# On the space-filling heptahedra

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MICHAEL GOLDBERG ON THE SPACE-F ILL ING HEPTAHEDRA ABSTRACT. A space-filling polyhedron is one whose replications can be packed to fill three-space completely. The space-filling tetrahedra, pentahedra and hexahedra have been previously investigated. The search is here extended to the convex space-filling heptahedra. 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 listed by the author [1], [2] for the tetrahedra, pen- tahedra and hexahedra. The findings of the previous works are an aid in deriving the sought space-filling heptahedra for tabulation and description in this work. 2. SYNTHETIC DERIVATION OF SPACE-FILLERS The plane pavings can be used to generate single layers of space-filling prisms. By the use of two layers, one shifted from the other and corrugating their interface, a two-layer honeycomb is generated. The bee-cell and the Fejes T6th cell are examples derived from hexagonal paving [7]. Other examples are obtained from quadrilateral and triangular pavings. The paving polygons need not be regular. However, the prisms and the honeycomb cells obtained from the regular polygons have an additional advantage. They may be further subdivided into congruent pieces which are also polyhedral space-fillers, usually of fewer faces. New polyhedral space-fillers can be obtained by combining two honeycomb cells into one by adjoining their congruent bases. The two component cells need not be congruent. They may be different cells like the bee-cell and the Fejes T6th cell. The square honeycomb cells are another of these combina- tions. Furthermore, these double-cells may be subdivided into congruent parts when the cells have some symmetry. The table lists all the space-filling heptahedra that were found by these methods. For each heptahedron, there is a pictorial representation, the number of edges, the methods of derivation, an approximately descriptive name, an assigned symbol and the Federico Number. The number of vertices is always five 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 pentagonal prism, we have the choice of independent angles and independent lengths. The special cases in which there is no variation in shape are designated as singular. The classes of seven-faced space-fillers described Geometriae Dedicata 7 (1978) 175-184. All Rights Reserved Copyright 1978 by D. Reidel Publishing Company, Dordrecht, Holland 176 MICHAEL GOLDBERG here are not mutually exclusive. The classes are examples of methods of derivation, and sometimes a particular shape can be derived by different methods. Furthermore, it is not known if the list is complete. 7-I, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 1 [5] 7-II, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 2 [5] 7-III, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 3 [5] 7-IV, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 4 [5] 7-V, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 5 [5] 7-VI, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 6 [5] 7-VII, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 7 [5] 7-VIII, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 8 [5] 7-IX, Pentagonal prism (right or inclined), F23, 15 edges, based on pentagon of Type 9 [6] 7-I-s, 7-II-s, 7-III-s, 7-IV-s, 7-IX-s, Truncated pentagonal prisms (right or inclined), F23, 15 edges, based on symmetric pentagons of Types 1, 2, 3, 4, 9, of pentagonal prisms 7-II-s-0, 7-III-s-0, 7-IX-s-0, Truncated pentagonal prisms, F24, 14 edges, based on symmetric pentagons of Types 2, 3, 9, of pentagonal prisms, eliminate one edge 7-X, Ungulated pentagonal prism, F20, 15 edges, of pentagonal prism 7-X-0, Ungulated pentagonal prism, F22, 14 edges, of pentagonal prism Pentagon A+B+C=27r A+B+D=2rr a=d A = C= D= 21r/3 a=b,d=c+e A = C = 7r/2 a=b,c=d A = ~r/3, C = 2~r/3 a=b,c=d A +B+ D=2rr , A =2C a=b=e,c=d (R.B. Kershner) 2B+ C=2D+A =2~r a=b=c=d (R.B. Kershner) 2A +B=2D + C=21r a=b=c=d (R. B. Kershner) E= ~r/2,,4 = I t - D B = (~ + D) [2 , C = ~r - D/2 e=a=d+b (Richard E. James, III) /,,IN ON THE SPACE-FILLING HEPTAHEDRA 177 7-XI, Truncated hexagonal prism, F11, 15 edges, of hexagonal prism 7-XII, Ungula of hexagonal prism, F23, 15 edges, of hexagonal prism 7-XII-0, Ungula of hexagonal prism, F30, 