In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a pentagon tiling is a tiling

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

 of the plane by pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

s. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle

Internal angle

In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon...

 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 tiling

Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane...

s:

Cairo pentagonal tiling Cairo pentagonal tiling In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name because it appears on the streets of Cairo and in many Islamic decorations. It is one of 14 known isohedral pentagon tilings....

Floret pentagonal tiling Floret pentagonal tiling In geometry, the floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 14 known isohedral pentagon tilings. It is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower...

Prismatic pentagonal tiling Prismatic pentagonal tiling In geometry, the prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 14 known isohedral pentagon tilings.Conway calls it a isopentille.It is the dual of the elongated triangular 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

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, with Schlafli symbol {5,3}, having 3 pentagons around reach vertex.

In the hyperbolic plane

Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional plane in hyperbolic geometry* A two-dimensional plane in Minkowski space...

, 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 Sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...	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 hypebolic plane pentagonal tilings

There are an infinite number of dual uniform tilings in hyperbolic plane

Uniform tilings in hyperbolic plane

There are an infinite number of uniform tilings on the hyperbolic plane based on the where 1/p + 1/q + 1/r ...

 with isogonal irregular pentagonal faces. They have face configuration

Face configuration

In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....

s 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
  Marjorie Rice
  Marjorie Rice is an American homemaker most famous for her discoveries in geometry. She lives in San Diego....
  
  , amateur mathematician who discovered four new types of tessellating pentagons

External links

• Pentagon Tilings
• The 14 Pentagons that Tile the Plane