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Realizability of Polyhedra by Walter Whiteley Realizability of Polyhedra by Walter Whiteley A great deal of our work employs surfaces which The Pattern for Convex Polyhedra Structural Topology (1) I@79 enclose sections of space (compact oriented 2- manifolds), constructed by joining plane polygons In many ways a model for the overall type of theory together in pairs across their edges. These finite we are seeking is offered by the most widely studied plane-faced coverings of the closed surface are the class of polyhedra, the convex mrr . These basic cells of the work on juxtapositions or space- polyhedra are formed in space by a series of plane fillings, and they also appear as the basic visual convex polygons, the faces, formed into a rough ball guide in our analysis of structural dependence and so that along each edge of aI polygon the face joins rigidity in plane frameworks. These polyhedra are one other face, and the plane of each facial polygon the essential building blocks of spatial geometry. cuts the three -dimensional space into two half- They have been scrutinized for centuries. In this spaces, one of which contains the entire polyhedron. century the study has been generalized to a topolo- Abstract gical theory of combinatorial polyhedra, but many of the fundamental problems, both solved and unsol- Such a polyhedron will be topologlcally like a ved, involve the realizability In space of the abstract sphere, and for a spherical polyhedron we have We address ourselves to three types of com- patterns within particular geometric constraints. Euler’s formula which gives a combinatorial rela- binatorial and projective problems, all of which tionship between the number of vertices V, edges E concern the patterns of faces, edges and and faces F: V - E t F = 2. When we have a set of vertices of polyhedra. These patterns, as com- vertices, edges and faces (each edge joining two binatorial structures, we call combinatorial vertices and separating two faces) which satisfies oriented polyhedra. Which patterns can be Each generation of workers has chosen particular thls formula, and thus forms a topological sphere*, realized in space with plane faces, bent along properties which they find desirable and then asked the skeleton formed by the edges and vertices forms every edge, and how can these patterns be - which polyhedra can be realized within these a planar graph, a graph* which can be drawn In the generated topologlcally? Which polyhedra are constraints? The constraints which we Investigate plane without any edges crossslng In their Interior. constructed in space by a series of single or within our research group arise directly or Indirectly (Flgure 1) double truncations on the smallest polyhedron from our prior concerns with space-fillings and of the type (for example from the tetrahedron rigidity. In addition, some constraints of obvious for spherical polyhedra)? Which plane line architectural or engineering significance - such as drawings portraying the edge graph of a com- metric regularity of angles, edge-lengths or face Combinatorial and Topological binatorial polyhedron are actually the projec- patterns - have been widely studied elsewhere and Problems tion of the edges of a plane-faced polyhedron will not be described here. Instead we will restrict our in space? Wherever possible known results attention to a series of questions of a combinatorial, and specific conjectures are given. topological and projective nature which represent An obvious question is: which planar graphs (with less traditional topics of study, the regions formed in the plane taken as the faces) 46 can be topologically reproduced as the faces, edges and the spatial construction by plane-truncations. projected lines labeled 12, 23, 31 be concurrent In and vertices of a convex polyhedron with plane faces the plane. Finally, to ensure the convexity of the and a real bend at each edge? Steinltz’ theorem proposed polyhedron we have found that two faces provides the answer - those planar graphs which Projective Problems which share a vertex but not an edge must have a are 3,.connected in a vertex sense and have more line of intersection which intersects the face poly- than three vertices (Grunbaum 1967, p. 235). Here 3- We now want to study in more detail the projective gons only at this vertex, and all the face polygons connected means that removing any two vertices construction of convex spatial polyhedron by plane must be convex (Flgure 4). Is this enough to guaran- (and the adjacent edges) cannot separate the graph slices on the tetrahedron. We have found that a tee the construction of a convex spatial polyhedron? into several components (Figure 1). useful way to observe this slicing Is in a single plane Yes. We have proven that all such consistent dia- projection where we record the projections of the grams of lines in the plane can be lifted back to lines where each