# Archimedean Tilings

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

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There are three uniform tilings of the plane by regular
polygons: with squares, triangles and hexagons. By uniform we
mean that every vertex is the same. (If we eliminate that
restriction there are an infinite number - we could stack bands
of squares and triangles in any sequence desired.) Thus we can
symbolize these three regular tilings as 333333, 4444, and 666,
where each symbol denotes one of the polygons at each vertex.

In addition there are other tilings with combinations of
polygons. The only possible combinations besides those above are:

- 33336 (left- and right-handed versions)
- 33344
- 33434
- 3463
- 3636
- 3-12-12
- 4-6-12
- 488

These plus the three regular tilings make a total of eleven,
called the *Archimedean Tilings.* Other sets of regular
polygons will fit around a single vertex, for example 5-5-10, but
they can't be extended to cover the plane completely.

## Even the Experts can be Fooled

More than one geometry text has claimed that tiling 43433 has
a right- and left- handed version. If we draw the pattern so that
one set of squares is upright, the other set appears to be tilted
to the right or left. In reality, the mirror planes, which run
along the edges shared by pairs of triangles, are just hard to
see in this orientation. The tiling actually has symmetry p4gm.

Tiling 63333 really is *enantiomorphic*, that is, lacks
mirror planes. It therefore has a left- and right- handed version
as shown below.

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*Created 25 Sep 1997, Last Update 2 July 1999*

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