apply the definitions given here to the simplest prismatoids the prisms and antiprisms.
3. The classification of prisms and antiprisms.
We begin by considering antiprisms, that is polyhedra with vertex-symbol (3.3.3.n), where n e" 3. We note that for a given n, all abstract antiprisms with vertex-symbol (3.3.3.n) are isomorphic. In any realization of an abstract antiprism by an isogonal polyhedron P, the bases of P have to be congruent regular polygons. For each regular polygon {n/d} there exists a continuous family of antiprisms with bases congruent to {n/d}; this family can be parametrized by a real-valued parameter. The parameter can be chosen to measure the twist of one of the bases with respect to the other; the two final antiprisms in each family (and only they) have reflective symmetry. In Figures 10, 11 and 12 the families of antiprisms with n = 3, 4 and 5 are illustrated. Either one or both extreme members of each such family has a representative with regular mantle faces, that is, is a uniform polyhedron and appears in the enumeration of Coxeter et al. [3]. Brckner [2] mentions such antiprisms in parts 3 and 4 of Section 140, while the existence of the other polyhedra in each family is mentioned briefly (and in very unclear and confused terms, without any details) in parts 7 and 8 of Brckner's Section 140. An antiprism with basis {n/d} can be aploic only if n and d are relatively prime; this is satisfied for the antiprisms in Figures 10, 11 and 12. Moreover, even if n and d are relatively prime, for certain values of the twist parameter the antiprism fails to be aploic due to coplanarity of distinct mantle faces. These cases are marked by asterisks in Figures 10, 11 and 12; the non-aploic polyhedra in each family that are marked by two asterisks seem to be particularly interesting.
According to the above definition, we can say that each of the continuous families of antiprisms contains five different geometric types of aploic polyhedra: the two extreme polyhedra have more symmetries than the other members of the family, and the two non-aploic polyhedra partition the intermediate members of the family into three types. If aploic, both extreme polyhedra can be represented by uniform polyhedra if and only if d>n/3.
Turning to prisms, with vertex-symbols (4.4.n), n e" 3, we note that, again, for each n there is a single combinatorial type of abstract prisms. Concerning geometric realizations by isogonal polyhedra, we find that for each isogonal n-gon with odd n there exist two (and only two) different prisms with bases congruent to this ngon; as mentioned above, this has to be a regular ngon {n/d}. For reasons which should be evident from the examples in Figure 13, we say that one of them has parallel bases, the other antiparallel bases. The former (which may be denoted by a "p" appended to the symbol of their bases) have rectangular mantle faces, the later (denoted similarly by an "a") have selfintersecting isogonal quadrangles as mantle faces. These prisms are aploic if and only if n and d are relatively prime; if d = 1 then {n/d}p is acoptic. Obviously, all prisms with parallel bases can be represented by regular-faced polyhedra.
For even n the situation is more interesting, due to the greater variety of isogonal polygons in this case, as well as to the greater flexibility possible for the choice of polygons that form the mantle. We consider here only the case in which n and d are relatively prime. Then the prism is aploic unless the basis is a polygon with coinciding vertices; that is, the prism is aploic if its basis is aploic. (If n and d have a common divisor k > 1, the appearance of the prisms is the same as for n' = n/k and d' = d/k, with k-tuples of vertices situated at each vertex of the prism corresponding to n' and d'; all such prisms are non-aploic.) For even n > 4, for every nonregular polygon of each family n/d there are four different prisms with this polygon as basis; each regular polygon {n/d} is the basis for two distinct prisms. The most interesting aspect of the situation is that for each pair (n,d) with d < n/4 all the prisms whose bases are in the family n/d form a single continuum of prisms. The "space" of these prisms, which can be parametrized by a real-valued parameter, is in fact homeomorphic to a circle. This is illustrated for n=6, d= 1 in Figure 14, and for n = 8, d = 1 in Figure 15; similar diagrams result for other pairs (n,d). In each case other than n = 4d, there are eight prisms which are the single members of their geometric type (only two of them are aploic), and eight geometric types that consist of a continuum of distinct aploic polyhedra; when d= 1 two of the latter families consist of acoptic prisms, which were considered also by Robertson and Carter [13] and Robertson [12]. Brckner [2] discusses prisms in Section 114 and in parts 1, 2, 5, 6 of Section 140; the sketchy presentation completely misses antiparallel prisms, as well as those prisms which are represented in the third row of each of our Figures 14 and 15.
