Journal of Geometry Voi.29 (1987)

0047-2468/87/020182-0951.50+0.20/0

(c) 1987 Birkh~user Verlag, Basel

SOME PROBLEMS ON POLYHEDRA*

Branko Griinbaura and Geoffre# O. Skephard

Jacob Steiner asked, more than 150 years ago, whether every convex polyhedron in Euclidean 3-space is isomorphic to one all vertices of which lie on a sphere. It is well known that the answer to this question is negative, but many related problems are still unsolved.

All the problems in this note concern conrez polyhedra in Euclidean space of three dimensions, that is to say, bounded subsets of E s which can be expressed as the intersection of a finite number of closed halfspaces. We shall use the word "polyhedron" without further qualification in this restricted sense. For detailed expositions of the theory of polyhedra in general, as well as background material for this note, see [1], [5], [15]. In view of the fact that such polyhedra have been studied for more than two millenia, it is surprising that many simple and tangible questions concerning them remain unanswered. In the case of the problems to be discussed here, this may be in part due to the fact that they depend essentially on the concept of "isomorphism~. To us this may seem a very natural concept, but it is worth recalling that even in implicit form it cannot be found in mathematical literature before the works of Euler and his eighteenth-century contemporaries. The difficulty of many of the problems stated in this paper may stem from the fact that, even today, we know of no systematic way in which to describe all polyhedra isomorphic to a given one. We begin by recalling some of the necessary definitions. Two polyhedra PI and P2 are

isomorphic (or corabinatoriall~l equivalent) if there exists a one-to-one correspondence between the elements (that is, vertices, edges and faces) of PI and the elements of P2 such that inclusion is preserved. On the other hand, if a one-to-one correspondence exists between the elements of PI and those of P2 which reverses inclusion, then PI and P2 are * Research supported by the National Science Foundation grant MCS8301971.

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called duals of each other. In particular, if S is a sphere (that is, the surface of a solid ball in E s) centered at an interior point of a polyhedron P, then the reciprocal (or polar) of P with respect to S is a polyhedron P* which is dual to P. It should be noted that, notwithstanding the frequently used definite article, a polyhedron P does not have a unique dual, but infinitely many duals all of which are mutually isomorphic, and are dual to each polyhedron isomorphic to P. In many publications duality is treated inadequately also in another respect: the definitions of duality given apply only to special polyhedra (such as Platonic or Archimedean solids), although they are often phrased as if more generally applicable. I1; is clear that if the polyhedron PI is obtained from a polyhedron P by means of a projective transformation, then P and PI are isomorphic. The converse--if P1 is any polyhedron isomorphic to P, then PI is a projective image of P - - i s true if and only if P has at most 9 edges (see [6, p. 1176]). As mentioned above, for more general polyhedra no precise description of all isomorphic ones is known. A polyhedron P is of iuscribable type ff there exists polyhedron P1 which is isomorphic to P and which can be inscribed in a sphere, that is, there exists a sphere which contains all the vertices of P1- Analogously we say that P is of circumscribable type if there exists Fk, isomorphic to P, which is circumscribed about a sphere, that is, all the faces of P1 a:re tangential to a sphere. By reciprocation we see immediately that a polyhedron P is of irtscribable type if and only if any dual polyhedron P* is of circumscribable type. In 1832, J. Steiner posed the question whether every polyhedron is of inscribabte type (see [~:0]), and for a long time it was considered self-evident that the answer is affirmative at least if all the faces of the polyhedron are triangles. (Such polyhedra are called simpticial, and their duals, in which three edges meet at each vertex, are called simple polyhedra.) The "proof" of this claim is quite convincing for the unwary: project the vertices of the given simplicial polyhedron onto a sphere centered at an interior point of P, and take the convex hull of the points so obtained [2, page 163, footnote 3]. Only in 1926 did E. Steinitz (see [21]) discover the first non-circumscribable types of polyhedra (and hence also the first non-inscribable ones). Actually, Steinitz described a very remarkable process that yields infinite families of such types, among them simplicial polyhedra. Examples of polyhedra of non-inscribable type obtainable by Steinitz' method include the rhombic dodecahedron, the triakis tetrahedron, the tetrakis hexahedron and the pentakis dodecahedron. These polyhedra, which are duals of Archimedean polyhedra, are described, for example, in [3] and [8].) Other methods of establishing that certain polyhedra are of non-inseribable

