Pentagonal tiling

Pentagonal tiling
Rank	3
Type	Regular
Space	Hyperbolic
Notation
Bowers style acronym	Peat
Coxeter diagram	x5o4o ()
Schläfli symbol	{5,4}
Elements
Faces	4N Pentagons
Edges	10N
Vertices	5N
Vertex figure	Square, edge length (1+√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{-1-\sqrt{5}}}{2} ≈ 0.89945 i$
Related polytopes
Army	Peat
Regiment	Peat
Dual	Order-5 square tiling
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	[5,4]
Convex	Yes

The order-4 pentagonal tiling, or simply pentagonal tiling or peat, is a regular tiling of the hyperbolic plane. 4 pentagons join at each vertex.

It can also be constructed as the rectified order-5 pentagonal tiling.

Representations

The pentagonal tiling has the following Coxeter diagrams:

• x5o4o (full symmetry)
• o5x5o (as rectified order-5 pentagonal tiling)

Related polytopes

o5o4o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Pentagonal tiling	peat	{5,4}	x5o4o
Truncated pentagonal tiling	topeat	t{5,4}	x5x4o
Tetrapentagonal tiling	tepet	r{5,4}	o5x4o
Truncated order-5 square tiling	topesquat	t{4,5}	o5x4x
Order-5 square tiling	pesquat	{4,5}	o5o4x
Small rhombitetrapentagonal tiling	srotepet	rr{5,4}	x5o4x
Great rhombitetrapentagonal tiling	grotepet	tr{5,4}	x5x4x
Snub tetrapentagonal tiling	stepet	sr{5,4}	s5s4s

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. "Peat".

• Wikipedia Contributors. "Order-4 pentagonal tiling".