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

O N T H E S P A C E - F I L L I N G O C T 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, pentahedra, hexahedra and heptahedra have been previously investigated. The search is here extended to the convex space-filling octahe- dra. The number of types is found to be at least 49.

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

The description and enumeration of the space-filling polyhedra, which are an answer to Hilbert's eighteenth problem, is still far from complete. Par- tial answers are given in papers by the author [3]-[7] for the tetrahedra, pentahedra, hexahedra, heptahedra and for other special cases. The findings of the previous works are an aid in deriving the sought space- filling convex octahedra for tabulation and description in this work.

2. T H E H E X A G O N A L PRISMS

There are three families of hexagons which can fill the plane by replication [9]. Right prisms or inclined prisms based on these hexagons are, therefore, eight-faced space-fillers. These are designated by 8-I, 8-II and 8-III. For those hexagons which have a line of symmetry, right prisms or inclined prisms based on them can be cut into two congruent truncated prisms which are also space-fillers. These are designated by 8-I-s, 8-II-s and 8-III-s. One edge can be eliminated by shortening the lateral edges so that the shortest edge is reduced to zero length. These are designated by 8-I-s-0, 8-II-s-0 and 8-III-s-0.

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Geometriae Dedicata 10 (1981) 323-335. 0046-5755/81/0103-0323501.95. Copyright �9 1981 by D. Reidel Publishin9 Co., Dordrecht, Holland, and Boston, U.S.A.

324 MICHAEL GOLDBERG

The inclined prism which has a regular hexagon for its normal section can be divided by plane cuts into six congruent truncated prisms. The cutting plane is advanced through the same distance along the axis be- tween cuts, while the prism is rotated about the axis through 60 ~ , as shown in Figure 1. New eight-faced space-fillers can be made of one, two or three segments. The first two are designated by 8-I-1 and 8-I-2. The third is the same as 8-I-s. If the lateral edges are shortened so that the shortest edge is reduced to zero length, then one edge is eliminated. These are designated by 8-I-1-0 and 8-I-2-0.

A symmetric hexagonal prism can be cut into two congruent portions by a plane through its center and which cuts the opposite bases. Each un- gulated portion is designated by 8-IV.

3. THE EQUILATERAL OCTAHEDRON AND ITS GENERALIZATION

The regular octahedron is not a space-filler. Minkowski [8], [10] showed that it can be packed to fill 18/19 of unbounded three-space.

Let three-space be filled by congruent cubes in the regular lattice arran- gement. Then, let each cube be exploded into six square pyramids by

Fig. 2.

ON THE S P A C E - F I L L I N G O C T A H E D R A 325

cutting planes from the center of each cube to its edges. If each pair of pyramids is joined at the common square face, then each pair becomes an equilateral octahedral space-filler. These octahedra have three orientations in the packing of space. The axes of these three orientations are equally inclined to the line x = y = z which is a main diagonal of the cube. Then all of these octahedra can be deformed in the same way by an expansion or a compression along the line x = y = z. This yields a one-parameter infinite family of octahedral space-filler, s. A typical example is shown in Figure 2 and is designated by 8-V.

4. G A B L E D R H O M B O H E D R O N

Congruent rhombohedra can be packed to make a plane slab. Then, one plane surface can be grooved to make a corrugated surface (or washboard surface) so that all the rhombohedra are modified in the same way. A similar process can be performed on the other plane surface in the other direction. The final result, shown in Figure 3, is a conversion of the six- faced rhombohedra into eight-faced gabled rhombohedra, designated by

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Fig. 3.

8-VI. Other identical layers of these eight-faced polyhedra can be adjoined by overturning the layers and orienting them so that the congruent surfaces match each other.

If the four lateral edges of the rhombohedron are reduced to zero, then the reduced gabled rhombohedron, designated by 8-VI-0, is obtained.

326 MICHAEL G O L D B E R G

5. F R A G M E N T A T I O N

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 most probable candidates for such consideration.

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 thirteen 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. If each of these capped cubes is divided into congruent parts by a plane through the axis, but not containing an edge, then the resultant parts are eight-faced space-fillers, designated by 8-VII-l, 8-VII-2 and 8-VII-3.