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types, and related results, can be found in [4], [5, Section 13.5], [7], [10], [11], [12], [13], [14, Chapter 4], [18]. By combining these methods it is possible to prove the existence of polyhedra of non-inscribable types and having various other properties; one example are polyhedra with all vertices of valence 5. Interesting questions on inscribable types arise in the metric theory of polyhedra in connection with dome structures, that is frameworks in which all nodes are on a sphere and the bars (edges) have only a small number of different sizes (see, for example, [22]). Once the existence of polyhedra of non-inscribable type is established, it is natural to inquire how few faces (or vertices) such a polyhedron can have. It is not hard to show that the non-inscribable polyhedron with the least number of faces is the singly-truncated cube shown in Figure 1 (see [10]). Clearly, similar quantitative questions could be asked in connection with each of the problems discussed below. We may note that a suitable combination of the polyhedron in Figure 1 with a polyhedron dual to it yields the (selfdual) polyhedron shown in Figure 2 which is neither of inscribable type nor of circumscribable type. Problems regarding polyhedra of inscribable type could be solved more readily if a complete determination of such polyhedra were known. Hence our first question is this:

P r o b l e m 1.

Characterize the polyhedra of inscribable type.

Obviously, a solution of Problem 1 would also yield a characterization of po!yhedra of circumscribable type. Although the terminology we have used so far is generally accepted, it is not convenient for describing the other problems we wish to discuss. For this reason we prefer to say "SVtype" instead of "inscribable type" and "SF-type" instead of "circumscribable type". Here the first letter S stands for "sphere", and the second letter V or F indicates whether vertices or faces touch the sphere. One of the advantages of this new terminology is that it permits modification and suggests generalizations. Thus we may say that a polyhedron P is of SE-type if there exists a polyhedron PI, isomorphic to P, such that all the edges of P1 are tangent to a sphere.

GrUnbaum and Shephard P r o b l e m 2.

185

Do there exist polyhedra which are not of SE-type?

In an attempt to solve Problem 2 it may be convenient to consider the following. Say Chat ~ polyhedron P is of weak SE-type if there exists a polyhedron PI isomorphic to P such that the (straight) lines determined by the edges of P1 are tangent to a sphere. Then one may ask whether every polyhedron of weak SE-type is also of SE-type, or if~ indeed, any polyhedra exist which are not of weak SE-type. A polyhedron P is said to be properly inscribed in a dual polyhedron P* if every vertex V of P is an interior point of the face of P* that corresponds to V in the duality. We say that P is of DV-type if there exists a polyhedron P1, isomorphic to P, which can be properly inscribed in one of its duals. Clearly, if P is of SV-type then it follows, by reciprocation in the sphere S containing the vertices of a polyhedron PI isomorphic to P, that P is of DV-type; the example of the polyhedron in Figure 1 can be used to show that eLpolyhedron can be of DV-type without being of SV-type. P r o b l e m 3.

Is every polyhedron of DV- type?

P r o b l e m 4.

Does every polyhedron of DV-type have a dual of DV-type?

P r o b l e m 5.

Which polyhedra of D r - t y p e are also of SV-type?

The corresponding problems for DF-type polyhedra (defined as DV-type polyhedra except that "inscribed" is replaced by "circumscribed") are equivalent to the above by duality. However, consideration of polyhedra of DE-type leads to additional open problems. (To be precise~ a polyhedron P is said to be of DE-type if there exists a polyhedron PI, ~omorphic to P, such that each edge of P1 intersects, in a single point, the corresponding edge of some polyhedron PI* dual to PI.) P r o b l e m 6.

Do there exist polyhedra which are not of DE-type?

I~ is also natural to ask whether every polyhedron of DE-type is necessarily of SE-type. If a polyhedron P has all its edges tangent to a sphere S, then not only will the polyhedron P* obtained from P by reciprocation in S have the same property and the corresponding edges of P and P* will meet in a single point, but also such edges will intersect at right angles. Hence we may define a polyhedron P to be of strong DE-type if there exists a polyhedron PI, isomorphic to P, such that each edge of PI intersects, at right angles, the corresponding edge of some dual polyhedron PI* of Pl.

186 P r o b l e m 7.

GrHnbaum and Shephard Do there exist polyhedra which are not of strong DE-type?

