Avenues for Polyhedral Research by Magnus J. Wenninger In the epilogue of my book Polyhedron Models I alluded Topologie Structurale #5, 1980 Structural Topology #5,1980 to an avenue for polyhedral research along which I myself have continued to journey, involving Ar- chimedean stellations and duals (Wenninger 1971). By way of introduction to what follows here it may be noted that the five regular convex polyhedra belong to a larger set known as uniform’ polyhedra. Their stellated forms are derived from the process of extending their R&urn6 facial planes while preserving symmetry. In particular the tetrahedron and the hexahedron have no stellated forms. The octahedron has one, the dodecahedron Cet article est le sommaire d’une etude et de three and the icosahedron a total of fifty eight (Coxeter recherches faites par I’auteur depuis une dizaine 1951). The octahedral stellation is a compound of two d’annees, impliquant les polyedres archimediens Abstract tetrahedra, classified as a uniform compound. The &oil& et les duals. La presentation en est faite a three dodecahedral stellations are non-convex regular partir d’un passe historique, et les travaux This article is a summary of investigations and solids and belong to the set of uniform polyhedra. Only d’auteurs recents sont brievement rappel& pour study done by the author over the past ten years, one icosahedral stellation is a non-convex regular la place importante qu’ils occupent dans I’etude involving Archimedean stellations and duals. The polyhedron and hence also a uniform polyhedron, but des formes polyedriques. L’auteur enonce une presentation is set into its historical background, several other icosahedral stellations are uniform com- regle &n&ale par laquelle on peut t,rouver un and the works of recent authors are briefly sket- pounds, namely the compound of five octahedra, five dual a partir de n’importe quel polyedre uniforme ched for their place in the continuing study of tetrahedra and ten tetrahedra. Their duals are also well non-convexe et il nous montre comment ap- polyhedral shapes. The author enunciates a known. pliquer la regle a I’aide de dessins. II nous general rule by which the dual of any non-convex presente egalement des photographies de uniform polyhedron can be discovered, and he modeles qu’il a fabriques lui-meme. C’est ainsi gives drawings to show how the rule applies. He The dual of any polyhedron is one which has the same que nous pouvons voir certains duals qu’ils a also presents photographs of models he has number of edges as the original from which it is derived, decouverts dernierement. II invite les techniciens made. These show some duals he has himself but there is an interchange in the number of faces and specialises en ordinateurs, les ingenieurs en recently discovered. He invites computer vertices. The kinds of faces and vertices are such, design et les artistes-decorateurs a considerer ces technicians, design engineers and decorator ar- however, that an n-sided polygon face in the original duals et les transformations possibles d’autres tists to consider these and other possible polyhedron yields an n-edged vertex in the dual. polyedres comme un champ de recherche a polyhedral transformations as an area for further developper et egalement comme elements research and for their usefulness as beautiful There are two so-called quasi-regular convex decoratifs aussi bien utiles que gracieux. decorative devices. polyhedra, the cuboctahedron and the 5 icosidodecahedron. Their stellated forms, especially Figure 1. The cuboctahedron and its complete stellation patterns. Figure 2. The truncated cube and its complete stellation patterns. 6 those of the latter, are extremely numerous, the total solid has more and more faces, the related stellation tahedral symmetry group. Computer graphics could number being unknown. But the stellation pattern, patterns take on more and-more intersecting lines, and produce motion pictures of such a transformation called also the ((complete face)), has intrinsic interest, hence more and -more cells are generated leading to process in action. Verheyen sees this process at work because it is from this pattern that new stellated forms greater and greater complexity in these stellated forms. together with rotational transformations along axes of are discoverable. Complete faces for some Ar- Yet those who, like me, have spent a good deal of time symmetry in the generation of all uniform polyhedra, chimedean solids are shown in Figures 1,2,3. The final investigating these myriad shapes cannot help but be and he alludes to the possibility of making computer stellation is of special interest for any given solid overawed by the beauty that lies hidden here, just generated movies (Verheyen 1979). because this is unique (Wenninger 1971). waiting to be discovered and brought to the attention of artists and design engineers. The investigation of duals opens an equally interesting Apart from what has appeared in (Coxeter 