The case n = 4, d = 1 is somewhat special; there is a single continuous family of prisms with quadrangular basis, as illustrated in Figure 16.
4. Other aploic prismatoids with bases.
Despite the great variety of forms possible for prisms and antiprisms, their abundance is negligible compared to the other isogonal prismatoids even if only aploic ones are considered. We start by discussing the prismatoids with a basis, restricting attention to those all mantle faces of which are quadrangles. Since we are interested in polyhedra other than prisms, the number of such quadrangular faces incident with each vertex must be at least three. It is most astonishing that none of these seems to have been mentioned anywhere in the literature; hence it is clear that they have no accepted names.
For aploic isogonal prismatoids that can be described collectively by the vertex-symbol (4.4.4.n), an investigation of the possible structure of the underlying abstract polyhedra leads to precisely three distinct combinatorial types; instead of a listing of faces in the general case, we describe and illustrate the three kinds in Figure 17 for n = 8. However, before continuing, it seems appropriate to acknowledge that the illustrations are hard to interpret, and to describe schematic diagrams useful both for understanding the structure of the polyhedra under discussion, and in establishing facts about them.
The diagrams in question, which are illustrated in Figure 18, take advantage of the fact that for isogonal prismatoids it is enough to specify which are the faces incident with one vertex. For ease of description we shall restrict the discussion to the case in which the bases are regular polygons. (In fact, there is no generality of structure lost by this simplification.) We start by choosing the n points that represent the vertices of one of the bases. Then, for one of the vertices we indicate all the faces incident with it, using the following conventions (loosely derived from the idea that we are looking at the prismatoid from far above the center):
(i) We indicate only the faces of the mantle.
(ii) A horizontal edge is indicated by a solid line, while an edge connecting vertices of the different bases is indicated by a dashed line; vertical edges are not shown.
(iii) A quadrangle with a pair of vertical edges is indicated by drawing a single bold line (either solid or dashed).
(iv) A vertical selfintersecting quadrangle with a pair of horizontal edges is indicated by a pair of parallel thin lines, one solid and one dashed.
(v) For nonvertical faces all four edges are indicated with thin lines, solid or dashed as appropriate.
With a little patience and some practice, these diagrams can be used with advantage for discussing the polyhedra they represent. For example, the two prisms in the leftmost column of Figure 14 are represented by the diagrams in Figure 18(a), while the prismatoids (4.4.4.8) of Figure 17 are represented by the three diagrams in Figure 18(b).
It is easily verified that these three kinds of polyhedra with vertex-symbols (4.4.4.n) exist whenever n > 4 is a multiple of 4. Polyhedra of the kind shown in Figure 17(a) are orientable, the other two kinds are non-orientable.
The examples in Figures 17 and 18 have as bases regular octagons, but this is just for ease of visualization: any isogonal n-gon with n=4k e" 8 can serve as basis for each of these three kinds of polyhedra, and again the polyhedra form continuous families. For the particular polyhedron of Figure 17(a) this family is illustrated by the diagrams in Figure 19. Details about the exact nature of these families have not been investigated so far.
After the relatively simple case of prismatoids with vertex-symbols (4.4.4.n) we turn to the more complicated polyhedra that have vertex-symbols (4.4.4.4.n). To begin with, there are polyhedra of this kind that are analogous to some extent to the ones of type (4.4.4.n) namely, involving a variety of shapes of quadrangles, including selfintersecting ones. Three examples of such polyhedra are shown in Figure 20 and described in its caption; the corresponding diagrams are shown in Figure 21; in this case, however, these examples are only some of the possibilities. Again, the use of regular octagons as bases is just for ease of visualization; any isogonal polygon could be used, and the polyhedra form continuous families. These have not been investigated in any detail, but it should be noted that if the basis is a non-regular isogonal polygon, the number of distinct quadrangles in the mantle may be larger than in Figure 20.
Besides these "strange" polyhedra, there are two families of isogonal prismatoids with vertex-symbol (4.4.4.4.n) that have mantles consisting entirely of simple quadrangles which, in suitable representatives, can all be made congruent. Thus they are, in a sense, closest to acoptic polyhedra and in particular to prisms and antiprisms. Prismatoids in the two families will be denoted by symbols of the form P(t0, t1; n) and A(t0, t1; n).
