Pentagonal icositetrahedron

# Pentagonal icositetrahedron

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Pentagonal icositetrahedron

Click ccw or cw for spinning versions.
Type Catalan
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

Face polygon irregular pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

Faces 24
Edges 60
Vertices 38 = 6 + 8 + 24
Face configuration
Face configuration
In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....

V3.3.3.3.4
Dihedral angle
Dihedral angle
In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...

136° 20'
Symmetry group O, [4,3]+, 432
Dual polyhedron
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

snub cube
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

Properties convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

, face-transitive, chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

Net
Net (polyhedron)
In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron...

In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a pentagonal icositetrahedron is a Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

which is the dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

of the snub cube
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

. It has two distinct forms, which are mirror image
Mirror image
A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...

s (or "enantiomorphs
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

") of each other.

If it has unit edge length, its surface area is and its volume is . Here is the tribonacci constant (see snub cube
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

).

## Related polyhedra and tilings

This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configuration
Face configuration
In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....

s (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.
 V3.3.3.3.3(332) and (532) V3.3.3.3.4Pentagonal icositetrahedronIn geometry, a pentagonal icositetrahedron is a Catalan solid which is the dual of the snub cube. It has two distinct forms, which are mirror images of each other....(432) V3.3.3.3.5Pentagonal hexecontahedronIn geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images of each other. It is also well-known to be the Catalan Solid with the most vertices...(532) V3.3.3.3.6(632) V3.3.3.3.7(732)