Augmented hexagonal prism

Augmented hexagonal prism
Type	Johnson J53 – J54 – J55
Faces	4 triangles 5 squares 2 hexagons
Edges	22
Vertices	13
Vertex configuration	2×4(42.6) 1(34) 4(32.4.6)
Symmetry group	C2v
Properties	convex
Net

In geometry, the augmented hexagonal prism is one of the Johnson solids (J₅₄). As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid (J₁) to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism (J₅₅), a metabiaugmented hexagonal prism (J₅₆), or a triaugmented hexagonal prism (J₅₇).

3D model of an augmented hexagonal prism

Construction

The augmented hexagonal prism is constructed by attaching one equilateral square pyramid onto the square face of a hexagonal prism, a process known as augmentation. ^[1] This construction involves the removal of the prism square face and replacing it with the square pyramid, so that there are eleven faces: four equilateral triangles, five squares, and two regular hexagons. ^[2] A convex polyhedron in which all of the faces are regular is a Johnson solid, and the augmented hexagonal prism is among them, enumerated as ${\displaystyle J_{54}}$. ^[3] Relatedly, two or three equilateral square pyramids attaching onto more square faces of the prism give more different Johnson solids; these are the parabiaugmented hexagonal prism ${\displaystyle J_{55}}$, the metabiaugmented hexagonal prism ${\displaystyle J_{56}}$, and the triaugmented hexagonal prism ${\displaystyle J_{57}}$. ^[1]

Properties

An augmented hexagonal prism with edge length ${\displaystyle a}$ has surface area ^[2] $${\displaystyle \left(5+4{\sqrt {3}}\right)a^{2}\approx 11.928a^{2},}$$ the sum of two hexagons, four equilateral triangles, and five squares area. Its volume ^[2] $${\displaystyle {\frac {{\sqrt {2}}+9{\sqrt {3}}}{2}}a^{3}\approx 2.834a^{3},}$$ can be obtained by slicing into one equilateral square pyramid and one hexagonal prism, and adding their volume up. ^[2]

It has an axis of symmetry passing through the apex of a square pyramid and the centroid of a prism square face, rotated in a half and full-turn angle. Its dihedral angle can be obtained by calculating the angle of a square pyramid and a hexagonal prism in the following: ^[4]

• The dihedral angle of an augmented hexagonal prism between two adjacent triangles is the dihedral angle of an equilateral square pyramid, ${\displaystyle \arccos \left(-1/3\right)\approx 109.5^{\circ }}$
• The dihedral angle of an augmented hexagonal prism between two adjacent squares is the interior of a regular hexagon, ${\displaystyle 2\pi /3=120^{\circ }}$
• The dihedral angle of an augmented hexagonal prism between square-to-hexagon is the dihedral angle of a hexagonal prism between its base and its lateral face, ${\displaystyle \pi /2}$
• The dihedral angle of a square pyramid between triangle (its lateral face) and square (its base) is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.75^{\circ }}$. Therefore, the dihedral angle of an augmented hexagonal prism between square-to-triangle and between triangle-to-hexagon, on the edge in which the square pyramid and hexagonal prism are attached, are $${\displaystyle {\begin{aligned}\arctan \left({\sqrt {2}}\right)+{\frac {2\pi }{3}}\approx 174.75^{\circ },\\\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.75^{\circ }.\end{aligned}}}$$.

References

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

External links

• Weisstein, Eric W., "Augmented hexagonal prism" ("Johnson Solid") at MathWorld .