13 edges, of hexagonal prism 7-XIII, Doubly truncated quadrilateral prism, F36, 13 edges, k of skewed double bee-cell 7-XIII-0, Doubly truncated quadrilateral prism, F44, 11 edges, of skewed double bee-cell, of space-filling octahedron 7-XIV, Three-fold truncation of triangular prism, F18, 13 edges, of double rhombic dodecahedron (augmented diamond-seven) 7-XIV-0, Three-fold truncation of triangular prism, F44, 11 edges, of double rhombic honeycomb (diamond-seven) 7-XV, Three-fold truncation of triangular prism, F22, 14 edges, of rhombic honeycomb 7-XV-0, Three-fold truncation of triangular prism, F44, 11 edges, of rhombic honeycomb 178 MICHAEL GOLDBERG 7-XVI, Ungulated symmetric quadrilateral prism, F12, 15 edges, of symmetric quadrilateral prism 7-XVII, Ungulated triangular pyramid, F12, 15 edges, of rhombohedron 7-XVIII, Three-fold truncation of triangular prism, F18, 13 edges, of unsymmetric dodeca- hedron 7-XVIII-0, F44, 11 edges 7-XIX, Three-fold truncation of triangular prism, F13, 15 edges, of skewed square honeycomb 7-XX, Doubly truncated rectangular prism, F23, 15 edges, of rectangular honeycomb 7-XX-0, Doubly truncated rectangular prism, F30, 13 edges, of rectangular honeycomb 7-XXI, Truncated pentagonal pyramid, F23, 15 edges, ~-~ of truncated octahedron 7-XXII, Doubly truncated quadrilateral pyramid, F20, 15 edges, ] of 14-faced truncated octa- hedron 7-XXII-0, Doubly truncated quadrilateral pyra- mid, F28, 14 edges, ~ of 14-faced truncated octahedron 7-XXIII, Triply truncated triangular pyramid, F29, 13 edges, singular, of ten-of-diamonds 7-XXIV, Triply truncated quadrilateral prism, F43, 11 edges, singular, of ten-of-diamonds ON THE SPACE-FILLING HEPTAHEDRA 179 7-XXV, Triply truncated triangular prism, F30, 14 edges, infinite family, of 7-XIV-0 I 7-XXVI, Triply truncated triangular prism, F25, 15 edges~ infinite family, of 7-XIV 7-XXVII, Doubly truncated quadrilateral prism, F42, 12 edges, ~ of square prism 7-XXVII-1, Doubly truncated quadrilateral prism, 14 edges, F28, add quadrilateral prism to base of 7-XXVII 7-XXVIII, Doubly truncated quadrilateral prism, F42, 12 edges, ~ of regular hexagonal prism 7-XXVIII-1, Doubly truncated quadrilateral prism, 14 edges, F28, add quadrilateral prism to base of 7-XXVIII 7-XXIX, Doubly truncated quadrilateral prism, F42, 12 edges, ~ of equilateral triangular prism 7-XXIX-1, Doubly truncated quadrilateral prism, F28, 14 edges, add quadrilateral prism to base of 7-XXIX 7-XXX, Truncated hexagonal prism, Fll, 15 edges, of inclined prism of regular hexagonal section 180 MICHAEL GOLDBERG 7-XXXI, Truncated hexagonal prism, Fll, 15 edges, of 7-XXX 7-XXXII, Hexagonal prismatoid, F18, 13 edges, of pencil cube 7-XXXIII, Ungula of hexagonal prismatoid, F22, 14 edges, of pencil cube 7-XXXIV, Doubly truncated quadrilateral prism, F36, 13 edges, ~ of pencil cube 7-XXXIV-1, Doubly truncated quadrilateral prism, F36, 13 edges, of capped quadrilateral prism 7-XXXIV-2, Doubly truncated quadrilateral prism, F36, 13 edges, of capped quadrilateral prism 7-XXXV, Truncated quadrilateral prism, F40, 12 edges, of pencil cube /N ON THE SPACE-FILLING HEPTAHEDRA 181 3. THE FEDERICO NUMBER Federico [3], [4] has prepared a table giving certain characteristics of the com- plete set of combinatorially distinct convex polyhedra with four to eight faces, with the corresponding Schlegel diagrams of each. A serial number has been assigned to each polyhedron. This number will be called the Federico Number of the polyhedron. This number is given in the tabulation of the space-filling heptahedra. Of the 34 possible types of heptahedra, only 16 types have been found among the space-fillers. The most common type is Federico Number 23, which corresponds to the pentagonal prism. The 13 space-fillers of this type differ in their metrical properties. 