Planigon

A convex polygon, or prototile, in the regular decomposition, or tiling, of the plane by equal polygons, i.e. in a decomposition for which there is a group of motions of the plane mapping the decomposition into itself and acting transitively on the set of tiles. In the Euclidean plane there are 11 combinatorial types of tilings, called Shubnikov–Laves tilings (see Fig.). However, the symmetry group for a single combinatorial type can act in different ways. The relationship between the combinatorial type and the symmetry group is characterized by the so-called adjacency symbol. In the Euclidean plane, there are 46 general regular tilings with different adjacency symbols.

Figure: p072820a

The planigons in the Lobachevskii plane are regular polygons with an arbitrary number of sides $k$ such that an arbitrary fixed number $\alpha$ of them meet at each vertex of the planigon. For numbers of sides $k=3,4,5,6$, and $>6$, one may choose a planigon such that $\alpha\geq7$, $\geq5$, $\geq4$, $\geq4$, and $\geq3$. The multi-dimensional analogue of a planigon is a stereohedron.

References

[1]	B.N. Delone, "Theory of planigons" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 3 (1959) pp. 365–386 (In Russian)
[2]	B.N. Delone, N.P. Dolbilin, M.I. Shtogrin, "Combinatorial and metric theory of planigons" Proc. Steklov Inst. Math. , 148 (1978) pp. 111–142 Trudy Mat. Inst. Steklov. , 148 (1978) pp. 109–140
[3]	, Symmetry designs , Moscow (1980) (In Russian; translated from English)

Comments

For more general types of periodic patterns the relationship between combinatorial type and symmetry group can be more conveniently described by the so-called Delone symbols, also spelled Delaney symbols, [a2]–[a3].

A general survey and a modern classification of tilings including planigons, i.e. prototiles of isohedral tilings, is given in [a1].

References

[a1]	B. Grünbaum, G.C. Shephard, "Tilings and patterns" , Freeman (1986)
[a2]	A. Dress, D. Huson, "On tilings of the plane" Geom. Ded. , 24 (1987) pp. 295–310
[a3]	A. Dress, "Presentations of discrete groups in terms of parametrized Coxeter matrices - A systematic approach" Adv. Math. , 63 (1987) pp. 196–212
[a4]	H. Weyl, "Symmetry" , Princeton Univ. Press (1952) (Translated from German)