User:Tomruen/Versatile (geometry)

In geometry, a versatile is a polygon that can express a monohedral tiling in many possible ways, periodically or aperiodically. The polygons may be equilateral or not, convex or not, connected edge-to-edge or not. The regular polygons, the square, equilateral triangle and regular hexagon do not qualify as a versatile because they only self-tile in one way.

The term was coined by Michael Hirschhorn in 1977, and used in 1979 by Branko Grünbaum and Geoffrey Colin Shephard in spiral tilings and expanded in 1981 by Marjorie Rice and Doris Schattschneider. for wider aperiod tiles.

All examples below are selected to each be able to fill a regular polygon, although that is not a requirement.

Quadrilateral examples

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  60° rhombus
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  Large hexagon dissected by 12 rhombi.
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  Tile as union of equilateral triangle and 45° right triangle
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  Dodecagon dissected by 12 quadrilaterals

Pentagon examples

See also: Equilateral pentagon and Pentagonal tiling

A reflexed pentagon, or crown tile	This equilateral pentagon can be seen as the union of a 30° rhombus and and equilateral triangle.	Hirschhorn pentagon. The pentagon can be seen as the union of an 80° rhombus and an equilateral triangle.
Dissection of regular decagon with ten reflexed pentagons.	Twelve can tile a regular dodecagon	18 Hirschhorn pentagons can dissect a regular octadecagon

Hexagon examples

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  A shield tile, a union of a square and triangles on to adjacent edges. It can also be constructed as two reflexed pentagons.
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  Six of them can tile a regular dodecagon.
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  Reflexed hexagon
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  Six of them can tile a large hexagon.
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  Skew concave hexagon as union of two rhombi
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  Six of them can tile a decagon
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  Decagon as union of 2 Penrose tiling rhombi
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  20 hexagons in a decagon

External links

• Spiral Tilings, Paul Gailiunas, 2000