4. DIVISION OF PRISMS Prisms, either right or inclined, based on the plane-fillers, are necessarily space-fillers. Therefore, the prisms based on the pentagonal plane-fillers are seven-faced space-fillers. A tabulation of the plane-filling pentagons was given by Kershner [5]. A new family of plane-filling pentagons is described by Schattschneider [6]. If such a prism has an axis of symmetry, then this prism can be cut by a plane through this axis into two directly congruent truncated prisms to yield a new space-filling heptahedron, These are designated by 7-I-s, 7-II-s, 7-III-s, 7-IV-s and 7-IX-s. If the cutting plane passes through two vertices, another type is derived with one edge less, shown as 7-II-s-0, 7-III-s-0 and 7-IX-s-0. If the cutting plane cuts all seven faces of the heptahedron, then two congruent eight-faced space-fillers are obtained. A rhombohedron (a rhombic prism, which may be a rectangular parallele- piped or a cube) has a center of symmetry. A plane through this center divides the rhombohedron into two congruent space-fillers. If the plane cuts only four faces, then congruent hexahedra are formed. If it cuts all six faces, then two congruent heptahedra are formed, shown as 7-XVI. A space-filling hexagonal prism, which has an axis parallel to two edges of the base, can be cut into two congruent truncated prisms of seven faces, shown as 7-XI and 7-XI-0. 5. OTHER METHODS OF DIVISION Two bee-cells, of regular hexagonal section but of different slopes for the caps, can be joined at their bases with different orientations to form a 12-faced space-filler. This polyhedron can be divided into three congruent heptahedra by a triplet of half-planes through the axis and the ridges of one cap, as shown in 7-XIII or 7-XIII-0. An unsymmetric rhombic dodecahedral space-filler can be divided by two skew planes through the axis into four congruent heptahedra, shown as 7-XVIII and 7-XVIII-0. An eight-faced space-filler can be made by joining two square honeycomb 182 MICHAEL GOLDBERG cells with crossing orientations. A plane through opposite parallel edges will divide this octahedron into two congruent heptahedra, shown as 7-XIX. A cell of a rectangular honeycomb is a nine-faced space-filler which is a rectangular prism capped by four rhombic faces which lie on a rectangular pyramid. A skew plane through the axis of symmetry divides the cell into two congruent heptahedra, shown as 7-XX or 7-XX-0. The 14-faced truncated octahedron can be divided into 12 congruent parts by planes through the lines from the center of the polyhedron to the centers of the eight hexagonal faces and bisecting the alternate edges of the hexagons and squares, shown as 7-XXI. The 14-faced truncated octahedron can be divided into two congruent parts by a plane midway between a pair of opposite square faces. Each of these parts can then be cut into four congruent parts by two skew planes through the axis at right angles to each other, shown as the infinite class 7-XXII. The ten-of-diamonds, described in [2], can be cut into four congruent parts, shown as 7-XXIII. The diamond-seven, shown as 7-XIV-0, is made of four space-filling tetra- hedra, as described in [2]. If a triangular prism is inserted between the halves of 7-XIV-0, we obtain the augmented diamond-seven, shown as 7-XIV. The cube can be divided into six congruent square pyramids by planes from the center to the edges. If two of these square pyramids are joined at their bases, a space-filling octahedron is obtained. If this is bisected by a plane through the center, then two congruent polyhedra are formed. If the plane passes through two vertices, then the congruent polyhedra are heptahedra, shown as 7-XIV-0. If the ten-of-diamonds is bisected by a plane which bisects each of the diamonds, then two congruent heptahedra are obtained, shown as 7-XXIV. If 7-XIV-0 is bisected by a plane through its axis, and does not pass through a vertex, then two congruent heptahedra are obtained, shown as 7-XXV. If 7-XIV is bisected by a plane through its axis and cuts five of its faces, then two congruent heptahedra are obtained, shown as 7-XXVI. A square prism can be divided into eight congruent heptahedra by 16 triangles which have a common vertex at the center of the prism. These tri- angles pass through midpoints or one-fourth points of the edges of the prism. Four triangles have a common vertex at the center of one square face, and four other triangles have a common vertex at the center of the opposite square, as shown in 7-XXVII. The special case of this space-filler for the cube was discovered by Bruce McPherson, a student of Professor Wood [8]. A similar division of the regular hexagonal prism into 12 congruent hepta- hedra is shown in 7-XXVIII; and a division of the equilateral triangle