A 16-faced space-filler can be made by modifying a regular tetrahedron. At each vertex of the tetrahedron, remove a tetrahedron whose edges are one-third of the edges of the large tetrahedron. Then, replace the removed tetrahedra by shallow tetrahedra of one-fourth of their height. The result- ing figure is bounded by six regular hexagons and twelve isosceles triangles. This 16-faced space-filler is found in nature as the fundamental domain of the diamond crystal.

The 16-faced space-filler can be divided into four congruent nine-faced space-fillers by planes from the center of the solid and normal to the edges of the original tetrahedron. Each of these parts can be further divided into three congruent parts by planes through the axis and making dihedral angles of 120 ~ . If these planes contain edges of the nine-faced space-filler, then the parts are seven-faced space-fillers which have not been previously reported. In accordance with the designations in [7], they are hereby designated 7-XXXVI, F36, 13 edges, 52-36.

If the three cutting planes do not contain edges of the nine-faced space- fillers, then the parts are eight-faced, and they are designated by 8-XXXII.

The 16-faced space-filler can be divided into two congruent ll-faced parts by the plane which is a perpendicular bisector of the common per- pendicular of a pair of opposite edges. These parts have an axis of symmetry. Hence, each of these parts can be divided into two congruent eight-faced parts in two different ways. One division yields 8-XXXIII, and the other division yields 8-XXXIV.

The 12-faced double bee-cell can be divided into two congruent parts by a plane through a pair of opposite edges. Each part is designated by 8-VIII.

The 12-faced rectangular honeycomb can be divided into two congruent parts by a plane through four edges. These parts are designated by 8-IX.

ON T H E S P A C E - F I L L I N G O C T A H E D R A 327

The ten-of-diamonds, described in [5], yields eight-faced space-fillers in two different ways. One is one-fourth of the ten-of-diamonds, and it is shown as 8-X. The other is one-half of the ten-of-diamonds, and it is shown as 8-XI. Other examples are:

8-XII - one-fourth of the 12-faced pencil cube; 8-XIII - one-fourth of the rhombic dodecahedron; 8-XIV - one-third of the 13-faced triangular prism honeycomb; 8-XV - one-fourth of the square honeycomb, cut through the hexagons; 8-XVI - one-fourth of the square honeycomb, cut through edges; 8-XVII - one-fourth of the square honeycomb, cut through quadrilateral and hexagons; 8-XVIII, 8-XIX, 8-XXXI - one-sixth of the truncated octahedron 14-1; 8-XXXIII, 8-XXXIV, 8-XXXVI - one-fourth of 16-1; 8-XX - one-half of the 12-faced space-filler 12-V; 8-XXIV - one-eighth of the square prism; 8-XXV - one-twelfth of the hexagonal prism; 8-XXVI - one-sixth of the triangular prism; 8-XXVII - one-sixth of the hexagonal prism; 8-XXVIII - one-half of the symmetric pentagonal prism; 8-XXIX - one-half of the deformed equilateral octahedron 8-V; 8-XXX - one-fourth of the regular hexagonal prism; 8 -XXXII - one-twelfth of 16-1; 8-XXXV - one-half of the 12-faced pencil cube.

6. FUSION

Sometimes, two known space-fillers can be fused together to form a new space-filler. Congruent triangular prisms can form a slab with plane paral- lel faces in which the plane faces are divided into congruent rectangles. Another shape of triangular prism, which has the same rectangle as a face, can make another slab. If the prisms of one slab are fused at their rec- tangles with the prisms of the other slab, then new eight-faced space-fillers are obtained, and are shown as 8-VI-0.

Two seven-faced space-fillers of type 7-XXVII of different heights can be combined to make 8-XXI. Similarly, two space-fillers of type 7-XXVIII, or of type 7-XXIX, can be joined to make 8-XXII and 8-XXIII. The insertion of prisms make 8-XXI-1, 8-XXII-1 and 8-XXIII-1.

7. OTHER MODIFICATIONS

Slices removed from the top and bot tom of 7-XXVII makes 8-XXIV, which is one-eighth of a square prism. Similarly, slices removed from the top and

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

bottom of 7-XXVIII makes 8-XXV, which is one-twelfth of a hexagonal prism. Slices removed from the top and bottom of 7-XXIX makes 8-XXVI, which is one-sixth of a triangular prism.