In Figure 22 we show some of the simplest representatives of the first family; the corresponding diagrams are shown in Figure 23. Since these polyhedra are just the smallest instances of prismatoids with more than three quadrangles incident with each vertex, we describe them by symbols that will be easily adaptable to the more general situations. Moreover, to simplify the exposition, we shall for the time being restrict attention to the case in which the basis is a regular polygon. Other possibilities will be considered later. We start by observing that each mantle face is a trapezoid T1; the parallel edges of T1 belong to the different basis planes of the prismatoid but only one of these edges is an edge of the basis polygon. All these trapezoids are congruent. In the symbol P(t0, t1; n) of a prismatoid of this kind, n is the number of vertices of each base, t0 is the span of that edge of T1 which is also an edge of the basis, and t1 is the span of each side-edge of T1. Here (and in the sequel) by "span" we mean across how many steps along the vertices that are in the basis does the edge in question reach; note that this refers to the vertices as they are encountered along the circle on which the lie, and not necessarily along the edges of the basis-polygon. The two bases are aligned, and the side-edges of T1, besides having span t1, also reach from one basis to the other. The fourth edge of T1 has span t0 + 2t1, and is a diameter of the n-gonal basis. Therefore n = 2t0 + 4t1. Since polyhedra are (by definition) connected, the positive integers t0 and t1 must be relatively prime, and t0 must be odd.
The construction of polyhedra in the second family, which are denoted by the symbol A(t0, t1; n), can be described as follows; again we consider here only the case in which each basis is a regular n-gon. We take these two n-gons in an antiprismatic position, and number all 2n vertices consecutively as they appear on an orthogonal projection. The mantle faces T1 are, as in the first family, trapezoids which share one of their parallel edges with a basis. That edge has span t0, which therefore must be an even positive integer, while each of the side-edges of the trapezoid has span t1, which is an odd positive integer. Here the relation between the parameters is n = t0 + 2t1. In Figure 24 are shown some examples of these prismatoids; the diagrams of three of them are contained in Figure 25.
The two families of polyhedra with vertex-symbols (4.4.4.4.n) are the starting members (for k = 1) of two "superfamilies" of isogonal prismatoids P(t0, t1,, tk; n) and A(t0, t1,, tk; n), with vertex-symbols (44k.n), where k is any positive integer, and n is determined in a suitable manner. (The "exponential" notation in the vertex-symbol is just an abridgement of the "product" notation.) The definition of these families is analogous to the one we have seen in case k = 1, except that now the "free" edge (of span t0 + 2t1) of the trapezoid T1, instead of being a diameter of the basis-polygon, is one of the parallel edges of a second trapezoid T2, whose side-edges have span t2. If k > 2, the fourth edge (of span t0 + 2t1 + 2t2) of T2 is one of the parallel edges of a trapezoid T3, the side-edges of which have span t3, and so on in a selfexplanatory manner. Hence the mantle consists of k kinds of trapezoids, each represented by 2n congruent copies. For prismatoids from the family P(t0, t1,, tk; n) the value of n is determined by n = 2t0 + 4(t1 + + tk), while for prismatoids from A(t0, t1,, tk; n) we have n = t0 + 2(t1 + + tk); the labelling of the vertices is in both cases the same as for k = 1. As in case k = 1, we need t0 to be positive, and odd in the Pfamilies, while even in the A-families. In both cases the parameters t0, t1, , tk cannot all have a common factor, and some additional conditions need to be satisfied in order to avoid unwanted coincidences. The precise conditions have not been determined, but it appears that if all parameters are positive it is sufficient to require that none equals the sum of any collection of the others. Examples of such polyhedra in case k = 2 are shown in Figures26 and 27.
In addition to the two families of isogonal prismatoids with vertex-symbols (4h.n), where h `" 0 (mod 4), there are two analogous families for which h `" 2 (mod 4). The polyhedra in these families will be denoted by symbols P(t0, t1,& , tk, t*; n) and A(t0, t1,& , tk, t*; n); they have vertex-symbols (44k+2.n). The polyhedra of these types are constructed in exactly the same way as indicated by those parts of their symbol which coincide with the symbols of the polyhedra discussed above. The only additional part, t*, indicates that the fourth edge of the trapezoid Tk is also an edge of a selfintersecting quadrangle T* which has a pair
of parallel edges, the distance between the parallel edges being t*, and the nonparallel edges having span n/2 in the first family and n in the second. Here n = 2t0 + 4(t1 + + tk) + 2t* for polyhedra with symbol P(t0, t1,, tk, t*; n), and n = t0 + 2(t1 + + tk) + t* for polyhedra with symbol A(t0, t1,, tk, t*; n). Again various conditions, which have not been completely determined so far, have to be satisfied by the parameters in order to avoid unwanted coincidences or disconnected "polyhedra". Examples of polyhedra of these types for k = 1 and shown in Figures 28 and 29.
