Biaugmented triangular prism

Biaugmented triangular prism
Type	Johnson J49 – J50 – J51
Faces	10 triangles 1 square
Edges	17
Vertices	8
Vertex configuration	${\displaystyle 2\times 3^{5}+2\times 3^{4}+4\times 3^{3}\times 4}$
Symmetry group	${\displaystyle C_{2\mathrm {v} }}$
Dihedral angle (degrees)	triangle-triangle: 109.5°, 144.5°, 169.4° triangle-square: 90°, 114.7°
Properties	convex, composite
Net

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. It can be found in stereochemistry in bicapped trigonal prismatic molecular geometry.

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 pyramids cover the square face of the prism, so the resulting polyhedron has 10 equilateral triangles and 1 square as its faces, 17 edges, and 8 vertices.² A convex polyhedron in which all faces are regular polygons is Johnson solid, after American mathematician Norman W. Johnson, who listed the 92 such polyhedra. The biaugmented triangular prism is among them, enumerated as the 50th Johnson solid ${\displaystyle J_{50}}$.³ By such a construction, it is composite.⁴

A biaugmented triangular prism has a surface area ${\displaystyle A}$ calculated by adding ten equilateral triangles and one square's area: ${\displaystyle A=10T+S}$. The area of an equilateral triangle and of a square are ${\textstyle T={\frac {\sqrt {3}}{4}}a^{2}}$ and ${\textstyle S=a^{2}}$, respectively, where ${\displaystyle a}$ is the edge length. All edges of the biaugmented triangular prism are equal in length. The biaugmented triangular prism's surface area is then:² $${\displaystyle A=10\left({\frac {\sqrt {3}}{4}}a^{2}\right)+a^{2}={\frac {5{\sqrt {3}}+2}{2}}a^{2}\approx 5.3301a^{2}.}$$

The volume of a biaugmented triangular prism ${\displaystyle V}$ is obtained by slicing it into a regular triangular prism and two equilateral square pyramids, and adding their volumes subsequently. The volumes of a triangular prism and pyramid are ${\textstyle {\frac {\sqrt {3}}{4}}a^{3}}$ and ${\textstyle {\frac {\sqrt {2}}{6}}a^{3}}$, respectively. Therefore, the volume of a biaugmanted triangular prism is:² $${\displaystyle V={\frac {\sqrt {3}}{4}}a^{3}+2\left({\frac {\sqrt {2}}{6}}a^{3}\right)={\frac {3{\sqrt {3}}+4{\sqrt {2}}}{12}}a^{3}\approx 0.904a^{3}.}$$

The biaugmented triangular prism has three-dimensional symmetry group of two-fold pyramidal symmetry ${\displaystyle C_{2\mathrm {v} }}$ of order 4. Its dihedral angle (i.e., the angle between two polygonal faces) can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:⁵

• The dihedral angle of a biaugmented triangular prism between two adjacent triangles is that of an equilateral square pyramid between two adjacent triangular faces, ${\textstyle \arccos \left(-1/3\right)\approx 109.5^{\circ }}$
• The dihedral angle of a biaugmented triangular prism between square and triangle is the dihedral angle of a triangular prism between the base and its lateral face, ${\textstyle \pi /2=90^{\circ }}$
• The dihedral angle of an equilateral square pyramid 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 ${\textstyle \pi /3=60^{\circ }}$. Therefore, the dihedral angle of a biaugmented triangular prism between a square (the lateral face of the triangular prism) and triangle (the lateral face of the equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the square face of the triangular prism, and between two adjacent triangles (the lateral face of both equilateral square pyramids) on the edge where two equilateral square pyramids are attached adjacently to the triangular prism, are $${\displaystyle {\begin{aligned}\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{3}}&\approx 114.7^{\circ },\\2\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{3}}&\approx 169.4^{\circ }.\end{aligned}}}$$
• The dihedral angle of a biaugmented triangular prism between two adjacent triangles (the base of a triangular prism and the lateral face of an equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the triangular prism, is: $${\displaystyle \arccos \left(-{\frac {1}{3}}\right)+{\frac {\pi }{2}}\approx 144.5^{\circ }.}$$

The biaugmented triangular prism can be found in stereochemistry, as a structural shape of a chemical compound known as bicapped trigonal prismatic molecular geometry. It is one of the three common shapes for transition metal complexes with eight vertices other than the chemical structure other than square antiprism and the snub disphenoid. An example of such structure is plutonium(III) bromide PuBr₃ adopted by bromides and iodides of the lanthanides and actinides.⁶