Elongated square pyramid

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Elongated square pyramid
TypeJohnson
J7J8J9
Faces4 triangles
1+4 squares
Edges16
Vertices9
Vertex configuration
Symmetry group
Dual polyhedronself-dual
Propertiesconvex
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. It is topologically (but not geometrically) self-dual.

Construction[edit]

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.[1]. 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 , the fifteenth Johnson solid.[2]

Properties[edit]

Given that 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 , and the height of an equilateral square pyramid is . Therefore, the height of an elongated square bipyramid is:[3]

Its surface area can be calculated by adding all the area of eight equilateral triangles and four squares:[4]
Its volume is obtained by slicing it into two equilateral square pyramids and a cube, and then adding them:[4]

3D model of a elongated square pyramid.

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube. In an equilateral square pyramid, the dihedral angle between square and triangle is , and that between two adjacent triangles is . The dihedral angle between two adjacent squares in a cube is . Therefore, for the elongated square pyramid, the dihedral angle of the triangle and square, on the edge where the equilateral square pyramid attaches the cube, is:[5]

Dual polyhedron[edit]

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

Dual elongated square pyramid Net of dual

Related polyhedra and honeycombs[edit]

The elongated square pyramid can form a tessellation of space with tetrahedra,[6] similar to a modified tetrahedral-octahedral honeycomb.

See also[edit]

References[edit]

  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. ^ 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.
  3. ^ Sapiña, R. "Area and volume of the Johnson solid ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.
  4. ^ a b 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. ^ 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. ^ "J8 honeycomb".

External links[edit]