Demiregular Tessellation

A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical. Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).

The number of demiregular tessellations is commonly given as 14 (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by "demiregular tessellation."

A more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65). There are 20 such tessellations, illustrated above, as first enumerated by Krötenheerdt (1969; Grünbaum and Shephard 1986, pp. 65-67).

REFERENCES

Critchlow, K. Order in Space: A Design Source Book. New York: Viking Press, 1970.

Ghyka, M. The Geometry of Art and Life. New York: Dover, 1977.

Grünbaum, B. and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986.

Krötenheerdt, O. "Die homogenen Mosaike -ter Ordnung in der euklidischen Ebene. I." Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Natur. Reihe 18, 273-290, 1969.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 75-76, 1999.

Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 35-43, 1979.