# On the space-filling enneahedra

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- MICHAEL GOLDBERG ON THE SPACE-F ILL ING ENNEAHEDRA ABSTRACT. A space-filling polyhedron is one whose replications can be packed to fill three-space completely. The space-filling polybedra of four to eight faces have been previously reported. The search is here extended to the convex space-fillers of nine faces. The number of types is found to be at least 40. 1. INTRODUCTION The description and enumeration of the space-filling polyhedra, which are an answer to Hilbert's eighteenth problem, is still far from complete. Partial answers are given in papers by the author [ 1 to 9]. The findings of the previous works are an aid in deriving some of the sought convex space-fillers of nine faces for tabulation and description in this work. A general method for obtaining all the solutions of Hilbert's problem has not yet been devised. Elke Koch [10] has programmed a computer to derive the Dirichlet regions of periodic point sets of the cubic lattice system. Those which employ only directly congruent regions will qualify as space-fillers as demanded by Hilbert's eighteenth problem. Some polyhedra fill space only by employing both directly congruent and mirror-image (enantimorphic) polyhedra. Furthermore, Dirichlet regions meet face-to-face without overlapping. But there are other qualifying polyhedra which fill space only by employing overlapping faces. 2. FRAGMENTATION If a known space-filler can be fragmented into several congruent parts, then each fragment is also a space-filler. The space-fillers which have a high degree of symmetry or regularity are the likeliest candidates for such con- sideration. The three-dimensional star-cluster of seven cubes, made by attaching a cube to each face of a cube, is a space-filler [11]. A star-cluster of 13 cubes, made by attaching two cubes to each face of a cube, is also a space-filler. Similarly, a cluster made by attaching half of a cube to each face of a cube is, also, a space-filler. When each of these three clusters is fragmented into six parts by dividing the central cube into six square pyramids, then each part is a nine-faced space-filler. In filling three-space with these parts, they assume six different orientations. But these parts are equally inclined to a main diagonal of the central cube. Therefore, an expansion or a compression Geometriae Dedicata 12 (1982) 297-306. 0046-5755/82/0123-0297501.50. Copyright t~ 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.
- 298 MICHAEL GOLDBERG in the direction of this main diagonal, will deform all the parts in the same way. These three infinite families are designated by 9-1-1, 9-1-2 and 9-1-3 in the tabulation. A cube can be fragmented into six square pyramids, which are also space- fillers. If two such pyramids are attached to the opposite faces of a cube, a 12-faced space-filler, known as a pencil cube is produced. A plane through the center of the pencil cube can divide it into two congruent nine-faced parts. This shape is designated by 9-11. If the cutting plane passes through two vertices of the pencil cube, then the parts have one edge less. This shape is designated by 9-11-0. A symmetric hexagonal prism, right or inclined, can be cut into congruent nine-faced parts by a plane through the center and cutting the bases. This shape is designated by 9-111. The 14-faced truncated octahedron is a well-known space-filler. This can be cut into four congruent parts by two planes which cut opposite squares into congruent quadrilaterals. This shape is designated by 9-IV. This 14-faced space-filler can be cut into two 10-faced parts by a plane which cuts a pair of opposite squares at their diagonals. Each of these parts can be cut into two congruent nine-faced parts by a plane through the centers of the two remaining squares and the triangles. This shape is designated by 9-V. The 16-faced space-filler 16-1 [-6] can be cut into four congruent nine- faced parts by four planes from the center and normal to the four edges which are the joins of the four hexagonal faces. This shape is designated by 9-VI. A plane through the axis of the ten-of-diamonds [-3] will cut it into two congruent nine-faced parts. This shape is designated by 9-VII. Three half-planes, 120 Â° apart, through an axis of three-fold symmetry of 16-1, will divide it into three congruent nine-faced parts. These are designa- ted by 9-VIII. Three half-planes, 120 Â° apart, through an axis of three-fold symmetry of 13-111 or 13-111-0, will divide each into three congruent nine-faced parts. These are designated by 9-IX and 9-IX-0. A plane through an axis of 12-1I I8] will divide it into two congruent nine-faced parts in two ways. These are designated by 9-X and 9-X-1. A plane will divide 12-II-0 into two congruent parts designated by 9-X-2. A plane through the axis of 12-X-0 1'8-1 will divide it into two congruent nine-faced parts designated by 9-XXX. A plane through an axis of 12-V 1'8-1 will divide it into two congruent nine- faced parts in two ways. These are designated by 9-XI-1 and 9-XI-2. The space-filler 12-V-0 is divided into two parts designated by 9-XI-0. The space-fillers 12-111 and 12-111-0 are divided into nine-faced parts designated by 9-XII and 9-XII-0.
