Space-filling polyhedron

Examples of space-filling polyhedra

cube gyrobifastigium 17-sided plesiohedron, the Voronoi diagram of a Laves graph trapezo-rhombic dodecahedron
Definition	a polyhedron that fill all of three-dimensional space via translations, rotations, or reflections

In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where filling means that; taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can, in fact, be generated by translating a primitive cell polyhedron.

Families and their examples

If a polygon can tile the plane, its prism is space-filling; examples include the cube, ^[1] triangular prism, ^[2] and the hexagonal prism. ^[3] Any parallelepiped tessellates Euclidean 3-space, as do the five parallelohedra (the cube, hexagonal prism, truncated octahedron, elongated dodecahedron, and rhombic dodecahedron). ^[3] Other space-filling polyhedra include the square pyramid, ^[4] plesiohedra, ^[1] and stereohedra, polyhedra whose tilings have symmetries taking every tile to every other tile, including the gyrobifastigium, ^[5] the triakis truncated tetrahedron, ^[6] and the trapezo-rhombic dodecahedron. ^[7]

The cube is the only Platonic solid that can fill space, although a tiling that combines tetrahedra and octahedra (the tetrahedral-octahedral honeycomb) is possible. ^[8] Although the regular tetrahedron cannot fill space, other tetrahedra can, including the Goursat tetrahedra derived from the cube,^{[ citation needed ]} and the Hill tetrahedra. ^[9]

Characteristics

Every polyhedron with Dehn invariant of zero that can space-fill. Relatedly, two different polyhedra can have Dehn invariants of zero, which may be able to tile each other. ^[10]

See also

• Hilbert's eighteenth problem, one of Hilbert's problems, consisting of three subproblems, one of which is a three-dimensional anisohedral tiling.
• Tridecahedron, a thirteen-sided polyhedron

References

1. Erdahl, R. M. (1999). "Zonotopes, Dicings, and Voronoi's Conjecture on Parallelohedra". European Journal of Combinatorics. 20 (6): 527–549. doi: 10.1006/eujc.1999.0294 . MR   1703597.. Voronoi conjectured that all tilings of higher-dimensional spaces by translates of a single convex polytope are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of zonotopes. But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see Grünbaum, Branko; Shephard, G. C. (1980). "Tilings with congruent tiles". Bulletin of the American Mathematical Society . New Series. 3 (3): 951–973. doi: 10.1090/S0273-0979-1980-14827-2 . MR   0585178.
2. Pugh, Anthony (1976). "Close-packing polyhedra". Polyhedra: a visual approach. University of California Press, Berkeley, Calif.-London. pp. 48–50. MR   0451161.
3. Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
4. Barnes, John (2012). Gems of Geometry (2nd ed.). Springer. p. 82. doi:10.1007/978-3-642-30964-9. ISBN   9783642309649.
5. Kepler, Johannes (2010). The Six-Cornered Snowflake. Paul Dry Books. Footnote 18, p. 146. ISBN   9781589882850.
6. Grünbaum, Branko; Shephard, G. C. (1980). "Tilings with congruent tiles". Bulletin of the American Mathematical Society . New Series. 3 (3): 951–973. doi: 10.1090/S0273-0979-1980-14827-2 . MR   0585178.
7. Lagarias, Jeffrey C. (2011). "The Kepler conjecture and its proof". The Kepler Conjecture: The Hales–Ferguson proof. Springer, New York. pp. 3–26. doi:10.1007/978-1-4614-1129-1_1. ISBN   978-1-4614-1128-4. MR   3050907. See especially p. 11
8. Posamentier, Alfred S.; Maresch, Guenter; Thaller, Bernd; Spreitzer, Christian; Geretschlager, Robert; Stuhlpfarrer, David; Dorner, Christian (2022). Geometry In Our Three-dimensional World. World Scientific. pp. 233–234. ISBN   9789811237126.
9. Sloane, N. J. A.; Vaishampayan, Vinay A. (2009). "Generalizations of Schöbi's Tetrahedral Dissection". Discrete & Computational Geometry. 41 (2): 232–248. arXiv: 0710.3857 . doi:10.1007/s00454-008-9086-6.
10. Lagarias, J. C.; Moews, D. (1995). "Polytopes that fill ${\displaystyle \mathbb {R} ^{n}}$ and scissors congruence". Discrete & Computational Geometry . 13 (3–4): 573–583. doi: 10.1007/BF02574064 . MR   1318797. See Equation 4.2 and the surrounding discussion.

• Space-Filling Polyhedron, MathWorld
• Arthur L. Loeb (1991). "Space-filling Polyhedra". Space Structures . Boston, MA: Birkhäuser. pp.  127–132. doi:10.1007/978-1-4612-0437-4_16. ISBN   978-1-4612-0437-4.