Biaugmented pentagonal prism

Biaugmented pentagonal prism
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Baupip
Elements
Faces	2+2+4 triangles, 1+2 squares, 2 pentagons
Edges	1+2+2+2+4+4+4+4
Vertices	2+2+4+4
Vertex figures	2 square, edge length 1
	4+4 irregular tetragons, edge lengths 1, 1, √2, (1+√5)/2
	2 isosceles triangles, edge lengths (1+√5)/2, √2, √2
Measures (edge length 1)
Volume	${\frac {4{\sqrt {2}}+3{\sqrt {25+10{\sqrt {5}}}}}{12}}\approx 2.19188$
Dihedral angles	3–4 join: $\arccos \left(-{\sqrt {\frac {13+{\sqrt {5}}+4{\sqrt {5-{\sqrt {5}}}}}{24}}}\right)\approx 162.73561^{\circ }$
	3–5 join: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)144.73561^{\circ }$
	3–3 pyramidal: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	4–4: 108°
	4–5: 90°
Central density	1
Number of external pieces	13
Level of complexity	23
Related polytopes
Army	Baupip
Regiment	Baupip
Dual	Paralaterobitruncated pentagonal tegum
Conjugate	Biaugmented pentagrammic prism
Abstract & topological properties
Flag count	92
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	K2×I, order 4
Convex	Yes
Nature	Tame

The biaugmented pentagonal prism is one of the 92 Johnson solids (J₅₃). It consists of 2+2+4 triangles, 1+2 squares, and 2 pentagons. It can be constructed by attaching square pyramids to two non-adjacent square faces of the pentagonal prism.

Vertex coordinates

A biaugmented pentagonal prism of edge length 1 has the following vertices:

• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}+2{\sqrt {5+{\sqrt {5}}}}}{8}},\,-{\sqrt {\frac {35-9{\sqrt {5}}+4{\sqrt {25-5{\sqrt {5}}}}}{160}}},\,0\right).$

External links

• Klitzing, Richard. "baupip".
• Quickfur. "The Biaugmented Pentagonal Prism".

• Wikipedia contributors. "Biaugmented pentagonal prism".
• McCooey, David. "Biaugmented Pentagonal Prism"