prism into six congruent heptahedra is shown as 7-XXIX. An inclined triangular prism whose normal section is an equilateral triangle, can be divided into three congruent space-filling tetrahedra. This prism can be ON THE SPACE-F ILL ING HEPTAHEDRA 183 trimmed at its three lateral edges to reduce the section from an equilateral triangle to a regular hexagon. The resulting hexagonal prism is also a space- filler. These three cuts add three new faces to each of the component tetra- hedra, converting them into congruent space-filling heptahedra, shown as 7-XXX. This polyhedron, 7-XXX, has an axis of symmetry which passes through the center of the parallelogram face and the midpoint of the opposite edge. A plane through this edge and the axis will divide the polyhedron into two congruent seven-faced space-fillers, shown as 7-XXXI. Other planes through this axis will produce an infinite family of six-faced space-fillers and an infinite family of eight-faced space-fillers. A cube, capped on two opposite faces by space-filling square pyramids, is a 12-faced space-filler. Critchlow [9] calls it a pencil cube. Since this polyhedron has a center of symmetry, any plane through this center will divide the poly- hedron into two congruent parts. These space-filling parts may have 7, 8, 9 or 10 faces. The singular seven-faced space-filler obtained in this manner is shown as 7-XXXII. Since 7-XXXII has an axis of symmetry, it may be further divided into two congruent parts by a plate through this axis. If this plane cuts five faces, the seven-faced family of parts are shown as 7-XXXIII. When the dividing plane is normal to the axis of 7-XXXII, the parts are nine-faced. Each part may now be divided into four congruent parts of five or seven faces. The family of seven-faced space-fillers is shown as 7-XXXIV. A three-dimensional cross can be made of seven congruent cubes by attaching a cube to each face of one of the cubes. This cross is a space-filler, as indicated by Stein [10]. This cross can be divided into six congruent parts by dividing the central cube into six square pyramids. Each part is then a cube capped by a square pyramid, and it is a nine-faced space-filler. Since it has an axis of four-fold symmetry, it can be divided into two congruent parts, or four congruent parts. The number of faces of each part may be 5, 6, 7 or 8. The seven-faced space-filler is shown as 7-XXXIV-1. It is similar to 7-XXXIV, except that the prismatic portion is twice as long. Another space-filling three-dimensional cross can be made of 13 congruent cubes. The arms of the cross consist of two cubes, instead of one cube as described in the foregoing example. As before, this cross can be divided into six congruent capped square prisms. These, also, can be divided to produce space-fillers of 5, 6, 7 or 8 faces. The seven-faced space-filler is shown as 7-XXXIV-2. It is similar to 7-XXXIV, except that the prismatic portion is four times as long. The pencil cube can be cut into two congruent eight-faced parts by a plane midway between two parallel square faces. Since this octahedron has an axis of symmetry, it can be cut into two congruent seven-faced parts, shown as 7-XXXV, by a plane through this axis and two parallel edges. 184 MICHAEL GOLDBERG B IBL IOGRAPHY 1. Goldberg, M., 'On the Space-filling Hexahedra', Geom. Dedicata 6, 99-108 (1977). 2. Goldberg, M., 'Several New Space-filling Polyhedra', Geom. Dedicata 5, 517-523 (1976). 3. Federico, P.J., 'Polyhedra with 4 to 8 Faces', Geom. Dedicata 3, 468-481 (1975). 4. Federico, P.J., 'The Number of Polyhedra', Philips Res. Repts. 30, 220-231 (1975). 5. Kershner, R.B., 'On Paving the Plane', Am. Math. Monthly 75, 839-844 (1968). 6. Schattschneider, D., 'General Paving Pattern of Richard James III ' , Private communication. 7. Fejes T6th, L., 'What the Bees Know and What They Do Not Know', Bull. Am. Math. Soc. 70, 468-481 (1964). 8. Wood, D.G., Space Enclosure Systems, The Orderly Sub-division of the Cube, The Ohio State University, 1973. 9. Critchlow, K., Order in Space, Viking Press, New York, 1970. 10. Stein, S.K., 'A Symmetric Star Body that Tiles as a Lattice', Proc. Am. Math. Soc. 36, 543-548 (1972). Author's address: Michael Goldberg, 5823 Potomac Ave., N.W., Washington, D.C. 20016 U.S.A. (Received November 9, 1976)