For many types of tilings, if the tiles are placed one by one, the position of a given tile does not determine the exact position of subsequent tiles.
For
other tilings, once one tile is placed, there is no choice for the placement of
any other tile.
For Kite and Dart tilings, sometimes a given tile
will force the position of other tiles. The simplest example of this is
that any dart is part of a configuration with 2 kites known as an
"ace".
The ace does not force any other tiles.
The
"star", pictured at left, forces the position of the tiles
immediately surrounding it, but it also forces the position of many other
tiles. These other tiles are sometimes called "empires".
All of the vertex configurations except the ace and the sun have an empire.
Below is pictured the "queen", and
part of her empire.
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