Elongated square pyramid

Elongated square pyramid
Type	Johnson J7 – J8 – J9
Faces	4 triangles 1+4 squares
Edges	16
Vertices	9
Vertex configuration	${\displaystyle 4\times (4^{3})}$ ${\displaystyle 1\times (3^{4})}$ ${\displaystyle 4\times (3^{2}\times 4^{2})}$
Symmetry group	${\displaystyle C_{4v}}$
Dihedral angle (degrees)	triangle-to-triangle: 109.47° square-to-square: 90° triangle-to-square: 144.74°
Dual polyhedron	self-dual[1]
Properties	convex, composite
Net

In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.

Construction

The elongated square pyramid is a composite, since it can be constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid.^[2][3] One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.^[4]

A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as ${\displaystyle J_{8}}$, the fifteenth Johnson solid.^[5]

Properties

Given that ${\displaystyle a}$ is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of an elongated square pyramid ${\displaystyle h}$ (i.e., the distance between the square pyramid's apex and the center of a square base) is the sum of the cube's side and the height of an equilateral square pyramid. Its surface area ${\displaystyle A}$ is the sum of four equilateral triangles and four squares' area. Its volume ${\displaystyle V}$ is the sum of an equilateral square pyramid and a cube's volume. With edge length ${\displaystyle a}$, the formulation for each is:^[6][4] $${\displaystyle {\begin{aligned}h&=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a,\\A&=\left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2},\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.\end{aligned}}}$$

3D model of an elongated square pyramid.

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group ${\displaystyle C_{4v}}$ of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:^[7]

• The dihedral angle of an elongated square pyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, ${\displaystyle \arccos(-1/3)\approx 109.47^{\circ }}$,
• The dihedral angle of an elongated square pyramid between two adjacent squares is the dihedral angle of a cube between those, ${\displaystyle \pi /2=90^{\circ }}$,
• The dihedral angle of an equilateral square pyramid between a square and a triangle is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}$. Therefore, the dihedral angle of an elongated square pyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is $${\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}$$

See also

• Elongated square bipyramid

References

1. Draghicescu, Mircea. "Dual Models: One Shape to Make Them All". In Torrence, Eva; Torrence, Bruce; Séquin, Carlo H.; McKenna, Douglas; Fenyvesi, Kristóf; Sarhangi, Reza (eds.). Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture (PDF). pp. 635–640.
2. Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
3. 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.
4. 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.
5. 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.
6. Sapiña, R. "Area and volume of the Johnson solid ${\displaystyle J_{8}}$". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.
7. 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., "Johnson solid" ("Elongated square pyramid") at MathWorld.