Gyroelongated square pyramid

The gyroelongated square pyramid is composite, since it can constructed by attaching one equilateral square pyramid to the square antiprism, a process known as the gyroelongation.{{r|timofeenko-2010|rajwade}} 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.{{r|berman}} 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 $J\_\{10\}$, the tenth Johnson solid.{{r|uehara}}

The surface area of a gyroelongated square pyramid with edge length $a$ is:{{r|berman}}

$$\backslash left(1\; +\; 3\backslash sqrt\{3\}\backslash right)a^2\; \backslash approx\; 6.196a^2,$$

the area of twelve equilateral triangles and a square. Its volume:{{r|berman}}

$$\backslash frac\{\backslash sqrt\{2\}\; +\; 2\backslash sqrt\{4\; +\; 3\backslash sqrt\{2\}\}\}\{6\}a^3\; \backslash approx\; 1.193a^3,$$

can be obtained by slicing the square pyramid and the square antiprism, after which adding their volumes.{{r|berman}}

It has the same three-dimensional symmetry group as the square pyramid, the cyclic group $C\_\{4v\}$ of order eight. Its dihedral angle can be derived by calculating the angle of a square pyramid and square antiprism in the following:{{r|johnson}}

• the dihedral angle of an equilateral square pyramid between two adjacent triangles, approximately $109.47^\backslash circ$
• the dihedral angle of a square antiprism between two adjacent triangles, approximately $127.55^\backslash circ$, and between a triangle to its base is $103.83^\backslash circ$
• the dihedral angle between two adjacent triangles, on the edge where an equilateral square pyramid is attached to a square antiprism, is $158.57^\backslash circ$, for which by adding the dihedral angle of an equilateral square pyramid between its base and its lateral face $54.74^\backslash circ$ and the dihedral angle of a square antiprism between two adjacent triangles.

In stereochemistry, the capped square antiprismatic molecular geometry can be described as the atom cluster of the gyroelongated square pyramid. An example is {{chem|[LaCl(H|2|O)|7|]|2|4+}}, a lanthanum(III) complex with a La–La bond.{{r|greenwood-earnshaw}}

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• Gyroelongated square bipyramid

• {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =GyroelongatedSquarePyramid| title =Gyroelongated square pyramid}}

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Category:Composite polyhedron

Category:Johnson solids

Category:Pyramids (geometry)