MICHAEL GOLDBERG

O N T H E S P A C E - F I L L I N G H E X A H E D R A

ABSTRACT. A space-filling polyhedron is one whose replications can be packed to fill three-space completely. The space-filling tetrahedra and pentahedra have been previously investigated. The search is here extended to the convex space-filling hexahedra.

1. INTRODUCTION

The description and enumeration of the space-filling polyhedra, which are an answer to Hilbert's eighteenth problem [I], is far from complete. Partial answers are given in papers by the author [2, 3, 4] for the tetrahedra and pentahedra. These papers give the references to the work of others in the field. The findings of the previous works are an aid in deriving the sought convex space-filling hexahedra for description and tabulation in this work.

2. SYNTHETIC DERIVATION OF SPACE-FILLERS

As in the past, the principal method of investigation is the use of plane-filling polygons to give space-filling prisms, and the division of the commonly known space-fillers into congruent parts to derive new space-fillers. Another method, which seems to have been overlooked, is the production of new space-fillers by the addition of two or more known space-fillers.

The table lists all the space-filling hexahedra that were found. For each hexahedron, there is a pictorial representation, the number of edges, the method of derivation, an approximately descriptive name, and an assigned symbol. The number of vertices is always four less than the number of edges, in accordance with Euler's formula.

There are seven varieties of general convex hexahedra [5]. Six of these are found among the space-filling hexahedra. The seventh, the pentagonal pyramid, is not represented among the space-filling hexahedra.

Most of the cases exhibited are members of infinite classes. For example, in selecting a quadrilateral prism, we have the choice of five independent angles, in addition to independent lengths. The special cases in which there is no variation of shape are designated as singular. The classes of six-faced space-fillers described 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.

Geometriae Dedicata 6 (1977) 99-108. All Rights Reserved Copyright © 1977 by D. Reidel Publishing Company, Dordrecht-Holland

100 MICHAEL GOLDBERG

6-I, Quadrilateral prism, 12 edges, every quadrilateral prism (right or inclined) is a space-filler

6-II-1, Truncated rhombic prism, 12 edges, ½ of rhombic prism

6-II-0, Truncated rhombic prism, 11 edges, ½ of rhombic prism

6-III-1, Truncated prism, 12 edges, ½ of isosceles trape- zoidal prism

6-111-0, Truncated prism, 11 edges, ½ of isosceles trape- zoidal prism

6-IV-l, Truncated square prism, 12 edges, ¼ of square prism

6-IV-0, Truncated square prism, 11 edges, ¼ of prism of square section

6-V-l, Truncated square prism, 12 edges, ½ of prism of square section

J 7 /

, 7

/ J'\ /

ON THE SPACE-FILLING HEXAHEDRA 101

6-V-0, Truncated square prism, 11 edges, ½ of prism of square section

w

6-VI, Double triangular pyra- mid, 9 edges, ¼ of cube

6-VII, Ungulated quadri- lateral prism, 12 edges,

of rhombohedron

6-VIII, Ungulated quadri- lateral pyramid, 11 edges,

of rhombohedron

6-IX, Double triangular pyra- mid, 9 edges, ~ of cube

6-X, Ungulated quadri- lateral pyramid, 10 edges, ½ of cube

I

6-XI, Double triangular pyramid, 9 edges, ¼ of space-filling octahedron. Same as 6-IX, in special case

6-XII, Truncated quadri- lateral pyramid, 12 edges,

of truncated octahedron

102 MICHAEL GOLDBERG

6-XIII, Ungulated quadri- lateral pyramid, 12 edges,

of 16-faced space-filler

6-XIV, Ungulated quadri- lateral pyramid, 12 edges, -1-- of 16-faced space-filler 1 2

6-XV, Doubly truncated prism, 12 edges, ½ of five- faced space-filler 5-XII

6-XVI, Double triangular pyramid, 9 edges, join of two Sommerville tetra- hedra No. 4

6-XVII, Ungulated quadri- lateral pyramid 10 edges, ½ of double triangular pyramid

6-XVIII, Truncated penta- gonal prism, 12 edges, ½ of prism of rectangular isosceles pentagon section

6-XIX, Ungulated triangular prism, 12 edges, ½ of prism of isosceles triangular section

ON THE SPACE-FILLING HEXAHEDRA 103

6-XX, Double triangular pyramid, 9 edges, combina- tion of Sommerville No. 2 and Sommerville No. 4, ½ of diamond-seven

6-XX-1, Doubly truncated isosceles right triangle prism, 11 edges, ¼ of square prism

6-XX-2, Doubly trtmcated isosceles right triangle prism, 10 edges, ¼ of square prism

6-XX-0, Double triangular pyramid, 9 edges, ¼ of square prism

6-XX-0-s, Double triangular pyramid, 9 edges, ¼ of square prism, combination of Sommerville No. 2 and Sommerville No. 4

3. DIVISION OF PRISMS

Prisms, either right or inclined, based on the plane-fillers, are necessarily space-fillers. Since every quadrilateral is a plane-filler, every quadrilateral prism is a space-filler of six faces. Sometimes, a quadrilateral prism can be divided into two directly congruent parts (not merely isometric) to give another shape of a six-faced space-filler. This is feasible when the normal section of the prism is a rectangle or a rhombus (both include squares). Every truncated prism of rhombic section with a plane of symmetry and with rhombic bases is a space-filler, and is shown as 6-II. Two of these can be joined at congruent bases to make an inclined prism. An isosceles trapezoidal prism of rectangular section can be divided into two congruent parts by a plane cut through its axis of symmetry. If the cutting plane cuts four faces, it produces truncated prisms of rectangular section and rectangular bases, and they are shown as 6-III-1 and 6-III-2. If it cuts through two vertices, it produces truncated trapezoidal prisms shown as 6-III-3 and 6-III-4.