1973, 1951) avenue for polyhedral research. The dual of the cuboc- and (Wenninger 1971) very little has been published An avenue of research as yet largely unexplored is that tahedron is the rhombic dodecahedron. The dual of the about stellated forms. So here is an avenue for of continuous transformations of polyhedral shapes. It icosidodecahedron is the rhombic triacontahedron. polyhedral research. Some closely related in- may be one thing to stellate the cuboctahedron, for just as the regular and semiregular convex solids can be vestigations have been done recently by Alexanderson, example, but a similar investigation of the truncated stellated by extending or producing their facial planes, Kerr and Wetzel; see (Alexanderson 1979, 1971), (Kerr cube and the truncated octahedron will show that these so these convex solids can be stellated by the same 1978) and (Wetzel 1978). Modern computer graphics are merely transformations of the cuboctahedron, process. A distinct advantage enters here because these can depict stellation patterns. In fact I have been which itself is a transformation of the cube and the oc- duals have each a single stellation pattern due to the greatly assisted in my work with such drawings supplied tahedron. This is already evident from the stellation pat- fact that dual forms are isohedral just as the originals are for my privately by J. Skilling, who has used computer terns shown in Figures 1, 2, 3. Such relationships for isogonal. analysis for his work; see (Skilling 1975, 1976). Com- the convex forms are beautifully shown in photographs puter graphics can also show individual polyhedral by (Holden 1971). In stellated forms the intersection of Stellations of the rhombic dodecahedron have been shapes, given data for vertices and their interconnec- facial planes for all of these is merely a parallel tran- studied by D. Luke; see (Cundy 1961). Interesting tedness; see the work of (Norman 1973) and (Smith slation into or away from a center of symmetry common varieties of interlocking puzzles, based on the rhombic 1973, 1974). In the stellation process as the original to all of them. In the example cited this is the oc- dodecahedron and rhombic triacontahedron, have 7 Figure 3. The truncated octahedron and its complete stellation patterns. been skilfully made in wood by (Coffin 1971). (Holden attempts to show how the Archimedean duals can be A third attempt to show the generation of convex duals 1971) depicts these shapes as rigid models. I have done geometrically derived by a process of erecting pyramids is that of (Fleurent 1979), who derived his inspiration some of these puzzle arrangemens as models in paper; bn the faces of a chosen basic polyhedron, these from Graziotti. This author seeks to introduce a suitable see (Wenninger 1963) and an unpublished typescript pyramids then determining the vertices of the related symbolism, and his work as yet unpublished could with diagrams in my files. dual. The heights of these pyramids are a crucial factor prove helpful to further understanding of the process of in this process. Unfortunately Graziotti has oversim- dualization as well as other relationships between con- Stellations of the rhombic triacontahedron have been plified the process to the extent that actual calculation vex solids. studied by (Ede 1958). An attempt at a systematic disqualifies seven of the thirteen cases he so beautifully enumeration of possible forms under some restrictive illustrates in his otherwise ingenious approach. rules came from the work of (Pawley 1973). Of special Thus, although the study of convex polyhedra and their interest is the fact that among the stellations of the Another more successful attempt is that of (Lalvani 1977), duals is still attracting attention today, myriads of non- rhombic triacontahedron is a uniform compound, the whose explosion-implosion process leads from the origi- convex forms and their duals remain largely unknown compound of five cubes, and the duals of two non- nal polyhedron through a stage he calls transpolyhedron and unexplored. M. Bruckner at the beginning of this convex uniform polyhedra (Coxeter 1973), (Cundy to the true dual form. In this work he alludes to the twentieth century was a pioneer explorer here, but it 1961) and (Holden 1971). The entire topic of uniform applicability of the process to uniform polyhedra and seems that the trail he blazed has not been taken up by compounds of uniform polyhedra has been thoroughly notes that this needs a more in-depth study. others. researched by (Skilling 1976). Skilling has also given the definitive enumeration of uniform polyhedra, bringing to a close the investigation of (Coxeter 1954) and (Skilling 1975). So besides the two quasi-regular convex solids and their duals, avenues for polyhedral research are still open for nine other semiregular convex solids and two convex snub polyhedra along with all their duals. Little is known about their stellated forms at the present time.