Modification of six 7-XXIX makes a hexagonal prism composed of six 8-XXVII. The removal of the bottom prism makes 8-XXVII-0.

8. T H E F E D E R I C O N U M B E R A N D P O L Y H E D R O N F O R M U L A S

Federico [1], [2] has prepared a table giving certain characteristics of the complete 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 Federico number will be called the F number of the polyhedron. Also listed are two numbers which are the face formula and the vertex formula.

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 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 of any particular degree exceeds nine, then the number is enclosed in parentheses.

Of the 257 possible types of octahedra, only 26 types have been found among the space-fillers. The most common type is F54, which corresponds to the hexagonal prism.

9. T A B U L A T I O N

The following table lists all the convex space-filling octahedra that have been found by these methods. For each octahedron there is an assigned symbol, an approximately descriptive name, the Federico number, the face formula, the vertex formula, the number of edges, the method of derivation and a pictorial representation. The number of vertices is always six 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 hexagonal prism, we have the choice of independent angles and independent lengths. The classes of eight-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 known if the list is complete.

ON THE S P A C E - F I L L I N G OCTAHEDRA 329

8-I, 8-II, 8-Ili: Hexagonal prism (right or inclined), F54, 18 edges, 2060-(12), based on hexagons of types 1, 2, 3 [9]

8-I-s, 8-II-s, 8-III-s: Truncated hexagonal prisms (right or inclined), F54, 18 edges, 2060-(12), based on symmetric hexagons of types 1, 2, 3, �89 of hexagonal prisms

8-I-s-0, 8-II-s-0, 8-III-s-0: Truncated hexagonal prisms, (right or inclined), F74, 17 edges, 2042-1(10), based on symmetric hexagons of types 1, 2, 3, -~ of hexagonal prisms, cut through vertex

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8-1-1, 8-1-2: Truncated hexagonal prism, F54, 18 edges, 2060-(12), ~ and �89 of uniformly sliced hexagonal prism of regular section

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8-IV: Ungulated hexagonal prism, F52, 18 edges, 2222-(12), �89 of symmetric hexagonal prism

8-V: Deformed equilateral octahedron, F300, 12 edges, 8-60

8-VI: Gabled rhombohedron, F58, 18 edges, 440-(12)

330 MICHAEL GOLDBERG

8-VI-0: Reduced gabled rhombohedron, F287, 14 edges, 44-44

8-VII-1: Gabled quadrilateral prism, F236, 15 edges, 143-36, �89 of nine-faced capped cube

8-VII-2:�89 of nine-faced capped double cube

8-VII-3 : �89 of nine-faced capped half-cube

8-VIII: Gabled quadrilateral prism, F137, 16 edges, 1052-28, �89 of double bee-cell

8-IX: Hexagonal prismoid, F54, 18 edges, 2060-{12), �89 of 12-faced rectangular honey- comb cell

8-X: Pentagonal prismoid, F133, 1052-109, �88 of ten-of-diamonds

16 edges,

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8-XI: Pentagonal prismoid, F222, 15 edges, 143-117, �88 of ten-of-diamonds

8-XII: Gabled quadrilateral prism, F282, 14 edges, 44-44, �88 of 12-faced pencil cube

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ON THE SPACE-FILLING OCTAHEDRA 331

8-XIII: Gabled quadrilateral prism, F233, 15 edges, 143-36, �88 of rhombic dodecahedron / / \ \

8-XIV: Gabled quadrilateral prism, F136, 16 edges, 1052-28, -} of 13-faced triangular prism honeycomb

8-XV: F80, 17 edges, 1232-1(10), �88 of square honeycomb, cut through hexagons, �88 of truncated octahedron 14-I

8-XVI: Fl17, 16 edges, 1214-28, �88 of square honeycomb, cut through edges -~ of truncated oetahedron 14-1 \ . < - - - y

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8-XVII: F55, 18 edges, 1412-(12), �88 of square honeycomb, cut through quadrilaterals and hexagons

