Английская Википедия:Gyroelongated square pyramid

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In geometry, the gyroelongated square pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in the chemistry such as square antiprismatic molecular geometry.

Construction

The gyroelongated square pyramid is constructed by attaching one equilateral square pyramid to the square antiprism, a process known as the gyroelongation.Шаблон:R This construction involves the covering of one of two square faces and replacing them with the four equilateral triangles, so that the resulting polyhedron has twelve equilateral triangles and one square.Шаблон:R The convex polyhedron in which all of the faces are regular is the Johnson solid, and the gyroelongated square pyramid is one of them, enumerated as <math> J_{10} </math>, the tenth Johnson solid.Шаблон:R

Properties

The surface area of a gyroelongated square pyramid with edge length <math> a </math> is:Шаблон:R <math display="block"> \left(1 + 3\sqrt{3}\right)a^2 \approx 6.196a^2, </math> the area of twelve equilateral triangles and a square. Its volume:Шаблон:R <math display="block"> \frac{\sqrt{2} + 2\sqrt{4 + 3\sqrt{2}}}{6}a^3 \approx 1.193a^3, </math> can be obtained by slicing the square pyramid and the square antiprism, after which adding their volumes.Шаблон:R

It has the same three-dimensional symmetry group as the square pyramid, the cyclic group <math> C_{4v} </math> of order eight. Its dihedral angle can be derived by calculating the angle of a square pyramid and square antiprism. The dihedral angle of a square pyramid between two adjacent triangles is approximately <math> 109.47^\circ </math>, and that angle between triangle and square is <math> 54.74^\circ </math>. The dihedral angle of a square antiprism between two adjacent triangles is <math> 127.55^\circ </math>, and that between square and triangle is <math> 103.83^\circ </math>. Therefore, for the attachment of both square pyramid and square antiprism, the dihedral angle between two adjacent triangles is <math> 54.74 ^\circ + 103.83^\circ = 158.57^\circ</math>.Шаблон:R

Applications

In stereochemistry, the capped square antiprismatic molecular geometry can be described as the atom cluster of the gyroelongated square pyramid. An example is Шаблон:Chem, a lanthanum(III) complex with a La–La bond.Шаблон:R

See also

• Gyroelongated square bipyramid

External links

• Шаблон:Mathworld2