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Difference between revisions of "Prototile"

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In the theory of [[tiling]]s a '''set of prototiles''', also called a '''protoset''',
 
In the theory of [[tiling]]s a '''set of prototiles''', also called a '''protoset''',
 
is a (finite or infinite) set of tiles that represent (by congruence) all the tiles of a tiling, or of a class of tilings.
 
is a (finite or infinite) set of tiles that represent (by congruence) all the tiles of a tiling, or of a class of tilings.
  
A protoset $ \P = \{P_k\} $ ''admits'' a tiling if there is a tiling $\T$ such
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A protoset $ \P = \{P_k\} $ ''admits'' a tiling if there is a tiling $ \T = \{T_i\} $ such
 
that all its tiles $T_i$ are congruent to some $P_k$.
 
that all its tiles $T_i$ are congruent to some $P_k$.
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An ''$n$-hedral'' tiling is a tiling in that $n$ (distinct) prototiles occur.
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For $n=1,2,3,...$ the terms ''monohedral'', ''dihedral'', ''trihedral'', etc. are used.
  
 
==== Tiling problem ====
 
==== Tiling problem ====

Latest revision as of 12:16, 12 December 2013

$ \def\T{\mathcal T} % tiling \def\P{\mathcal P} % protoset $

In the theory of tilings a set of prototiles, also called a protoset, is a (finite or infinite) set of tiles that represent (by congruence) all the tiles of a tiling, or of a class of tilings.

A protoset $ \P = \{P_k\} $ admits a tiling if there is a tiling $ \T = \{T_i\} $ such that all its tiles $T_i$ are congruent to some $P_k$.

An $n$-hedral tiling is a tiling in that $n$ (distinct) prototiles occur. For $n=1,2,3,...$ the terms monohedral, dihedral, trihedral, etc. are used.

Tiling problem

The tiling problem, i.e., the problem to determine whether a protoset admits a tiling is (algorithmically) undecidable, both in general and for many special cases. Normally, a protoset will (if at all) admit either a single tiling or uncountably many tilings, but intermediate numbers are also possible. If a protoset admits (up to congruence) only a single tiling then it is called monomorphic, if it admits precisely two distinct tilings it called dimorphic, trimorphic for three tilings, etc., $n$-morphic for $n$ tilings, and $\sigma$-morphic if there are countably many distinct tilings.

How to Cite This Entry:
Prototile. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prototile&oldid=20981