*Forma,* Vol. 21 (No. 3), pp. 177-196, 2006

*Original Paper*

## How Many Facets on Average can a Tile Have in a Tiling?

Nikolai Dolbilin^{1}* and Masaharu Tanemura^{2}

^{1}Steklov Mathematical Institute, Gubkin 8, Moscow 117966, Russia

^{2}The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

*E-mail address: dolbilin@mi.ras.ru

(Received May 9, 2005; Accepted May 26, 2006)

**Keywords: **
Tiling, Spacefiller, Stereotope, Delone (Delaunay) Set, Voronoi Tiling, Delone (Delaunay) Tiling, Face-to-Face Tiling, Monotypic Tiling

**Abstract. **
A question, how many faces can have a convex polytope which tiles space by its copies, is long standing and intruiging. Another, more general question, how many faces on average can convex tiles have in a face-to-face tiling, is related to the concept of the complexity of a tiling. In the paper it will be said out about basic results in this field. In particular, it will be shown that in Euclidean 3D-space there are periodic tilings whose all tiles are pairwise combinatorially isomorphic and have arbitrarily large number of faces, and also there are periodic Voronoi tilings whose each tile has faces as many as you like.

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