Pentagonal pyramid

Pentagonal pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Peppy
Coxeter diagram	ox5oo&#x
Conway notation	Y5
Stewart notation	Y5
Elements
Faces	5 triangles, 1 pentagon
Edges	5+5
Vertices	1+5
Vertex figures	1 pentagon, edge length 1
	5 isosceles triangles, edge lengths 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$
Volume	${\frac {5+{\sqrt {5}}}{24}}\approx 0.30150$
Dihedral angles	3-3: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	3-5: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
Height	${\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573$
Central density	1
Number of external pieces	6
Level of complexity	4
Related polytopes
Army	Peppy
Regiment	Peppy
Dual	Pentagonal pyramid
Conjugate	Pentagrammic pyramid
Abstract & topological properties
Flag count	40
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Flag orbits	4
Convex	Yes
Net count	15
Nature	Tame

The pentagonal pyramid (OBSA: peppy) is a pyramid with a pentagonal base and 5 triangles as sides. The version with equilateral triangles as sides is the second of the 92 Johnson solids (J₂). In what follows, unless otherwise specified, this what will be meant by a "pentagonal pyramid", even though other variants with isosceles triangles as sides exist.

It is the vertex-first cap of the icosahedron. A regular icosahedron can be constructed by attaching two pentagonal pyramids to the bases of a pentagonal antiprism.

It is one of three regular polygonal pyramids to be CRF. The others are the regular tetrahedron (triangular pyramid) and the square pyramid.

Vertex coordinates

A pentagonal pyramid of edge length 1 has the following vertices:

• $\left(0,\,\pm {\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}},\,0\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,{\frac {1}{2}}\right)$.

These coordinates are obtained as a subset of the vertices of the regular icosahedron.

Alternatively, starting from the coordinates of a regular pentagon in the plane, we obtain the pyramid with the following coordinates:

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0\right)$,
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0\right)$,
• $\left(0,\,0,\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)$.

Related polyhedra

Two pentagonal pyramids can be attached at their bases to form a pentagonal tegum.

A pentagonal prism can be attached to the base of a pentagonal pyramid to form the elongated pentagonal pyramid. If a pentagonal antiprism is attached instead, the result is the gyroelongated pentagonal pyramid.

General variant

For the general pentagonal pyramid with base edges of length b and lacing edges of length l, its height is given by ${\sqrt {l^{2}-b^{2}{\frac {5+{\sqrt {5}}}{10}}}}$, its circumradius by ${\frac {l}{2{\sqrt {1-{\frac {(5+{\sqrt {5}})b^{2}}{10l^{2}}}}}}}$, and its volume is given by ${\frac {\sqrt {25+10{\sqrt {5}}}}{12}}b^{2}{\sqrt {l^{2}-b^{2}{\frac {5+{\sqrt {5}}}{10}}}}$.

Pentagonal pyramids occur as vertex figures of 6 uniform polychora, including the convex truncated hexacosichoron, truncated great hecatonicosachoron, truncated great faceted hexacosichoron, quasitruncated great stellated hecatonicosachoron, icosahedral prism, and small stellated dodecahedral prism.

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#4 under ike).

• Klitzing, Richard. "peppy".
• Quickfur. "The Pentagonal Pyramid".

• Weisstein, Eric W. "Pentagonal Pyramid" ("Johnson solid") at MathWorld.

• Wikipedia contributors. "Pentagonal pyramid".
• McCooey, David. "Pentagonal Pyramid"

• Hi.gher.Space Wiki Contributors. "Pentagonal pyramid".