Augmented triangular prism

49th Johnson solid

Augmented triangular prism

Type	Johnson J₄₈ – J₄₉ – J₅₀
Faces	6 triangles 2 squares
Edges	13
Vertices	7
Vertex configuration	${\begin{aligned}&2\times (3\times 4^{2})\,+\\&1\times (3^{4})\,+\\&4\times (3^{3}\times 4)\end{aligned}}$
Symmetry group	$C_{2\mathrm {v} }$
Properties	convex
Net

In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.

Construction

The augmented triangular prism can be constructed from a triangular prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation.^[1] This square pyramid covers the square face of the prism, so the resulting polyhedron has 6 equilateral triangles and 2 squares as its faces.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the augmented triangular prism is among them, enumerated as 49th Johnson solid $J_{49}$ .^[3]

Properties

An augmented triangular prism with edge length $a$ has a surface area, calculated by adding six equilateral triangles and two squares' area:^[2]

{\frac {4+3{\sqrt {3}}}{2}}a^{2}\approx 4.598a^{2}.

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

{\frac {2{\sqrt {2}}+3{\sqrt {3}}}{12}}a^{3}\approx 0.669a^{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 and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectively:^[4]

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

Application

In the geometry of chemical compounds, a polyhedron may commonly visualize an atom cluster surrounding a central atom. The capped trigonal prismatic molecular geometry describes clusters for which this polyhedron is an augmented triangular prism.^[5] An example of such compound is the potassium heptafluorotantalate.^[6]

References

^ 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.
^ Hoffmann, Roald; Beier, Barbara F.; Muetterties, Earl L.; Rossi, Angelo R. (1977). "Seven-coordination. A molecular orbital exploration of structure, stereochemistry, and reaction dynamics". Inorganic Chemistry. 16 (3): 511–522. doi:10.1021/ic50169a002.
^ Kaupp, Martin (2001). ""Non-VSEPR" Structures and Bonding in d(0) Systems". Angew Chem Int Ed Engl. 40 (1): 3534–3565. doi:10.1002/1521-3773(20011001)40:19<3534::AID-ANIE3534>3.0.CO;2-#. PMID 11592184.