Elongated triangular pyramid

Elongated triangular pyramid
Type	Johnson J6 – J7 – J8
Faces	4 triangles 3 squares
Edges	12
Vertices	7
Vertex configuration	1(33) 3(3.42) 3(32.42)
Symmetry group	C3v, [3], (*33)
Rotation group	C3, [3]+, (33)
Dual polyhedron	self
Properties	convex
Net

Johnson solid J₇.

In geometry, the elongated triangular pyramid is one of the Johnson solids (J₇). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.

Construction

The elongated triangular pyramid is constructed from a triangular prism by attaching regular tetrahedron onto one of its bases, a process known as elongation. ^[1] The tetrahedron covers an equilateral triangle, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three squares as its faces. ^[2] A convex polyhedron in which all of the faces are regular polygons is called the Johnson solid, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid ${\displaystyle J_{7}}$. ^[3]

Properties

An elongated triangular pyramid with edge length ${\displaystyle a}$ has a height, by adding the height of a regular tetrahedron and a triangular prism: ^[4]

$${\displaystyle \left(1+{\frac {\sqrt {6}}{3}}\right)a\approx 1.816a.}$$

Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares: ^[2]

$${\displaystyle \left(3+{\sqrt {3}}\right)a^{2}\approx 4.732a^{2},}$$

and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up: ^[2] :

$${\displaystyle \left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\approx 0.551a^{3}.}$$

It has the three-dimensional symmetry group, the cyclic group ${\displaystyle C_{3\mathrm {v} }}$ of order 6. Its dihedral angle can be calculated by adding the angle of the tetrahedron and the triangular prism: ^[5]

• the dihedral angle of a tetrahedron between two adjacent triangular faces is ${\textstyle \arccos \left({\frac {1}{3}}\right)\approx 70.5^{\circ }}$;
• the dihedral angle of the triangular prism between the square to its bases is ${\textstyle {\frac {\pi }{2}}=90^{\circ }}$, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is ${\textstyle \arccos \left({\frac {1}{3}}\right)+{\frac {\pi }{2}}\approx 160.5^{\circ }}$;
• the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle ${\textstyle {\frac {\pi }{3}}=60^{\circ }}$.

Dual polyhedron

Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.

Dual elongated triangular pyramid	Net of dual

Related polyhedra and honeycombs

The elongated triangular pyramid can form a tessellation of space with square pyramids and/or octahedra. ^[6]

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. Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN   978-981-15-4470-5. S2CID   220150682.
4. Sapiña, R. "Area and volume of the Johnson solid ${\displaystyle J_{8}}$". Problemas y Ecuaciones (in Spanish). ISSN   2659-9899 . Retrieved 2020-09-09.
5. 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.
6. "J7 honeycomb".

External links

• Weisstein, Eric W., "Johnson solid" ("Elongated triangular pyramid") at MathWorld .