An arbitrary plane cuts a square prism in a parallelogram. If the plane is

104 MICHAEL GOLDBERG

held fixed, and the prism is rotated about its axis through an angle of 90 ° while it advances along the axis a fixed distance (this is a twist), the plane will again cut the prism in a congruent parallelogram. If this operation is repeated three more times, the square prism has four congruent segments between the initial cut and the final cut. One of these segments is designated as 6-IV. If two adjacent segments are combined into one piece, it is designated as 6-V. If the parallel edges of the prism are reduced so that the shortest edge vanishes, we have the special cases 6-IV-O and 6-V-O.

4. DIVISION OF CUBES AND RHOMBOHEDRA

Because of the symmetry of the cube and the rhombohedron, they can be divided into congruent parts in several ways. Some divisions are singular, giving only one shape. Others yield members of a continuous infinity of shapes because of the arbitrary position of the set of cuts.

Three half=planes, which have the axis in common, and making angles of 120 ° with each other, will divide them into three congruent six-faced parts shown as 6-VL This is an infinite class of divisions.

A plane through the center at right angles to the principal diagonal (or axis) divides them into two congruent parts which have three-fold symmetry. Three half-planes, which have the axis in common and making angles of 120 ° with each other, will divide them into six-faced parts. One division cuts the small triangles adjoining the hexagon into two parts, shown as 6-VII. Another division does not cut these triangles, and is shown as 6-VIII. Each of these divisions is an infinite class of divisions.

A cube can be divided into 12 congruent parts, one for each edge. The other vertices of a part are the centers of the adjoining faces and the center of the cube, as shown in 6-1X. This is a singular division.

A cube can be cut into four congruent parts, as shown in 6-X. The vertices of a part are a vertex of the cube, the three vertices at the other ends of the face diagonals, and the center of the cube. This is a singular division.

The isosceles rectangular pentagon is a plane-filler. The prism based on this pentagon is a seven-faced space-filler. Because this prism has an axis of symmetry, it can be cut by a plane through this axis into two congruent six-faced space-fillers, shown as 6-XVIII. The prism may be a right prism or an inclined prism, provided the bases are at right angles to the plane contain- ing the base-edges of the pentagons.

Similarly, the prism whose section is an isosceles triangle, can be cut into two congruent six-faced space-fillers by a plane through its axis of symmetry, shown as 6=XIX. Because the bases are cut, the polyhedron obtained is called an ungulated triangular prism. Again, the prism may be a right prism or an inclined prism, provided the bases are perpendicular to the plane containing the base-edges of the isosceles triangles.

ON THE SPACE-FILLING HEXAHEDRA 105

5. DIVISION OF THE OCTAHEDRAL DOUBLE PYRAMID

If the cube is exploded from the center, it produces six square pyramids whose bases are the faces of the cube. This square pyramid was noted by Aristotle as a space-filler. If two of these are joined at their bases, then an octahedral space-filler is formed. However, it is not a regular octahedron, since its height is less than the height of a regular octahedron.

Two planes, through the axis and at right angles to each other, will divide the double pyramid into four six-faced parts, shown as 6-XI. This is an infinite class of divisions.

6. DIVISION OF THE 14-FACED T R U N C A T E D OCTAHEDRON

One of the better known multi-faced space-fillers is the truncated octahedron. It occurs in nature as the shape of some cells in plants and animals, and as a crystal. It can be produced by removing small square pyramids at the vertices of a regular octahedron. The length of the edge of the removed pyramid is one-third of the length of the edge of the octahedron.

The truncated octahedron can be exploded into 24 congruent six-faced parts. The vertices of a part are the center and a vertex of the truncated octahedron, and the centers of the faces adjacent to this vertex, as shown in 6-XII. This is a singular division.

7. DIVISION OF THE 16-FACED SPACE-FILLER

One of the multi-faced space-fillers, not commonly known, is the 16-faced polyhedron that is made by cutting away small tetrahedra from the corners of a large regular tetrahedron, and then replacing the small tetrahedra by shallower tetrahedra. The length of an edge of the removed tetrahedra is one-third of the length of the edge of the large tetrahedron. The faces of the large tetrahedron become regular hexagons. The height of the replaced tetrahedra is one-third of the height of the removed tetrahedra.