- ON THE SPACE-F ILL ING ENNEAHEDRA 299 The space-filler 13-1-0 is divided into two congruent parts in two ways, designated by 9-XIII and 9-XIV. The space-filler 13-1 is divided into two congruent parts in two ways, designated by 9-XV and 9-XVI. The space-filler 14-1 is divided into six congruent parts, designated by 9-XIX. The space-filler 16-1 is divided into four congruent parts designated by 9-XX. 3. FusION Sometimes a new space-filler can be obtained by combining two identical or two different space-fillers. Two space-fillers 8-XXVII [9-] of the same height or of different heights can be fused at their bases to produce 9-XVII. Similarly, two space-fillers 8-XXVII-0 of the same height or of different heights can be fused at their bases to produce 9-XIX. A square prism can be divided into eight congruent parts, shown as 7-XXVII or 7-XXVII-1 in [5]. Another square prism can be divided into parts which are mirror images of the first set. A collection of square prisms divided into these two ways can be arranged in a layer in a checkerboard pattern so that the parts of one set meet face-to-face with the parts of the adjacent sets which are mirror images of them. If each part is joined with its mating mirror image, then new space-fillers of nine faces are produced. They are designated as 9-XXI and 9-XXI-1. 4. COMPUTER DERIVATION Several examples were selected from the Koch dissertation because they employed only directly congruent polyhedra [10]. Each of those listed here as 9-XXV (9/8-2), 9-XXVI (9/10-2) and 9-XXVII (9/12-2) has a plane of symmetry. Therefore, each is congruent with its mirror image. Each of those listed as 9-XXIV (9/8-1) and 9-XXIII (9/10-1) has an axis of symmetry and has both right and left forms. However, in packing of these, only one form is used. Each of the space-filling polyhedra with an axis of symmetry can be divided into two congruent parts by a plane containing this axis. If the plane cuts seven of the faces, then each part is a new nine-faced space-filler. These are designated as 9-XXVIII and 9-XXIX. Planes cutting fewer faces produce new space-fillers of seven and eight faces. 5. THE BRUCKNER NUMBER AND POLYHEDRON FORMULAS The face formula of a polyhedron expresses the number of faces of each type. The digit on the right is the number of triangular faces. The next digit is
- 300 MICHAEL GOLDBERG the number of quadrilateral faces; the succeeding digit is the number of pentagonal faces; etc. The vertex formula of a polyhedron expresses the number of vertices of each type. The digit on the right is the number of vertices of degree three (three edges at a vertex); the next digit is the number of vertices of degree four; etc. When the number of vertices exceeds nine, then the number is enclosed in a parenthesis. Federico [12] has reported that there are 2606 combinatorially different polyhedra of nine faces. Of these, Brtickner [13] and Grace [-14] have tabulated and numbered the 50 nine-faced polyhedra all of whose vertices are of degree three. Of the 2,606 possible types of nine-faced polyhedra, only 31 have been found among the space-fillers. The most common type has the polyhedron formulas 54-54. There are 30 different polyhedron formulas, and at least 31 different combinatorial types. 6. TABULAT ION The following table lists all the convex space-filling nine-faced polyhedra that have been found. For each polyhedron, there is an assigned symbol, an approximately descriptive name, the Briickner number if applicable, the face formula, the vertex formula, the number of edges, the method of derivation and a pictorial representation. The number of vertices is always seven less than the number of edges, in accordance with Euler's formula. Most of the cases exhibited are members of infinite classes. For example, in selecting a parallelogram prism, we have the choice of independent angles and independent lengths. The classes of nine-faced space-fillers described here are not mutually exclusive. These classes are examples of methods of derivation, and sometimes a particular shape can be derived by different methods. Furthermore, it is not believed that the list is complete. 9-I-1: Deformed capped cube, 16 edges, 54-54, 1/6 of deformed 7-cube star-cluster 9-I-2: Deformed capped double cube, 16 edges, \ 54-54, 1/6 of deformed 13-cube star-cluster \ 9-I-3: Deformed capped half-cube, 16 edges, 54-54, 1/6 of deformed 4-cube star-cluster
