Pentagonal icositetrahedron

(Click ccw or cw for rotating models.)
Type Catalan
Conway notation gC
Coxeter diagram
Face polygon
irregular pentagon
Faces 24
Edges 60
Vertices 38 = 6 + 8 + 24
Face configuration V3.3.3.3.4
Dihedral angle 136° 18' 33'
Symmetry group O, ½BC3, [4,3]+, 432
Dual polyhedron snub cube
Properties convex, face-transitive, chiral

Net

In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron[1] is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.[2][3]

It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

## Construction

The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual. Square pyramids are added to the six square faces of the snub cube, and triangular pyramids are added to the eight triangular faces that do not share an edge with a square. The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube. The result is the pentagonal icositetrahedron.

## Cartesian coordinates

Denote the tribonacci constant by ${\displaystyle t\approx 1.839\,286\,755\,21}$. (See snub cube for a geometric explanation of the tribonacci constant.) Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the origin, are as follows:

• the 12 even permutations of (±1, ±(2t+1), ±t2) with an even number of minus signs
• the 12 odd permutations of (±1, ±(2t+1), ±t2) with an odd number of minus signs
• the 6 points (±t3, 0, 0), (0, ±t3, 0) and (0, 0, ±t3)
• the 8 points (±t2, ±t2, ±t2)

The convex hulls for these vertices[4] scaled by ${\displaystyle t^{-3))$ result in a unit circumradius octahedron centered at the origin, a unit cube centered at the origin scaled to ${\displaystyle R\approx 0.9416969935}$, and an irregular chiral snub cube scaled to ${\displaystyle R}$, as visualized in the figure below:

## Geometry

The pentagonal faces have four angles of ${\displaystyle \arccos((1-t)/2)\approx 114.812\,074\,477\,90^{\circ ))$ and one angle of ${\displaystyle \arccos(2-t)\approx 80.751\,702\,088\,39^{\circ ))$. The pentagon has three short edges of unit length each, and two long edges of length ${\displaystyle (t+1)/2\approx 1.419\,643\,377\,607\,08}$. The acute angle is between the two long edges. The dihedral angle equals ${\displaystyle \arccos(-1/(t^{2}-2))\approx 136.309\,232\,892\,32^{\circ ))$.

If its dual snub cube has unit edge length, its surface area and volume are:[5]

{\displaystyle {\begin{aligned}A&=3{\sqrt {\frac {22(5t-1)}{4t-3))}&&\approx 19.299\,94\\V&={\sqrt {\frac {11(t-4)}{2(20t-37)))}&&\approx 7.4474\end{aligned))}

## Orthogonal projections

The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.

Projectivesymmetry Image Dualimage [3] [4]+ [2]

### Variations

Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths.

This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces.

 Snub cube with augmented pyramids and merged faces Pentagonal icositetrahedron Net

## Related polyhedra and tilings

This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry
n32
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Snub
figures
Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞
Gyro
figures
Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞

The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞

The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35

## References

1. ^ Conway, Symmetries of things, p.284
2. ^ "Promorphology of Crystals I".
3. ^ "Crystal Form, Zones, & Habit". Archived from the original on 2003-08-23.
4. ^ Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions". Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.
5. ^ Weisstein, Eric W., "Pentagonal icositetrahedron" ("Catalan solid") at MathWorld.