several new space-filling polyhedra

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MICHAEL GOLDBERG SEVERAL NEW SPACE-FILLING POLYHEDRA 1. INTRODUCTION Only partial answers have been offered to the question that Hilbert asked in his eighteenth problem [1]: 'What convex polyhedra exist for which a com- plete filling of space is possible by juxtaposition of congruent copies ?' Summaries of the results for tetrahedra and pentahedra are given in papers by the author [2, 3, 4]. In his search for space-filling hexahedra, several new types have been derived. These have enabled the derivation of new space- fillers of 6, 7, 8, 9, and 10 faces. 2. THE HALF-SHELL AND THE CLAM Most of the previously described space-fillers have been derived by sub- division of the space-filling prisms into congruent parts. An illustration of this method is shown in the following paragraphs where Sommerville's space- filling tetrahedra are described. A prism, whose normal section is an equilateral triangle, is divided into three congruent tetrahedra, as shown in Figure 1. If the opposite long edges of this tetrahedron, shown as heavy lines, have length 2, then the other four short edges have length ~. This space-filling tetrahedron is designated as Sommerville No. 1. Note that the three tetrahedra can be assembled to make a prism in which the long edges are the chords of either a right-hand helix or Fig. 1. Three Sommerville tetrahedra No. 1 make either a left-hand prism or a right- hand prism. Geometriae Dedicata 5 (1976) 517-523. All Rights Reserved Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland

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Page 1: Several new space-filling polyhedra

M I C H A E L G O L D B E R G

S E V E R A L N E W S P A C E - F I L L I N G P O L Y H E D R A

1. I N T R O D U C T I O N

Only partial answers have been offered to the question that Hilbert asked in his eighteenth problem [1]: 'What convex polyhedra exist for which a com- plete filling of space is possible by juxtaposition of congruent copies ?' Summaries of the results for tetrahedra and pentahedra are given in papers by the author [2, 3, 4]. In his search for space-filling hexahedra, several new types have been derived. These have enabled the derivation of new space- fillers of 6, 7, 8, 9, and 10 faces.

2. THE HALF-SHELL AND THE CLAM

Most of the previously described space-fillers have been derived by sub- division of the space-filling prisms into congruent parts. An illustration of this method is shown in the following paragraphs where Sommerville's space- filling tetrahedra are described.

A prism, whose normal section is an equilateral triangle, is divided into three congruent tetrahedra, as shown in Figure 1. If the opposite long edges of this tetrahedron, shown as heavy lines, have length 2, then the other four

short edges have length ~ . This space-filling tetrahedron is designated as Sommerville No. 1. Note that the three tetrahedra can be assembled to make a prism in which the long edges are the chords of either a right-hand helix or

Fig. 1. Three Sommerville tetrahedra No. 1 make either a left-hand prism or a right- hand prism.

Geometriae Dedicata 5 (1976) 517-523. All Rights Reserved Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland

Page 2: Several new space-filling polyhedra

m m

Fig. 2.

518 MICHAEL GOLDBERG

The front and side elevations of the hinged model of three Sommerville tetra- hedra No. 1.

a left-hand helix. A model of the three tetrahedra is shown in Figure 2. They are hinged at the long edges. Then the tetrahedra can be folded to make a prism with either a right-hand helix or a left-hand helix.

Each Sommerville tetrahedron No. 1 can be exploded from its center into

four congruent shallow fragments of height V~]4 as shown in the model of Figure 3. Each fragment is another space-filling tetrahedron designated as

Fig. 3. The hinged model of four Sommerville tetrahedra No. 4.

Sommerville No. 4. In the drawing of the model, the doubled edges are hinges about which the parts may be folded to make a Sommerville No. 1.

If both right-hand and left-hand prisms of Figure 1 are assembled in the pattern shown in Figure 4, then each right-hand prism makes face contact with three left-hand prisms, and each left-hand prism makes contact with three right-hand prisms. If the prisms are adjusted so that the faces of the component Sommerville tetrahedra No. 1 are matched in indentical positions, then the tetrahedra can be exploded into Sommerville tetrahedra No. 4 so that a No. 4 of each prism has a face in common with a No. 4 of another

Page 3: Several new space-filling polyhedra

Fig. 4. The arrangement of right-hand and left-hand prisms to obtain the clam space- filler.

SEVERAL NEW SPACE-FILLING POLYHEDRA 519

prism. Similarly, in each prism, the faces of the component pairs of Sommer- ville tetrahedra No. 4 are matched. These two No. 4 tetrahedra can be fused at this face, like the two half-shells of a clam, to make a new space-filling double pyramid of six faces. Thus, to the technique of making space-fillers by fission can be added the technique of making space-fillers by fusion.

3. DERIVATIVES OF THE CLAM

The six-faced double pyramid (the clam) has two planes of symmetry, shown as AB and CD in Figure 5. Their intersection is an axis of symmetry. Plane

Fig. 5.

C F

The space-filling double pyramid (the clam) and its derivatives.

Page 4: Several new space-filling polyhedra

520 MICHAEL G O L D B E R G

AB contains the common faces of the two component Sommerville tetra- hedra No. 4. The other plane of symmetry, CD, divides the clam into two congruent five-faced quadrilateral pyramids. This space-filler seems to be new; it has not been found in the previous literature. The base of the pyramid is composed of two isosceles triangles and it is sometimes called a kite.