8-XVIII: Fl17, 16 edges, 1214-28, x 6 of 14-faced truncated octahedron

8-XIX: F58, 18 edges, 440-(12), -~ of 14-faced truncated octahedron

332 MICHAEL GOLDBERG

8-XX: Gabled quadrilateral prism, F54, 18 edges, 2060-(12), �89 of 12-faced space-filler 12-V

8-XXI: Gabled quadrilateral prism, F271, 14 edges, 125-44, join of two 7-XXVII of different heights

8-XXI-I: F137, 16 edges, 1052-28, insert prism into 8-XXI

8-XXII: F271, 14 edges, 125-44, join of two 7-XXVIII of different heights

8-XXII-I: F137, 16 edges, 1052-28, insert prism into 8-XXII

8-XXIII: F271, 14 edges, 125-44, join of two 7-XXIX of different heights

8-XXIII-I: F137, 16 edges, 1052-28, insert prism into 8-XXIII

8-XXIV: F66, 17 edges, 2204-1(10), -~ of square prism, slices removed from top and bottom of 7-XXVII

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ON THE S P A C E - F I L L I N G OCTAHEDRA 333

8-.XXV: F66, 17 edges, 2204-1(10), ~ of hexa- gonal prism, slices removed from top and bottom of 7-XXVIII

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8-XXVI: F66, 17 edges, 2204-1(10), -~ of triangu- lar prism, slices removed from top and bottom of 7-XXIX

8-XXVII: F95, 17 edges, 341-1(10), -~ of hexa- gonal prism cut from triangular prism of six 7-XXIX, -~ of truncated octahedron 14-1

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8-XXVII-0: F206, 15 edges, 224-36, bottom of 8-XXVII removed to eliminate two edges

8-XXVIII: Ungulated pentagonal prism, F46, 18 edges, 11213-(12), �89 of symmetric pentagonal prism

334 MICHAEL GOLDBERG

8-XXIX: Hexagonal prismoid, F191, 15 edges, 1034-36, �89 of deformed equilateral octahedron 8-V

8-XXX: F50, 18 edges, 3113-(12), �88 of regular hexagonal prism

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8-XXXI: F171, 16 edges, 161-28, -~ of truncated octahedron

8-XXXII: F235, 15 edges, 143-36, �89 of nine-faced space-filler, ~ of 16-faced space-filler

8-XXXIII: Modified triangular prism, F215, 15 edges, 224-36, �88 of 16-faced space-filler

8-XXXIV: F265, 14 edges, 125-44, �88 of 16-faced space-filler

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

8-XXXV: F299, 13 edges, 26-52, -~ of pencil cube

8-XXXVI: F105, 16 edges, 2024-28, �88 of 16-faced space-filler, �89 of 11-faced space-filler

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

1. Federico, P.J., ' Polyhedra with 4 to 8 Faces', Geom. Ded. 3, 468-481 (1975). 2. Federico, P.J., 'The Number of Polyhedra', Philips Res. Repts 30, 220-231 (1975). 3. Goldberg, M., 'Three Infinite Families of Tetrahedral Space-fillers', J. Comb. Theory 16,

348-354 (1974). 4. Goldberg, M., 'The Space-filling Pentahedra', J. Comb. Theory 13, 437-443 (1972); 17,

375-378 (1974). 5. Goldberg, M., ' Several New Space-filling Polyhedra ', Geom. Ded. 5, 517-523 (1976). 6. Goldberg, M., 'On the Space-filling Hexahedra', Geom. Ded. 6, 99-108 (1977). 7. Goldberg, M., 'On the Space-filling Heptahedra', Geom. Ded. 7, 175-184 (1978). 8. Hancock, H., Development of the Minkowski Geometry of Numbers, Vol. 2, 1939, pp.

740-741. 9. Kershner, R.B., 'On Paving the Plane', Am. Math. Monthly 75, 839-844 (1968).

10. Minkowski, H., Gesammelte Abhandlungen, Vol. 1, p. 354. 11. Stein, S.K., 'A Symmetric Body that Tiles but not as a Lattice', Proc. Am. Math. Soc. 36,

543-548 (1972).

(Received July 5, 1979)

Author's address:

M i c h a e l G o l d b e r g 5823 P o t o m a c Avenue , N , W . Washington, D.C., 20016 U.S.A.