geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the ''demiregular tilings'' are a set of Euclidean

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...

s made from 2 or more

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

faces. Different authors have listed different sets of tilings. A more systematic approach looking at symmetry orbits are the 2-uniform tilings of which there are 20. Some of the demiregular ones are actually 3-uniform tilings.

20 2-uniform tilings

Grünbaum and Shephard enumerated the full list of 20 2-uniform tilings in ''Tilings and Patterns'', 1987:

Ghyka's list (1946)

Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.

Steinhaus's list (1969)

Steinhaus gives 5 examples of non-homogeneous tessellations of regular polygons beyond the 11 regular and semiregular ones.Steinhaus, 1969, p.79-82. (All of them have 2 types of vertices, while one is 3-uniform.)

Critchlow's list (1970)

Critchlow identifies 14 demi-regular tessellations, with 7 being 2-uniform, and 7 being 3-uniform. He codes letter names for the vertex types, with superscripts to distinguish face orders. He recognizes A, B, C, D, F, and J can't be a part of continuous coverings of the whole plane.

References

* Ghyka, M. ''The Geometry of Art and Life'', (1946), 2nd edition, New York: Dover, 1977. * Keith Critchlow, ''Order in Space: A design source book'', 1970, pp. 62–67 * pp. 35–43 * Steinhaus, H. ''Mathematical Snapshots'' 3rd ed, (1969), Oxford University Press, and (1999) New York: Dover * p. 65 *
In Search of Demiregular Tilings
Helmer Aslaksen

External links

* {{MathWorld , urlname=DemiregularTessellation , title=Demiregular tessellation

Brian Galebach Tessellation Semiregular tilings