As stated above, the discussion of the P and Afamilies was conducted assuming the basis polygons to be regular. However, the symmetry of these polyhedra allows various modifications of the bases and of the polyhedra themselves, without loss of their character as isogonal prismatoids. As is easily verified, in the case of polyhedra P(t0, t1,, tk; n) or P(t0, t1,, tk, t*; n), any isogonal polygon can serve as basis; in fact, these polyhedra give rise to continuous families such as the ones in Figures 14, 15, 16, 19. This is illustrated in Figure 30 for the polyhedron P(1, 1; 6). Analogously, the bases of polyhedra A(t0, t1,, tk; n) can be twisted with respect to each other, and then each trapezoid can be replaced by two triangles; if the replacement is done systematically, isogonal prismatoids with vertex-symbol (36k.n), where k e" 1, are obtained. These can be considered as natural relatives of the traditional antiprisms. As an example, one isogonal prismatoid with vertex-symbol (36.4), obtained from the polyhedron A(2,1;4) by a slight twist of one basis polygon and replacement of each trapezoid by two triangles, is illustrated in Figure 31. Clearly, the same method can be applied as well to all isogonal prismatoids P(t0, t1,, tk; n) with regular polygons as basis.
5. Basis-free aploic prismatoids.
There are several interesting families of basis-free aploic polyhedra, some of which we shall describe next. However, these are only a small part of the possible polyhedra of this kind. A systematic investigation would seem to be a challenging but rewarding task.
(i) The simplest (and most widely known) are the sphenoids, that is, isogonal tetrahedra, examples of which are shown in Figure 32. If we had included "digons" among polygons, sphenoids would be antiprisms with digonal bases. All sphenoids are isohedral as well (that is, their symmetry group acts transitively on the faces); in the terminology of Grnbaum [7] such polyhedra are called "noble". The sphenoids and the Platonic (regular) polyhedra are the only noble polyhedra that are acoptic; below we shall describe additional noble polyhedra which are not acoptic. There are two geometric types of aploic sphenoids, distinguished by their symmetry group. The less symmetric sphenoids form a continuous family that depends on one real parameter.
(ii) A general method for the generation of basis-free isogonal prismatoids uses the Boolean sum of two or more suitable isogonal prismatoids with bases. The prismatoids have to be chosen so that their bases coincide in pairs, and those pairs are deleted from the new polyhedron. With appropriate choices, the resulting isogonal prismatoids are aploic; in fact, in some special cases they are even acoptic. The method of Boolean sums is an open-ended one, since the number of prismatoids with bases that can be involved in the formation of a basis-free prismatoid can be arbitrarily large; hence there is no hope of giving a complete enumeration. Some of the possibilities are discussed in the following paragraphs.
The simplest case is the Boolean sum of two antiprisms, or two prisms, or one prisms and one antiprism, which share both bases; eliminating these bases gives a basis-free isogonal prismatoid. This is illustrated in Figure 33. More complicated Boolean sums, each involving four prisms, are shown by the examples in Figure 34. The basisfree prismatoids in Figures 33 and 34 are aploic, those in Figure 34 are orientable. In fact, it is easy to verify that, understanding the faces of the prismatoid in Figure 34(a) as determining quadrangular regions, one obtains a realization of the regular toroidal map {4,4}4,0. The results of analogous Boolean sums involving four antiprisms are shown in Figure 35. The polyhedron in Figure 35(c) is not aploic, but it is noble; this type of noble polyhedra was described in Grnbaum [7] under the name "wreath polyhedra".
