Pentagon tiling

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.

Contents

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

References

External links