Elongated square pyramid explained

Type:	Johnson
Faces:	4 triangles 1+4 squares
Edges:	16
Vertices:	9
Symmetry:	C _{4v}
Vertex Config:	\begin{align} 4 x ( 4 ^{3)} & + \\ 1 x ( 3 ^{4)} & + \\ 4 x ( 3 ^{2} x 4 ^{2) \end{align}}
Net:	Elongated_Square_Pyramid_Net.svg

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. It is topologically (but not geometrically) self-dual.

Construction

The elongated square bipyramid is constructed by attaching two equilateral square pyramids onto the faces of a cube that are opposite each other, a process known as elongation. This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight equilateral triangles and four squares as their faces.. 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

J _{15} , the fifteenth Johnson solid.

Properties

Given that

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 a cube is the same as the given edge length a , and the height of an equilateral square pyramid is ( 1 /\sqrt{ 2 })a . Therefore, the height of an elongated square bipyramid is:$$a\; +\; \backslash fraca\; =\; \backslash left(1\; +\; \backslash frac\backslash right)a\; \backslash approx\; 1.707a.$$Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:$$\backslash left(5\; +\; \backslash sqrt\backslash right)a^2\; \backslash approx\; 6.732a^2.$$Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:$$\backslash left(1\; +\; \backslash frac\backslash right)a^3\; \backslash approx\; 1.236a^3.$$C _{4v} of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:

• The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces,

\arccos( -1 / 3 ) ≈ 109 . 47 ^{\circ}

• The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those,

\pi/ 2

• The dihedral angle of an equilateral square pyramid between square and triangle is

\arctan \left(\sqrt{ 2 }\right) ≈ 54 . 74 ^{\circ} . Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is $$\backslash arctan\backslash left(\backslash sqrt\backslash right)\; +\; \backslash frac\; \backslash approx\; 144.74^\backslash circ.$$

Dual polyhedron

The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square, and 4 trapezoidal.

Related polyhedra and honeycombs

The elongated square pyramid can form a tessellation of space with tetrahedra,^[1] similar to a modified tetrahedral-octahedral honeycomb.

See also

• Elongated square bipyramid

Notes and References

1. Web site: J8 honeycomb.