If such polyhedra exist, it is natural to ask whether every polyhedron of strong DE-type is necessarily also of SE-type. It is worth noting, in this connection, that in certain naive discussions of duality in the literature it is claimed, without justification, that every polyhedron is of strong DE-type. Sometimes it is even claimed that for every polyhedron P there exists a dual P* whose edges intersect the corresponding edges of P at right angles. That this is not so can be seen from an example such as that of an n-gonal antiprism in which the distance between the two n-goual faces is very small compared to the length of sides of these n-gonal faces. Properties related to, but weaker than, those considered above arise from considering supporting planes to the polyhedra. We recall that a plane L supportt a polyhedron P at one of its elements (a vertex V, an edge E or a face F) if P lies entirely in one of the closed haifspaces bounded by L, and the intersection of L with P is precisely V, E or F respectively. The outtoard normal to a supporting plane L of P is the normal to L that lies in the halfspace not occupied by P. Supporting planes L1 and L2 of the polyhedra P1 and P2 are called parallel if their outward normals point in the same direction. If there exists a dual P2 of PI such that, for each face F of PI, the supporting plane of P2 parallel to the plane determined by F intersects P2 in a vertex which corresponds to F under the duality, then PI is said to have the parallel supporting plane property. In this case it is not hard to see that for every pair of elements GI of PI and G2 o f / 2 , that correspond under the duality, there exist parallel supporting planes LI of P1 and L2 of P2 such that L1 n P1 = G1 and L2 0 P2 = G2. A polyhedron P is said to be of PSP-type if there exists a polyhedron/>1, isomorphic to P, which has the parallel supporting plane property. Clearly a polyhedron of DF-type is necessary also of PSP-type. P r o b l e m 8.

Are all polyhedra of PSP-type?

Part of the interest of Problem 8 is in connection with the construction of four-dimensional antiprisms. We recall that a convex four-dimensional polytope A is called an antiprism

with ba~es P, P* if A is the convex hull of dual polyhedra P, P* lying in parallel threedimensional hyperplanes in such a way that every three-dimensional face of A (other than P and P* ) is the convex hull of an element G of P and the element of P* that corresponds to G in the duality between P and P*. An affirmative answer to Problem 8 is clearly equivalent to the statement that every polyhedron is isomorphic to a base of some four-dimensional antiprism. A polyhedron P is said to be of PF-type if there exists a point O, and a polyhedron P1,

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187

isomorphic to P, such that for each face F of P1 the perpendicular from 0 to the plane determined by F meets that plane in a relatively interior point of F. In a similar way we define

PE-type; here the perpendiculars from O onto the lines determined by the edges

of P1 meet those lines in relatively interior points of the edges. It is clear that every polyhedron of SF-type is also of PF-type, and every polyhedron of SE-type is also of PE-type (for we may choose O, in each case, as the center of the sphere that touches the faces or edges of P~). Besides asking whether the converse statements are true, we can pose the following questions. P r o b l e m 9. P r o b l e m 10.

Are all polyhedra of PF-type? Are all polyhedra of PE=type?

We shall say that a polyhedron P is of

CV-type or EV-type if for some polyhedron

PI, isomorphic to P, each face is inscribable in a circle or an ellipse, respectively. It is easy to see that a

simplepolyhedron is of CV-type if and only if it is of SV-type. This is

the crucial step in the method used in [4] and [10] to establish that certain polyhedra are not of SV- type. Problem U.

Are all polyhedra of l~V- type?

P r o b l e m 12.

Is every simple polyhedron of EV-type necessarily of SV-type?

Analogously, we shall say that a polyhedron P is of CE-type or of EE-tvpe if for some 'polyhedron PI, isomorphic to P, each face of P1 is circumscribed about a circle or an ellipse, respectively. P r o b l e m 13.

Are all polyhedra of CE-type, or at least of EE-type?

A polyhedron is of strong EE-type if it is of EE-type and the two ellipses which touch each ~dge of/'1 do so at the same point. P r o b l e m 14.

Is every polyhedron of EE-type also of strong EE-type?

We conclude by two problems on self-dual polyhedra. (For information about some properties of self-dual polyhedra see [9].) ]Problem 15. Is every self-dual polyhedron P isomorphic to a polyhedron P1 which has a reciprocal polyhedron P2 congruent to P1 ?