- ON THE SPACE-F ILL ING ENNEAHEDRA 301 9-1I: Capped quadrilateralprism, 16edges,54-54, 1/2 of 12-faced deformed pencil cube 9-I1-0: Capped quadrilateral prism, 15 edges, 36-143, 1/2 of 12-faced deformed pencil cube 9-111: Ungulated hexagonal prism, 21 edges, 110223-(14), Briickner 4a, 1/2 of symmetric hexagonal prism 9-IV: 21 edges, 630-(14), Briickner 33, 1/4 of 14-1, 1/2 of 10-faced space-filler I L_._ I I V__"__J 9-V: 21 edges, 4032-(14). Briickner 21, I/4 of 14-I, 1/2 of 10-faced space-filler i.-V 9-VI: 18 edges, 333-38, 1/4 of 16-I #1\ //I ~ t i / 9-VII: 17 edges, 1044-208, 1/2 often-of-diamonds + 9-VIII: 19 edges, 3105-2(10), 1/3 of 16-I 6
- 302 MICHAEL GOLDBERG 9-IX: 19 edges, 2133-10(11), 1/3 of 13-III 9-IX-O: 18 edges, 333-119, 1/3 of 13-Ill-0 9-X: Parallelogram honeycomb, 20 edges, 450-1 (12), 1/2 of 12-1I 9-X- 1 : Parallelogram honeycomb, 18 edges, 171-38, 1/2 of 12-11 9-X-2: Parallelogram honeycomb, 16 edges, 54-54, 1/2 of 12-II-0 9-XI-I : 20 edges, 450-1(12), 1/2 of 12-V 9-XI-2:19 edges, 270-2(10), !/2 of 12-V 9-XI-O: 16 edges, 54-54, 1/2 of 12-V-O
- ON THE SPACE-F ILL ING ENNEAHEDRA 303 9-XII: 19 edges, 3024-2(10), 1/2 of 12-II1 9-XII-0:16 edges, 54-54, 1/2 of 12-111-0 9-XIII: 20 edges, 10413-1(12), 1/2 of 13-1-0 9-XIV: 19 edges, 513-2(10), 1/2 of 13-1-0 \ 9-XV: 21 edges, 10512-(14), Briickner 20, 1/2 of 13-1 9-XVI: 21 edges, 630-(14), Briiekner 33, 1/2 of 13-I 9-XVII: 19edges, 1242-2(10),joinoftwo8-XXVIl 9-XVIII: 17 edges, 315-46,join of two 8-XXVII-0
- 304 MICHAEL GOLDBERG 9-XIX: 20 edges, 3123-1(12), 1/6 of 14-I, 1/2 of I 0-faced space-filler 9-XX: 17 edges, 1125-127, 1/4 of 16-I, 1/2 of 11-faced space-filler 9-XXI: 16 edges, 135-54, Join of two 7-XXVII, opposite hand 9-XXI-l: 19 edges, 270-2(10), Join of two 7-XXVII-1, opposite hand 9-XXII: 17 edges, 153-46, 1/2 of ll-faced space- filler made of four 7-XXVII (two opposite hand) 9-XXII-I: 19 edges, 270-2(10), 1/2 of ll-faced space filler made of four 7-XXVII-1 (two opposite hand) 9-XXIII: Pentagonal prismatoid 17 edges, 234-46, Koch 9/10-1
- ON THE SPACE-F ILL ING ENNEAHEDRA 305 9-XXIV: 15 edges, 36-224, Koch 9/8-1 9-XXV: 15 edges, 36-305, Koch 9/8-2 9-XXVI: 17 edges, 72-127, Koch 9/10-2 9-XXVII: 19 edges, 1080-2(10), Koch 9/12-2 9-XXVIII: Prismatoid 19 edges, 10152-2(10) 1/2 ofg-xxII I 9-XXIX: 18 edges, 10134-119, 1/2 of 9-XXIV 9-XXX: 17 edges, 1044-46, 1/2 of 12-faced space- filler (66-92) made of four 7-XXIX (two opposite-hand) \ \ /\1 / \1 / /
- 306 MICHAEL GOLDBERG BIBL IOGRAPHY 1. Goldberg, M. : 'Three Infinite Families of Tetrahedral Space-Fillers'. J. Comb. Theory 16 (1974), 348-354. 2. Goldberg, M. : 'The Space-Filling Pentahedra'. J. Comb. Theory 13 (1972), 437-443; 17 (1974), 375-378. 3. Goldberg, M. : 'Several New Space-Filling Polyhedra'. Geometriae Dedicata 5 (1976), 517-523. 4. Goldberg, M. : 'On the Space-Filling Hexahedra'. Geometriae Dedicata 6 (1977), 99-108. 5. Goldberg, M. : 'On the Space-Filling Heptahedra'. Geometriae Dedicata 7 (1978), 175-184. 6. Goldberg, M. : 'Convex Space-Fillers of More than Twelve Faces'. Geometriae Dedicata 8 (1979), 491-500. 7. Goldberg, M. : 'Some Overlooked Convex polyhedral Space-Fillers'. Geometriae Dedicata 9 (1980), 375-379. 8. Goldberg, M. : 'On the Dodecahedral Space-Fillers'. Geometriae Dedicata 10 (1981), 79-89. 9. Goldberg, M. : 'On the Space-Filling Octahedra'. Geometriae Dedicata 10 (1981) 323-335. 10. Koch, Elke: 'Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubschen Gitterkomplexen'. Dissertation, Philipps-Universit~it Marburg/Lahn, 1972. 11. Stein, S. K. : 'A Symmetric Star Body that Tiles But Not as a Lattice'. Proc. Amer. Math. Soc. 36 (1972), 543-548. 12. Federico, P. J. : 'The Number of Polyhedra'. Philips Res. Reports 30 (1975), 220-231. 13. Briickner, Max : Vielecke und Vielflache. B. G. Teubner, Leipzig, 1900, Table III 14. Grace, D, W. : 'Computer Search for Non-isomorphic Convex Polyhedra'. Tech. Report CS 15, Computer Science Dept., Stanford University (1965). Author's Address: Michael Goldberg, 5823 Potomac Avenue, N.W., Washington, D.C. 20016, U.S.A. (Received July 11, 1980)