Pentagon tiling

The 14 known pentagon tilings

In geometry, a pentagon tiling is a tiling of the plane by pentagons. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108, is not a divisor of 360. There are fourteen known types of convex pentagon that tile the plane; it is not known if this list is complete.

Dual uniform tilings

There are 3 isohedral pentagonal tilings generated as duals of the uniform tilings:

Cairo pentagonal tiling

Floret pentagonal tiling

Prismatic pentagonal tiling

Regular pentagonal tilings in non-Euclidean geometry

A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schlafli symbol {5,3}, having 3 pentagons around reach vertex.

In the hyperbolic plane, there are tilings of regular pentagons, for instance order-4 pentagonal tiling, with Schlafli symbol {5,4}, having 4 pentagons around reach vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.

Sphere	Hyperbolic plane
Dodecahedron {5,3}	order-4 pentagonal tiling {5,4}	order-5 pentagonal tiling {5,5}	order-6 pentagonal tiling {5,6}	order-7 pentagonal tiling {5,7}	...{5,∞}

Irregular hyperbolic plane pentagonal tilings

There are an infinite number of dual uniform tilings in hyperbolic plane with isogonal irregular pentagonal faces. They have face configurations as V3.3.p.3.q.

Order p-q floret pentagonal tiling

7-3	8-3	9-3	...	5-4	6-4	7-4	...	5-5
V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.9	...	V3.3.4.3.5	V3.3.4.3.6	V3.3.4.3.7	...	V3.3.5.3.5	...

See also

• Marjorie Rice, amateur mathematician who discovered four new types of tessellating pentagons

References

• Grünbaum, Branko; Shephard, G. C. (1987). "Tilings by polygons". Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 0-7167-1193-1. 
• Gardner, Martin (1988). "Tiling with Convex Polygons". Time travel and other mathematical bewilderments. New York: W. H. Freeman and Company. ISBN 0-7167-1925-8. 

External links

• Weisstein, Eric W., "Pentagon Tiling" from MathWorld.
• Pentagon Tilings
• The 14 Pentagons that Tile the Plane