Any plane through the axis of symmetry, like EF, cuts the clam into two congruent fragments. Except for the case of the kite pyramids, these fragments have six faces. Since the orientation of the cutting plane is arbitrary, a new infinite family of six-faced space-fillers is obtained in this way.

4. T H E T E N - O F - D I A M O N D S AND ITS DERIVATIVES

In the search for other space-fillers that could be made by fusion, several new space-fillers were found. One of them is made by combining four clams. Another way of making this polyhedron is by fusing a Sommerville No. 4 to each of the four faces of a Sommerville No. 1. The result is a ten-faced polyhedron, shown in Figure 6, eight of whose faces are congruent isosceles triangles, while the other two faces are congruent rhombs (or diamonds). It can be described as a space-filling diamond antiprism of ten faces, but it will be hereafter called the ten-of-diamonds. It has been discovered independently in another way by Michael A. Liscano, an engineering student of Professor Donald G.Wood [5]. It can fill space by placing its copies appropriately in six different orientations. If these polyhedra are joined in pairs at their diamond faces, then the axis of each pair is parallel to one of the three axes of a cubic lattice, as shown in Figure 7. To simplify the diagram, only the diamond faces are shown.

Because of the symmetry of the ten-of-diamonds, it is possible to derive a series of new space-fillers by fragmentation. A plane, midway between the diamond faces, divides the polyhedron into two congruent polyhedra, also of ten faces. The new face at the section is an octagon. Let us call this polyhedron the octagon-ten.

Any plane through the axis of symmetry divides the ten-of-diamonds into two congruent fragments. If this plane passes through two vertices, then the fragments are seven-faced. If it does not pass through vertices, then the fragments are nine-faced.

The octagon-ten can be further divided into two congruent fragments by a plane through the axis. If the plane passes through either diagonal of the diamond, then the fragments are seven-faced. If it passes through other vertices, or no vertices, then the fragments are eight-faced.

In Figure 6, the number of faces of the polyhedron is given by F. The notation 1"/2 indicates that the displayed polyhedron is half of the ten-of- diamonds; the notation P/4 indicates that the displayed polyhedron is one-

Page 5: Several new space-filling polyhedra

S E V E R A L N E W S P A C E - F I L L I N G P O L Y H E D R A 521

P - T£N -OF - DIAMONDS; F : i0

P/~; F = 9

\ . , /

P/k; ~ = 8

P/2 ; F -- 7

F i g . 6.

P/2; F = i0

/N / \

P/a; F= 8

/ /

// \\

\ \ P/~; F : 7

? /4 ; F = 7

The ten-of-diamonds space-filler and its derivatives of F faces.

fourth of the ten-of-diamonds. The light dashed lines are the ghost outlines of the missing twins of the polyhedra.

5. THE D I A M O N D - S E V E N A N D I T S D E R I V A T I V E S

Another 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. The result is a seven-faced space-filler, shown in Figure 8. When two of these are joined at

Page 6: Several new space-filling polyhedra

522 MICHAEL GOLDBERG

Fig. 7.

/

The locations and orientations of the diamond faces of the space-filling ten-of- diamonds with respect to a cubic lattice.

their diamond faces, the pairs can be stacked by alternating their orientations along a prism of square section. This seven-faced space-filler can be further sub-divided into congruent parts by a plane through the axis of symmetry. If the plane does not pass through a vertex, then the parts are seven-faced. A plane through the short diagonal of the diamond will divide the polyhedron into two congruent five-faced space-fliers. A plane through the long diagonal

Fig. 8.

~\ y/~\ /7- \\// \ ~,.__ / i~,, fU-\

The diamond-seven and its packing into an infinite square prism.

Page 7: Several new space-filling polyhedra

SEVERAL NEW SPACE-FILLING POLYHEDRA 523

of the diamond will divide the polyhedron into two congruent six-faced space- fillers. This double pyramid is composed of a Sommerville No. 4 fused to a Sommerville No. 2.

Both this seven-faced space-filler and the ten-of-diamonds are only special cases of an infinite family of topologically similar polyhedral space-fliers.

BIBLIOGRAPHY

1. Hilbert, D.: Mathematische Probleme, Nachr. Gesellschaft Wiss. G5ttingen, Math.- Phys. Klasse, 1900; Bull. Amer. Math. Soc. 8 (1901-1902), 437-479; Gesammelte Abhandlungen, Vol. 3, 290-329, Springer, Berlin, 1935.

2. Goldberg, Michael: 'Throe Infinite Families of Tetrahedral Space-fillers', J. Com- binatorial Theory, Sect. A16 (1974), 348-354.

3. Goldberg, Michael: 'The Space-filling Pentahedra', J. Combinatorial Theory, Sect. A13 (1972), 437-443.

4. Goldberg, Michael: 'The Space-filling Pentahedra, II', J. Combinatorial Theory, Sect. A17 (1974), 375-378.

5. Wood, Donald G.: Space Enclosure Systems---The orderly sub-division of the cube, The Ohio State University, 1973.

(Received September 2, 1975)

Author's address:

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