A space-filling polyhedron is a Polyhedron which can be used to generate a Tessellation of space. There exists one 16-sided space-filling Polyhedron, but it is unknown if this is the unique 16-sided space-filler. The Cube, Rhombic Dodecahedron, and Truncated Octahedron are space-fillers, as are the Elongated Dodecahedron and hexagonal Prism. These five solids are all ``primary'' Parallelohedra (Coxeter 1973).

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).

Keller's Conjecture, Parallelohedron, Prism, Rhombic Dodecahedron, Schmitt-Conway Biprism, Tessellation, Tiling, Truncated Octahedron

**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.

Eppstein, D. ``Re: Aperiodic Space-Filling Tile?.'' http://www.ics.uci.edu/~eppstein/junkyard/biprism.html.

Holden, A. *Shapes, Space, and Symmetry.* New York: Dover, pp. 154-163, 1991.

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.

© 1996-9

1999-05-26