Order-5 pentagonal tiling

Order-5 pentagonal tiling
Rank	3
Type	Regular
Space	Hyperbolic
Notation
Bowers style acronym	Pepat
Coxeter diagram	x5o5o ()
Schläfli symbol	{5,5}
Elements
Faces	2N pentagons
Edges	5N
Vertices	2N
Vertex figure	Pentagon, edge length (1+√5)/2
Measures (edge length 1)
Circumradius	$\sqrt{-\frac{\sqrt5}{8}} ≈ 0.52869 i$
Related polytopes
Army	Pepat
Regiment	Pepat
Dual	Order-5 pentagonal tiling
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	[5,5]
Convex	Yes

The order-5 pentagonal tiling or pepat is a regular tiling of the hyperbolic plane. 5 pentagons join at each vertex. It is self-dual.

It is the simplest self-dual regular hyperbolic tiling.

It can be formed by alternating the order-5 square tiling.

Representations

The order-5 pentagonal tiling has the following Coxeter diagrams:

• x5o5o (main symmetry)
• s4o5o (as alternated order-5 square tiling)

Related polytopes

Two uniform polyhedra, the dodecadodecahedron and the ditrigonary dodecadodecahedron, along with their duals, the medial rhombic triacontahedron and medial triambic icosahedron, are quotients of this tiling.

o5o5o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Order-5 pentagonal tiling	pepat	{5,5}	x5o5o
Truncated order-5 pentagonal tiling	topepat	t{5,5}	x5x5o
Pentapentagonal tiling = Pentagonal tiling	peat	r{5,5}	o5x5o
Truncated order-5 pentagonal tiling	topepat	t{5,5}	o5x5x
Order-5 pentagonal tiling	pepat	{5,5}	o5o5x
Small rhombipentapentagonal tiling = Tetrapentagonal tiling	tepet	rr{5,5}	x5o5x
Great rhombipentapentagonal tiling = Truncated pentagonal tiling	topeat	tr{5,5}	x5x5x
Snub pentapentagonal tiling	spepat	sr{5,5}	s5s5s

External links

• Klitzing, Richard. "Pepat".

• Wikipedia Contributors. "Order-5 pentagonal tiling".