Biaugmented_triangular_prism Knowpia

Biaugmented triangular prism
Biaugmented triangular prism
Type	Johnson; J49 – J50 – J51
Faces	10 triangles; 1 square
Edges	17
Vertices	8
Vertex configuration
Symmetry group
Properties	convex
Net

In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid.

Construction edit

The biaugmented triangular prism can be constructed from a triangular prism by attaching two equilateral square pyramids onto its two square faces, a process known as augmentation.^[1] These square pyramid covers the square face of the prism, so the resulting polyhedron has 10 equilateral triangles and 1 square as its faces.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the biaugmented triangular prism is among them, enumerated as 50th Johnson solid $J_{50}$ .^[3]

Properties edit

A biaugmented triangular prism with edge length $a$ has a surface area, calculated by adding ten equilateral triangles and one square's area:^[2]

{\frac {2+5{\sqrt {3}}}{2}}a^{2}\approx 5.3301a^{2}.

Its volume can be obtained by slicing it into a regular triangular prism and two equilateral square pyramids, and adding their volumes subsequently:^[2]

{\frac {59+24{\sqrt {6}}}{144}}a^{3}\approx 0.904a^{3}.

It has three-dimensional symmetry group of the cyclic group $C_{2\mathrm {v} }$ of order 4. Its dihedral angle can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism. The dihedral angle of an equilateral square pyramid between two adjacent triangular faces is ${\textstyle \arccos \left(-1/3\right)\approx 109.5^{\circ }}$ , and that between a triangular face and its base is ${\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.7^{\circ }}$ . The dihedral angle of a triangular prism between two adjacent square faces is the internal angle of an equilateral triangle ${\textstyle \pi /3=60^{\circ }}$ , and that between square-to-triangle is ${\textstyle \pi /2=90^{\circ }}$ . Therefore, the dihedral angle of the augmented triangular prism between square-to-triangle, between triangle-to-triangle on the edge where an equilateral square pyramid and a triangular prism is attached, and between triangle-to-triangle on the edge where two square pyramids and a triangular prism are attached, is:^[4]

{\begin{aligned}\arccos \left(-{\frac {1}{3}}\right)+{\frac {\pi }{3}}&\approx 104.5^{\circ },\\\arccos \left(-{\frac {1}{3}}\right)+{\frac {\pi }{2}}&\approx 144.5^{\circ },\\2\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{3}}&\approx 169.4^{\circ }.\end{aligned}}

References edit

^ Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
^ ^a ^b ^c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
^ Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.

External links edit

Weisstein, Eric W. "Johnson Solid". MathWorld.
- Weisstein, Eric W. "Biaugmented triangular prism". MathWorld.

[rajwade-1] Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.

[berman-2] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[francis-3] Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.

[johnson-4] Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.

[1]

[2]

[3]

[4]

Summary

Construction edit

Properties edit

See also edit

References edit

External links edit

Biaugmented triangular prism

Type	Johnson $J 49 - J 50 - J 51$
Faces	10 triangles 1 square
Edges	17
Vertices	8
Vertex configuration	$2\times (3^{5})+2\times (3^{4})+4\times (3^{3}\times 4)$
Symmetry group	$C_{2\mathrm {v} }$
Properties	convex
Net