new face plane meets all of the produce convex polyhedra in space. Closely related to this result, and used In some previous planes (Flgure 3). We then attempt to proofs, is the fact that all 3-connected planar graphs reverse the procedure and ask when a pattern (combinatorial convex polyhedra) can be created of lines drawn in the plane represents the To fix this spatial polyhedron, we need to specify our from the graph of the tetrahedron by successively construction of a convex polyhedron In point of projection and the position of the projection splitting faces in one of three natural ways (Figure 2) space. Of course we must identify, among all these plane. For convenience we wilt think of orthogonal (Lyusternik 1963, p.75). We also know, from their lines, the 3-connected planar graph which we want projection along the direction of the r-axis (from the definition, that any convex polyhedron can be sliced to be the projected edges of the proposed polyhe- point at infinity on the end of this axis) onto the xy- out of an underlying tetrahedron by a series of dron. In addition we must label each of the other plane. With this frame of reference specified, If a plane-cuts or truncations* - one for each face lines as the projected intersection of two faces of the plane diagram lifts to one convex polyhedron in beyond the original four faces of the tetrahedron. proposed polyhedron; We know that three planes In space, then it lifts to a four parameter family of One central theme for our further investigations will space 1, 2 and 3 will have a point of intersection (or, polyhedra - three parameters for the location of an be the connections between these three approaches as a degenerate case, a line of lntersectlon) so the initial face-plane, and the fourth for a factor which - the graphic characterization of a class of spatial three lines 12, 23, 31 are concurrent in space. Thus represents stretching along the z-axis, towards the polyhedra, the topological evolution of these graphs, in our labeled diagram it is necess;ary that the three point of projection. If one polyhedron of this four- parameter family is convex, then all of these equiva- lent realizations will also be convex. If we simplify the initial information In the plane and begin only with a drawing, with straight lines, of a 3- L3 connected planar graph, we have the obvious ques- E=3 tion - what possible complete diagrams of lntersec- F=2 A . tion lines for constructing. a convex polyhedron can A . VA E = 10 F=6 6 VA E= 12 F= 6 C . Figure 1. Both 3-connected planar graphs (C), and planar graphs Figure 2. Face splitting by a new edge adds new faces to a Figure 3. A convex polyhedron Is shown In proJectIon, and all Ilnes 47’ which are only P-connected, create spherical polyhedra combinatorial polyhedron. of Intersection of pairs of faces are drawn. The Intersection of face regions of the plane are chosen as faces. 1 and face 2 Is labeled 12. be produced as extensions of this initial drawing? the bent edges of the spatial polyhedron. (Whiteley 013 1976). 234= 23~34 Equivalently what convex polyhedra exist in space 124= 12~14 which project to the initial plane drawing? We will 24= 124~234 return to this basic problem shortly, but we note that n in many cases (E852V-2) this question can be answe- 013 =OlAO3A13 red by a series of direct geometric constructions General Concepts 024 = 02 ~04 A 24 from projective* (or descriptive) geometry. Consider the examples illustrated in flgun 5. Figure SA - shows the completion of a projected triangular prism Combinatorial Polyhedra by a sequence of steps, where points 013 and 024 are shown to confirm the consistency of the diagram. We want to use a similar approach to analyze other A . If one of these points of concurrence of three lines plane-faced space enclosures. A first step is to exists, then the other point must also exist, as a abstract a basic pattern of faces, edges and vertices result of Desargues’ theorem on perspective trian- which is found in the convex examples. bles* Flgure SB shows the completion of a second figure, with point 025 drawn as a check of consis- tency. Flgure SC shows an identical graph, where An orlented comblnatorlrl polyhedron is a set of the point 025 has broken up, and thus no consistent vertices vl,...,v, and a set of faces F,,...,Fk such completion (or spatial polyhedron) is possible. that (I) each face is assigned a polygonal cycle of distinct We also recall that a thorough understanding of the vertices ul,... ,un (at least 3). statics of frameworks in the plane would provide a 024=24/\04 solution to the problem of polyhedral completions, (2) the edgea are identified as unordered pairs of 02=024 v 012 since a convex polyhedron projects, via Maxwell’s vertices which are adjacent in the polygon of some 014=OlAO4 theorem, into a static stress on the plane framework face (including the last and first vertices of the cycle).
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