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Mathematical Questions Concerning Zonohwlral Space-Filling by Henry Crapo

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Mathematical Questions Concerning Zonohwlral Space-Filling by Henry Crapo Mathematical Questions concerning Zonohwlral Space-Filling by Henry Crapo Introduction application, have already been pursued during the Structural Topology #2,1979 years since “Polyhedral Habitat” was written, by In his article “Polyhedral Habitat”, Janos Baracs Nabil Macarios, Yves Dumas, Pierre Granche, Vahe introduces a technique for synthesizing zonohedral Emmian, and other members and associates of the space fillings. It is an effective drafting technique, Structural Topology research group. The problems since it enables the user quickly to arrive at a single outlined in the present article are those which arise correct plane projection of a spatial framework, from when one begins to piece together a certain mathe- which the required spatial information may then be matical theory of space-filling, a theory designed to surmised, either by intuition, or by calculation based shed some light on space-filling polyhedra which are upon a very few free choices. The technique pro- formed as concave clusters of zonohedra. Recent ceeds in three stages: work in this direction was carried forward during meetings of the research group, mainly through the A) successive splittings of a plane tessellation, to efforts of Janos Baracs, Walter Whiteley, Marc produce tessellations by concave clusters of convex Pelletier, and the present author. As we outline the zonagons. problems which we see arising, we will point out B) staggerlng (relative parallel transport) of two those areas where we have been able to make some copies of the resulting tessellation, to arrive at a progress, and will give some references to what we projection of a space-filling. Abstract feel is relevant mathematical literature, chiefly to the C) llftlng the projection into space, to arrive at a papers of Coxeter, Grunbaum, Shephard and space-filling by concave clusters of convex zonohe- McMullen. An exposition of the basic mathematical pro- dra. perties of zonohedra and their diagrams, ar- Associated with this technique, but not analyzed in ranged so as to bring into focus the many as much detail in the abovementioned article, is the challenging unsolved problems involved in method of Zonohedra implementing the procedures for synthesizing D) beginning with a known space-filling by a convex space-fillings which were described in the parallelohedron, to arrive, by successive spatial A zonohedron is a convex polyhedron expressible as article “Polyhedral Habitat”. splittings, at space-fillings using more general con- the convex sum of a finite set of line segments (see cave parallelohedra. Grunbaum 1967). This construction is as follows. Given n vectors at the origin in three-dimensional Note aux lecteurs francophones: Each of these procedures raises a number of inte- space, vectors not lying in a single plane through the Une traduction franGaise de cet article est resting and challenging problems, with which we origin, their convex sum is the set of points whose disponible sur demande. Un bon de com- hope to tempt some of our mathematically-minded position vectors may be written as linear combina- mande est inclus. readers. Certain lines of inquiry, primarily those tions of the given vectors, using scalars chosen closest to the domain of intended architectural between 0 and 1, inclusive. The boundary of this convex sum is a convex polyhedron, the zonohe- compass construction). This line LO’ gives the true converse is also true: at any vertex of the zonohe- dron, and the original array of vectors we call the length of the line LO, because LK is the hypotenuse dron, find a plane which touches the zonohedron star of that zonohedron. See Flgure 1. of a right triangle with altitude 1, and base LO. By only at that vertex. The plane parallel to this plane, dropping a perpendicular to MN through the point A and passing through the origin, produces the requi- Given a 3-dimensional model of the star, it is easy to until it meets the line O’M, we find the location A’ of red partition of the vectors. generate a set of parallelograms and other 2n-gons, the point A as the face OAB is rotated about the line using in turn all possible pairs of vectors, or other MN until it lies in the plane of projection. Doing the Every edge of the zonohedron, considered as a free larger coplanar sets of vectors, in the star. These same construction at B, we find B’, and completing vector, is equal to one of the vectors in the star. If we zonogons, cut from cardboard, are precisely the the parallelogram, we find a picture of the face OAB fix our attention on all the edges parallel to any one faces