Four planes, bisecting the dihedral angles of the edges common to two adjoining hexagons, will cut the 16-faced space-filler into four congruent seven-faced parts of three-fold symmetry. Three half-planes, having the axis of the seven-faced part in common, will divide it into three congruent six- faced parts. I f these planes cut the small triangles adjoining the hexagon, then the parts shown in 6-XIII are produced. If the planes do not cut these triangles, then the parts become as shown in 6-XIV. Each of these divisions is an infinite class of divisions.

8. DIVISION OF A FIVE-FACED SPACE-FILLER

One of the five-faced space-fillers is a symmetrically truncated prism whose section is an equilateral triangle. It is shown as 5-XII in [4]. Because of its

106 MICHAEL G O L D B E R G

symmetry, this can be cut through its axis into two congruent parts. If this plane cuts all three of the remaining parts of the edges of the prism, it produces other five-faced space-fillers. However, if it cuts only one of the remaining edges, it produces the six-faced parts shown as 6-XV. This is an infinite class of divisions.

9. T H E SYMMETRIC DOUBLE PY RA MID

The most symmetric tetrahedral space-filler, designated in [2] as Sommerville No. 1, has four congruent isosceles faces. A pair of opposite edges have length

2, while the other four edges have length V'3. This tetrahedron can be exploded from its center into four congruent tetrahedra, each designated as Sommer- ville No. 4. These can be packed to make triangular prisms whose normal sec- tion is an equilateral triangle. This packing can be done in two isometric ways, each being the mirror image of the other. These prisms can then be assembled to fill space so that each large face of a Sommerville No. 4 in each prism coincides with a large face of a Sommerville No. 4 in the adjacent prism. Therefore, two tetrahedra of type Sommerville No. 4 can be joined at their common faces to produce the new double pyramid six-faced space-filler shown as 6-XVI. This is a singular combination.

Because this double pyramid has two planes of symmetry, it has an axis of symmetry. A plane cut through this axis and not containing an edge will divide the double pyramid into two congruent six-faced space-fillers shown as 6-XVII. This is an infinite class of divisions.

A plane cut through the axis and the short edges will divide the double pyramid into two congruent parts of five faces. This is a new five-faced space- filler, hereby designated 5-XV, which is not included in the lists given in [3] and [4].

10. O T H E R COMBINATIONS OF S P A C E - F I L L E R S

The double pyramid of paragraph 9 is only one of several cases described in the foregoing in which two identical space-fillers are combined to produce another space-filler. A seven-faced space-filler can be made by fusing a Sommerville No. 4 to each of the two faces adjoining a long edge of a Sommerville No. 1. A plane through the long diagonal of the diamond face of this seven-faced polyhedron will divide it into two congruent six-faced space-fillers, shown as 6-XX in the table. It can be produced, also, by com- bining a Sommerville No. 2 with a Sommerville No. 4. Other cases, combining two or more different space-fillers, are possible. For example, 6-VII combined with 6-VIII gives 5-IV which is one third of a rhombohedron. Also, 6-XIII combined with 6-XIV gives 5-Ill which is a rhombic pyramid and is one- sixth of the 16-faced space-filler. These are illustrated in Figures 1 and 2.

ON THE SPACE-FILLING HEXAHEDRA 107

Fig. 1. (6-XIII) + (6-XIV) = (5-IID.

Fig. 2. (6-VII) + (6-VIII) = (5-IV).

11. SUGGESTIONS FOR FUTURE INVESTIGATION

There is no assurance that the list o f space-filling hexahedra is complete. This lack o f assurance holds, also, for the tetrahedra and pentahedra. More in- tensive investigation is needed in all o f these cases.

Space-filling polyhedra o f more than six faces have been only sparsely studied. There are examples o f space-filling n-hedra up to n = 16, with the puzzling exception o f n -- 15. There is no p roo f that n may not exceed 16. Much more work is needed to answer Hilbert 's eighteenth problem in full.

B I B L I O G R A P H Y

1. Hilbert, D.: Mathematische Probleme, Nachr. Gesellschaft Wiss. G6ttingen, Math.- Phys. Klasse, 1900; Bull. Am. Math. Soc. 8 (1901-1902), 437-479; 'Gesammelte Abhandlungen', vol. 3, pp. 290-329, Springer, Berlin, 1935.

2. Goldberg, M.: 'Three Infinite Families of Tetrahedral Space-fillers', or. Combinatorial Theory 16 (1974), 348-354.

108 MICHAEL GOLDBERG

3. Goldberg, M. : 'The Space-filling Pentahedra', Y. Combinatorial Theory 13 (1972), 437-443.

4. Goldberg, M. : 'The Space-filling Pentahedra', II, J. Combinatorial Theory 17 (1974), 375-378.

5. Steiner, J. : Gesammelte Werke, Vol. 1, pp. 227 and 454. 6. Wood, W.G.: 'Space Enclosure Systems', Bulletin 203, 22-25, Engineering Experi-

ment Station, Ohio State University, Columbus, Ohio, 1968.

(Received September 25, 1975)

Author's address:

Michael Goldberg, 5823 Po tomac Ave., N.W., Washington, D.C. 20016, U.S.A.