As a particular cases of Boolean sums, if two suitable acoptic antiprisms are used, or an antiprism and a prism, acoptic isogonal tori can be obtained. Examples of such toroidal polyhedra, with vertexsymbol (3.3.3.3.3.3) are shown in Figures 36 and 37. Like the antiprisms or prisms used in their construction, these polyhedra depend on various parameters; beyond certain parameter values they cease to be acoptic. Polyhedra of this kind were described (along with isogonal acoptic polyhedra of higher genera) in Grnbaum & Shephard [8]. According to a private communication from Prof. J. M. Wills, some of these toroidal prismatoids were described earlier, by U. Brehm at a meeting in Oberwolfach in 1977; however, the only published account of this presentation (Brehm & Khnel [1], page 438) contains no specifics, and does not mention isogonality.
Boolean sums of suitable prisms with prismatoids of type P(t0, t1,, tk; n) or P(t0, t1,, tk, t*; n) can yield aploic basis-free isogonal prismatoids with vertexsymbol (42h), where h e" 6. Other combinations using the same technique are possible as well; for example, from antiprisms and prismatoids of type A(t0, t1,& , tk; n) or A(t0, t1,& , tk, t*; n) one can obtain aploic basis-free isogonal prismatoids with vertex symbols (3.3.3.42h), where h e" 4, and many other polyhedra.
(iii) Another remarkable family of basis free isogonal prismatoids are the crown polyhedra; they were first described by Edmund Hess (see references in Grnbaum [7]) under the name "stephanoids" (from the Greek word for "crown"). Like the sphenoids, all crown polyhedra are isohedral as well (thus they are noble). The crown polyhedra are of two kinds, which can be called the prismatic and the antiprismatic, each kind depending on three positive integers as parameters; for appropriate values of these parameters, the crown polyhedra are aploic. In all cases, the faces are congruent selfintersecting quadrangles. The two kinds are illustrated in Figure 38 by one aploic representative each. For more details see Grnbaum [7, Section 6].
6. Comments and open questions.
(i) The present paper developed from the observation that the presentation of the classification of isogonal prisms and antiprisms in Brckner [2] is confused and incomplete. Since there appears to be no other treatment of this topic, I decided to write up my observations in what became Section 3 of this note. However, further investigation showed that Brckner was not only deficient in the treatment of isogonal prisms and antiprisms, but that he missed completely the huge collection of isogonal prismatoids discussed in the above Sections 4 and 5. (The only exception to this is his mention of the crown polyhedra ("stephanoids") which, curiously enough, he did not discuss in the presentation of isogonal polyhedra.) In the beginning I believed that Brckner (unconsciously ???) wished to avoid including in the discussion of isogonal polyhedra those that have selfintersecting quadrangles as faces, or polyhedra that do not have isohedral polar polyhedra; but the existence of polyhedra of type P(t0, t1,, tk; n), which he also failed to mention, disproves this idea. Now I am inclined to think that he simply did not look for any isohedra except those that are isomorphic to uniform polyhedra. Why he would have thought this appropriate (in particular, without mentioning it), and especially in view of the fact that he was aware of the existence of various noble polyhedra discovered by Hess, is really mystifying. But on the other hand, it is equally hard to understand that during the almost full century since the publication of Brckner's book no attempt was made to rectify his omission in fact, to the best of my knowledge, no mention of the shortcomings of his enumeration of isogonal polyhedra made its way into print !!! Naturally, since his enumeration of the isogonal prismatoids was so faulty, one has to wonder whether his enumeration of the other isogonal polyhedra is complete. Since Brckner in this context again deferred the discussion of some noble polyhedra to a later section, and since his list of noble polyhedra is incomplete, a negative answer is obvious. A thorough investigation of these questions would appear to be long overdue.
(ii) Non-aploic isogonal prismatoids arise not only as limiting cases of aploic families, but in many other ways as well. One general method, which is essentially again Boolean addition, is to replace one polygon P by an edge-sharing family of polygons that have free edges coinciding with those of P, and that have, as a family, the same symmetry as P. This is illustrated by the examples in Figure 39. Another application of the same idea is the observation that given a rectangle and the two selfintersecting quadrangles which have the same vertices, any one of these three polygons can be replaced by the family consisting of the other two, or can be used to replace that family. For example, if the polyhedron in Figure 17(c) is combined in the manner discussed in Section 5(ii) with an octagonal prism (with the deletion of the bases of both), the resulting polyhedron is not aploic since four faces of the original polyhedron have the same vertices as four of the eight mantle faces of the prism. However, if each such pair is replaced by the other selfintersecting quadrangle with the same vertices, the aploic isogonal prismatoid
shown in Figure 40 is obtained; this polyhedron has vertex-symbol (4.4.4.4), and it is orientable with genus 0. Many other such examples can be formed, resulting in either aploic or non-aploic polyhedra.
