Английская Википедия:Elongated square pyramid
Шаблон:Short description Шаблон:Infobox polyhedron
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.Шаблон:R. 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 <math> J_{15} </math>, the fifteenth Johnson solid.Шаблон:R
Properties
Given that <math> a </math> 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 <math> a </math>, and the height of an equilateral square pyramid is <math> (1/\sqrt{2})a </math>. Therefore, the height of an elongated square bipyramid is:Шаблон:R <math display="block"> a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a. </math> Its surface area can be calculated by adding all the area of eight equilateral triangles and four squares:Шаблон:R <math display="block"> \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. </math> Its volume is obtained by slicing it into two equilateral square pyramids and a cube, and then adding them:Шаблон:R <math display="block"> \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. </math>
The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group <math> C_{4v} </math> 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 <math> \arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>, and that between two adjacent triangles is <math> \arccos(-1/3) \approx 109.47^\circ </math>. The dihedral angle between two adjacent squares in a cube is <math> \pi/2 </math>. 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:Шаблон:R <math display="block"> \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. </math>
Dual polyhedron
The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square, and 4 trapezoidal.
| Dual elongated square pyramid | Net of dual |
|---|---|
| Файл:Dual elongated square pyramid.png | Файл:Dual elongated square pyramid net.png |
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
References
External links
Шаблон:Johnson solids navigator
