Augmented truncated tetrahedron

Augmented truncated tetrahedron
Type	Johnson J64 – J65 – J66
Faces	8 triangles 3 squares 3 hexagons
Edges	27
Vertices	15
Vertex configuration	2x3(3.62) 3(3.4.3.4) 6(3.4.3.6)
Symmetry group	C3v
Properties	convex
Net

3D model of an augmented truncated tetrahedron

In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto a truncated tetrahedron. It is an example of a Johnson solid.

Construction

The augmented truncated tetrahedron is constructed from a truncated tetrahedron by attaching a triangular cupola. ^[1] This cupola covers one of the truncated tetrahedron's four hexagonal faces, so that the resulting polyhedron's faces are eight equilateral triangles, three squares, and three regular hexagons. ^[2] Since it has the property of convexity and has regular polygonal faces, the augmented truncated tetrahedron is a Johnson solid, denoted as the sixty-fifth Johnson solid ${\displaystyle J_{65}}$. ^[3]

Properties

The surface area of an augmented truncated tetrahedron is: ^[2] $${\displaystyle {\frac {6+13{\sqrt {3}}}{2}}a^{2}\approx 14.258a^{2},}$$ the sum of the areas of its faces. Its volume can be calculated by slicing it off into both truncated tetrahedron and triangular cupola, and adding their volume: ^[2] $${\displaystyle {\frac {11{\sqrt {2}}}{4}}a^{3}\approx 3.889a^{3}.}$$

It has the same three-dimensional symmetry group as the triangular cupola, the pyramidal symmetry ${\displaystyle C_{3\mathrm {v} }}$. Its dihedral angles can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following: ^[4]

• its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°;
• its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
• its dihedral angle between triangle and square is as in the triangular cupola's angle: 125.3°
• its dihedral angle between triangle and square, on the edge where the triangular cupola and truncated tetrahedron are attached, is the sum of both triangular cupola's square-hexagon angle and the truncated tetrahedron's triangle-hexagon angle: approximately 164.17°; and
• its dihedral angle between triangle and hexagon, on the edge where triangular cupola and truncated tetrahedron are attached, is the sum of the dihedral angle of a triangular cupola and truncated tetrahedron between that: approximately 141.3°;

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 truncated tetrahedron" ("Johnson solid") at MathWorld .