(iii) A complete determination of isogonal prismatoids with vertex-symbols (4.4.4.4.n), n e" 5, would be desirable, although probably quite hard. On the other hand, even for small values of n there are interesting other questions that one may pursue. For example, it is easy to verify that the prismatoids A(2, 3; 8) and A(6, 1; 8) shown in Figure 24 are isomorphic (have the same underlying abstract polyhedron). The same relation exists between A(2, 3, 1*; 7) and A(4, 1, 5*; 7), as well as between P(1,2;10) and P(3, 1; 10). This is clearly a consequence of some relations in the modular arithmetic; however, the general behavior of the various prismatoids with respect to isomorphism has not been clarified so far. For example, there is a bijection between the vertices of A(2, 1, 3*; 7) and A(4, 1, 1*; 7) which maps faces to faces, but is not an isomorphism. Also open are questions regarding the character of the continuous families that can be derived from the various P and Aprismatoids.
References.
[1] U. Brehm and W. Khnel, Smooth approximation of polyhedral surfaces regarding curvatures. Geometriae Dedicata 12(1982), 435 - 461.
[2] M. Brckner, Vielecke und Vielflache. Teubner, Leipzig 1900.
[3] H. S. M. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller, Uniform polyhedra. Philos. Trans. Roy. Soc. London (A) 246(1953/54), 401 - 450.
[4] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups. 4th ed. Springer, Berlin 1980.
[5] B. Grnbaum, Regular polyhedra. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness, ed. Routledge, London 1994. Vol. 2, pp. 866 - 876.
[6] B. Grnbaum, Metamorphoses of polygons. Proc. Strens Memorial Conference on Recreational Mathematics & its History, R. K. Guy and R. E. Woodrow, eds. Math. Assoc. of America 1994, pp. 35 - 48.
[7] B. Grnbaum, Polyhedra with hollow faces. Proc. NATO - ASI Conference "POLYTOPES: Abstract, Convex and Computational", Toronto 1993. T. Bisztriczky, P. McMullen, R. Schneider and A. Ivic' Weiss, eds. Kluwer Acad. Publ., Dordrecht 1994. Pp. 43 - 70.
[8] B. Grnbaum and G. C. Shephard, Polyhedra with transitivity properties. Math. Reports Acad. Sci. Canada 6(1984), 61 - 66.
[9] Z. Har'El, Uniform solutions for uniform polyhedra. Geometriae Dedicata 47(1993), 57 - 110.
[10] E. Hess, Ueber gleicheckige und gleichkantige Polygone. Schriften der Gesellschaft zur Befrderung der gesammten Naturwissenschaften zu Marburg, Vol. 10(1874), 611 - 743 + plates.
[11] A. L. F. Meister, Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus. Novi Comm. Soc. Reg. Scient. Gotting. 1(1769/70), pp. 144 - 180 + plates.
[12] S. A. Robertson, Polytopes and Symmetry. London Math. Soc. Lecture Notes Series #90. Cambridge Univ. Press 1984.
[13] S. A. Robertson and S. Carter, On the Platonic and Archimedean solids. J. London Math. Soc. (2) 2(1970), 125 - 132.
[14] E. Schnhardt, ber die Zerlegung von Dreieckspolyedern in Tetraeder. Mathematische Annalen 98(1927), 309 - 312.
[15] J. Skilling, The complete set of uniform polyhedra. Philos. Trans. Roy. Soc. London (A) 278(1975), 111 - 135.
[16] S. P. Sopov, Proof of the completeness of the enumeration of uniform polyhedra. Ukrain. Geom. Sbornik 8(1970), 139 - 156.
[17] E. Steinitz, Polyeder und Raumeinteilungen. Encykl. math. Wiss. Vol. 3 (Geometrie), Part 3AB12, pp. 1 - 139. (1922).
[18] Webster's Third New International Dictionary of the English Language. Encyclopdia Britannica, Chicago 1966.
[19] C. Wiener, ber Vielecke und Vielflache. Teubner, Leipzig 1864.
Department of Mathematics
University of Washington GN-50, Seattle, WA 98195
e-mail: grunbaum@math.washington.edu
Research supported in part by NSF grant DMS-9300657.
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