Gyroelongated square pyramid

Gyroelongated square pyramid
Type	Johnson J9 – J10 – J11
Faces	12 triangles 1 square
Edges	20
Vertices	9
Vertex configuration	${\displaystyle 1\times 3^{4}+4\times 3^{3}\times 4+4\times 3^{5}}$
Symmetry group	${\displaystyle C_{4v}}$
Properties	convex, composite
Net

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 chemistry; for example, the capped square antiprismatic molecular geometry.

Construction

The gyroelongated square pyramid is composite, since it can be constructed by attaching one equilateral square pyramid to the square antiprism, a process known as gyroelongation. ^{[1] [2]} 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. ^[3] Any convex polyhedron in which all of the faces are regular is a Johnson solid, and the gyroelongated square pyramid is one of them, enumerated as ${\displaystyle J_{10}}$, the tenth Johnson solid. ^[4]

Properties

The surface area of a gyroelongated square pyramid with edge length ${\displaystyle a}$ is: ^[3] $${\displaystyle \left(1+3{\sqrt {3}}\right)a^{2}\approx 6.196a^{2},}$$ the area of twelve equilateral triangles and a square. Its volume: ^[3] $${\displaystyle {\frac {{\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}}{6}}a^{3}\approx 1.193a^{3},}$$ can be obtained by slicing the square pyramid and the square antiprism, after which adding their volumes. ^[3]

3D model of a gyroelongated square pyramid

It has the same three-dimensional symmetry group as the square pyramid, the cyclic group ${\displaystyle C_{4v}}$ of order eight. Its dihedral angle can be derived by calculating the angle of a square pyramid and square antiprism in the following: ^[5]

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

Applications

In stereochemistry, the capped square antiprismatic molecular geometry can be described as the atom cluster of the gyroelongated square pyramid. An example is [LaCl(H
_²O)
_⁷]⁴⁺
₂, a lanthanum(III) complex with a La–La bond. ^[6]

References

1. Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
2. Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. doi:10.1007/978-93-86279-06-4. ISBN   978-93-86279-06-4.
3. 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.
4. 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.
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. Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 917. ISBN   978-0-08-037941-8.

See also

• Gyroelongated square bipyramid

External links

• Weisstein, Eric W., "Gyroelongated square pyramid" ("Johnson solid") at MathWorld .