[go: up one dir, main page]
TOPICS
Search

Space-Filling Polyhedron

SpaceFillingPolyhedra

A space-filling polyhedron is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not. Several space-filling polyhedra are illustrated above.

Having Dehn invariant 0 is a necessary but not sufficient condition for a polyhedron to be space-filling.

The cube is the only Platonic solid possessing this property (Gardner 1984, pp. 183-184). However, a combination of tetrahedra and octahedra do fill space (Steinhaus 1999, p. 210; Wells 1991, p. 232). In addition, octahedra, truncated octahedron, and cubes, combined in the ratio 1:1:3, can also fill space (Wells 1991, p. 235). In 1914, Föppl discovered a space-filling compound of tetrahedra and truncated tetrahedra (Wells 1991, p. 234).

There are only five space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron (Steinhaus 1999, pp. 185-190; Wells 1991, pp. 233-234), and gyrobifastigium (Johnson 2000). The rhombic dodecahedron (Steinhaus 1999, pp. 185-190; Wells 1991, pp. 233-234) and elongated dodecahedron, and trapezo-rhombic dodecahedron appearing in sphere packing are also space-fillers (Steinhaus 1999, pp. 203-207), as is any non-self-intersecting quadrilateral prism. The cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron are all "primary" parallelohedra (Coxeter 1973, p. 29).

A stereohedron is a convex polyhedron that is isohedrally space-filling, meaning the symmetries of a tiling of copies of a stereohedron take any copy to any other copy. A plesiohedron is a space-filling polyhedron which has special symmetries that take any copy of the plesiohedron in the space-filling honeycomb to any other.

In the period 1974-1980, Michael Goldberg attempted to exhaustively catalog space-filling polyhedra. According to Goldberg, there are 27 distinct space-filling hexahedra, covering all of the 7 hexahedra except the pentagonal pyramid. Of the 34 heptahedra, 16 are space-fillers, which can fill space in at least 56 distinct ways. Octahedra can fill space in at least 49 different ways. In pre-1980 papers, there are forty 11-hedra, sixteen dodecahedra, four 13-hedra, eight 14-hedra, no 15-hedra, one 16-hedron originally discovered by Föppl (Grünbaum and Shephard 1980; Wells 1991, p. 234), two 17-hedra, one 18-hedron, six icosahedra, two 21-hedra, five 22-hedra, two 23-hedra, one 24-hedron, and a believed maximal 26-hedron. In 1980, P. Engel (Wells 1991, pp. 234-235) then found a total of 172 more space-fillers of 17 to 38 faces, and more space-fillers have been found subsequently. P. Schmitt discovered a nonconvex aperiodic polyhedral space-filler around 1990, and a convex polyhedron known as the Schmitt-Conway biprism which fills space only aperiodically was found by J. H. Conway in 1993 (Eppstein).

Schmitt (2016) gives a summary of space-filling polyhedra via an investigation of which Dirichlet-Voronoi stereohedraStereohedron the tetragonal, trigonal, hexagonal, and cubic groups can generate.

See also

Cube, Cuboctahedron, Dehn Invariant, Elongated Dodecahedron, Escher's Solid, Keller's Conjecture, Kelvin's Conjecture, Octahedron, Parallelohedron, Plesiohedron, Primary Parallelohedron, Prism, Rhombic Dodecahedron, Schmitt-Conway Biprism, Sphere Packing, Stereohedron, Tessellation, Tetrahedron, Tiling, Triangular Orthobicupola, Truncated Octahedron

Explore with Wolfram|Alpha

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973.Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970.Devlin, K. J. "An Aperiodic Convex Space-Filler is Discovered." Focus: The Newsletter of the Math. Assoc. Amer. 13, 1, Dec. 1993.Engel, P. Geometric Crystallography: An Axiomatic Introduction to Crystallography. New York: Springer-Verlag, 1986.Eppstein, D. "Re: Aperiodic Space-Filling Tile?." http://www.ics.uci.edu/~eppstein/junkyard/biprism.html.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984.Goldberg, M. "Convex Polyhedral Space-Fillers of More than Twelve Faces." Geom. Dedicata 8, 491-500, 1979.Grünbaum, B. and Shephard, G. C. "Tilings with Congruent Tiles." Bull. Amer. Math. Soc. 3, 951-973, 1980.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, 1999.Holden, A. Shapes, Space, and Symmetry. New York: Dover, pp. 154-163, 1991.Johnson, N. W. Uniform Polytopes. Cambridge, England: Cambridge University Press, 2000.Kramer, P. "Non-Periodic Central Space Filling with Icosahedral Symmetry Using Copies of Seven Elementary Cells." Acta Cryst. A 38, 257-264, 1982.Pearce, P. Structure and Nature as a Strategy for Design. Cambridge, MA: MIT Press, 1978.Schmitt, M. W. "On Space Groups and Dirichlet-Voronoi Stereohedra." Dr. rer. nat. thesis. Berlin: Freien Universität Berlin, 2016. https://refubium.fu-berlin.de/handle/fub188/10176.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-190, 1999.Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.Thompson, D'A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992.Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, pp. 567 and 723, 1964.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 232-236, 1991.Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979.

Cite this as:

Weisstein, Eric W. "Space-Filling Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Space-FillingPolyhedron.html

Subject classifications