188 P r o b l e m 16.

Grfinbaum and Shephard Does there exist a self-dual polyhedron which has a center of symmetry? $

$

$

In conclusion we remark that many of the problems we have stated above have analogues in higher dimensions. For example, n-dimensional polytope P is m-inseribable if there exists a polytope PI, which is isomorphic to P, such that all its m-dimensional faces touch an {n - 1)-sphere in E '~. Problem 2 is equivalent to asking whether there exist any 3dimensional polytopes which are not 1-inscribable. The only results in this direction seem to be those of Schulte [17] who proved that if n > 4, then for each m with 0 < m < n there exist n-dimensional polytopes which are not m-inscribable. The case n = 3, m = 1 is therefore the only case yet to be decided. The problems discussed here are of special t y p e - - t h e y are related to isomorphism and duality of polyhedra. Problems of other kinds are implicit in many books and papers on polyhedra; explicit formulations of some unsolved problems can be found, for example, in [16] and [19]. Clearly there remain many possibilities for contributions to this "most tangible" area of mathematics.

GrUnbaum and Shephard

(a)

189

(b)

Figure 1. The polyhedron of non-inscribable type with the smallest possible number of f~ces. (a) view, (b) Sehlegel diagram.

Figure 2. A Schlegel diagram of a polyhedron which is of non-inscribable type as well as of non-circumscribabte type.

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REFERENCES

[1] A. Brcndsted, An Introduction to Convex Polytopes. Springer-Verlag, New York 1983. M. Brfickner, Vielecke und Vielflaehe. Teubner, Leipzig 1900. H. M. Cundy and A. P. Rollett, Mathematical Models. Second ed. Clarendon Press, Oxford 1961. [4] B. Grfinbaum, On Steinitz's theorem about non-inscribable polyhedra. Ned. Akad. Weteneehap. Proe. Ser. A, 66(1983), 452-455. B. Griinbaum, Polytopes, graphs, and complexes. Bull. Araer. Math. Soc. 76(1970), 1131-1201. [7] B. Grfinbaum and E. Jucovi~, On non-lnscribable polytopes. Cz~hosloeak Math. J.

23(1974), 424-429.

[8] B. Griinbaum and G. C. Shephard, Patterns on the 2-sphere. Matheraatika 28(1981), 1-35. [9] E. Jncovi~, Self-dual K-polyhedra. [In Russian, with summary in German] Matematielco-.fyzih~ln# ~aeopis 12(1962~, 1-22. [10] E. JucovM, O mnohostenoch bez opisanej gulovej plochy. IOn non-inscribable polyhedra.] Matematicko-yyzik~lnu ~asopis 15(1965), 90-94. [11] E. Jucovi~, O mnogostenoch bez opisanej gulovej plochy. H. [On non-inscribable polyhedra. H.] Matematicko-.f#zikdlny ~asopie 16(1966), 229-234. [12] E. Jucovi~, Bemerkung zu einem Satz yon Steinitz. Elemeate der Math. 22(1967), 39. [13] E. Jueovid, On polyhedral surfaces which are not inscribable in spherical shells. Probl~mes Combinatoires et Th4orie des Graphes, Colloques Internat. C. N. R. S. No. 260. Paris 1978, pp. 251-253. [14[ E. Jucovid, Konvexne mnohosteny. [Convex polyhedra.] Slovenska Akademia Vied, Bratislava 1981. [lSl P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture. London Math. Soc. Lecture Note Series Vol. 3, Cambridge Univ. Press 1971. [16] W. Moser, Research Problems in Discrete Geometry. Mimeographed notes, McGiU University, 1981. [IT] E. Schulte, Higher-dimensional analogues of Steinitz's theorem about non-inscribable polytopes. (To appear). [18] S. ~evec, On the non-inseribability of certain families of polyhedra. IIn Russian] Math. Slova~a 32(1982), 23-34. [19] G. C. Shephard, Twenty problems on convex polyhedra. Math Gazette 52(1968), 136-147 and 359-367. [20] J. Steiner, Systematische Entwickelung der Abh~ngigkeit geometrischer Gestalten yon einander. Fincke, Berlin 1832. (-- Gesammelte Werke, Vol. 1, Reimer, Berlin 1881, pp. 229-458; see, in particular, p. 454.) [21] E. Steinitz, Uber isoperimetrische Probleme bei konvexen Polyedern. J. reiae augew. Math. 158(1927), 129-153 and 159(1928), 133-143. [22] T. Tarnai, Spherical grids of triangular network. Aeta Teehu. Aead. Sei. Huugar. 76(1974), 307-336.

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