needed for a model of the zonohedron.But this with its true edge lengths and angles. vector, we find they form a belt, or zone around the is not the way we would proceed in practice, be- zonohedron. The zonohedron is centrally- symmetric, about a point whose position vector is cause there is really no need to construct a spatial The lengths of vectors in a star can be changed exactly half the sum of all the vectors in the star. This model of the star. Say instead we have a bit more freely without affecting the topology of its zonohe- symmetry is visible in plane projection, by virtue of information in a plane diagram: not only a plane dron, so the lengths can all be set equal to one the fact that the upper cap and lower cap are projection of the star, but also the intersection points without loss of generality. The number of distinct obtained from one another by central reflection of the lines of the star with the plane of projection, as faces needed for the construction of a zonohedron (Figure 4). This will be important when we come to in Figure 2. We may assume that the projection is an can be reduced by assuring that there are relatively study staggerings and lifting, below. We have also orthogonal projection, and that the actual centre of few different angles between pairs of vectors in the used Figure 4 to illustrate how hexagonal faces are the star is at distance 1 above its image in the plane. star. The vectors might be taken as pointing outward formed when we take three vectors coplanar in the Then standard constructions in descriptive geome- toward the vertices of a regular polyhedron, for star, here bdf and ace. try provide correct plane drawings of the real lengths example. and angles of the faces. We see this construction for a typical parallelogram face in Flgure 3. Here the Given a plane through the origin, a plane containing (Coxeter 1962) simplifies the development of zono- lines of the vectors OA and OB meet the plane of none of the vectors of the star, we obtain a partition hedra one step further, by studying the figure projection on points M and N respectively. The point of the vectors into two subsets: those on one side of obtained by intersecting the lines of the vectors in L is the foot of the perpendicular through 0 to the line the plane, those on the other. By adding the vectors the star with a plane not through the origin, such as MN, the line OK is parallel to MN, the segment OK has in these two subsets, we obtain the position vectors in Figure 2, above. He calls this plane figure the length 1, and the line LO’ is equal in length to LK (a of a pair of opposite vertices of the zonohedron. The (primal) projectlve dlagram of the zonohedron. bcdef abcdef bef b d a Figure 2. Intersection of all lines of a spatial star with a plane Figure 1. Each vertex of the zonohedron is reached by adding some (indicated, in a few cases) subset of the set of vectors in its star. produces a (primal) p rejective diagram for its zonohedron. 54 that bipartition of the points. For instance, using not even topologically a zonohedron. By contracting Figure 5, we know that from the vertex df we can any one zone in Flgure 7A, and moving the resulting reach adjacent vertices by adding the vector c, meridian to the perimeter, we obtain a subdivision adding a, or subtracting f. into parallelograms of the zonogon with eight pairs of sides, a subdivision which cannot be the top cap If we contract to zero along itself any one vector of a of a (convex) zonohedron (Flgure 7C). By putting the star, we will contract to zero the width of one zone of word “convex” in parentheses we wish merely to the zonohedron. What remains of the zone will be a underline that strict convexity is part of the definition meridian path, which traverses a sequence of edges of zonohedra. There are, as it is easy to show, many and faces of the contracted zonohedron, using each concave polyhedra having Flgurer 7 as their parallel remaining vector or its negative once (either by projections, concave polyhedra which resemble zo- going along an edge in that direction, or diagonally nohedra in most respects. Figure 3. From a diagram like Figure 2, it is easy to construct the across a face involving that zone), to reach the vertex true dimensions of each face by “rabattement”, rotating the face opposite to the starting point, then continuing on the These facts derive from a dual projective diagram down into the plane of projection, using its intersection with the also invented by Coxeter, and from the theory of plane of projection as hinge for the rotation. opposite route (opposite in the sense of central symmetry) back around to the starting point (Flgure arrangements of lines and pseudo-lines (Grunbaum Planes through the origin, in the above procedure 6).
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