Hexapentakis truncated icosahedron

Last updated
Hexapentakis truncated icosahedron
Conway polyhedron K6k5tI.png
Conway notation ktI
Geodesic polyhedron {3,5+}3,0
Faces180
Edges270
Vertices92
Face configuration (60) V5.6.6
(120) V6.6.6
Symmetry group Icosahedral (Ih)
Dual polyhedron Truncated pentakis dodecahedron
Properties convex

The hexapentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron. It is geodesic polyhedron {3,5+}3,0, with pentavalent vertices separated by an edge-direct distance of 3 steps.

Contents

Construction

Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (A tessellation on a sphere) with true geodesic curved edges on the surface of a sphere. and spherical triangle faces.

Conway u3I = (kt)I(k5)k6tI(k)tISpherical ktI
Image Conway polyhedron flat ktI.png K5k6tI.png Conway polyhedron flat2 ktI.png Kised truncated icosahedron spherical.png
Form3-frequency subdivided
icosahedron
1-frequency subdivided
hexakis
truncated icosahedron
1-frequency subdivided
truncated icosahedron
Spherical polyhedron
Polyhedron Truncated Icosahedron #Pentakis truncated Icosahedron #Hexakis truncated Icosahedron Hexapentakis truncated Icosahedron
Image Uniform polyhedron-53-t12.svg Pentakis truncated icosahedron.png Hexakis truncated icosahedron.png Conway polyhedron K6k5tI.png
Conway tIk5tIk6tIk5k6tI

Pentakis truncated icosahedron

Pentakis truncated icosahedron
Pentakis truncated icosahedron.png
Conway notation k5tI
Faces132:
60 triangles
20 hexagons
Edges90
Vertices72
Symmetry group Icosahedral (Ih)
Dual polyhedron Pentatruncated pentakis dodecahedron
Properties convex

The pentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron, adding pyramids to the 12 pentagonal faces, creating 60 new triangular faces.

It is geometrically similar to the icosahedron where the 20 triangular faces are subdivided with a central hexagon, and 3 corner triangles.

Dual

Its dual polyhedron can be called a pentatruncated pentakis dodecahedron, a dodecahedron, with its vertices augmented by pentagonal pyramids, and then truncated the apex of those pyramids, or adding a pentagonal prism to each face of the dodecahedron. It is the net of a dodecahedral prism.

Dual pentakis truncated icosahedron.png

Hexakis truncated icosahedron

Hexakis truncated icosahedron
Hexakis truncated icosahedron.png
Conway notation k6tI
Faces132:
120 triangles
12 pentagons
Edges210
Vertices80
Symmetry group Icosahedral (Ih)
Dual polyhedron Hexatruncated pentakis dodecahedron
Properties convex

The hexakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron, adding pyramids to the 20 hexagonal faces, creating 120 new triangular faces.

It is visually similar to the chiral snub dodecahedron which has 80 triangles and 12 pentagons.

Dual

The dual polyhedron can be seen as a hexatruncated pentakis dodecahedron, a dodecahedron with its faces augmented by pentagonal pyramids (a pentakis dodecahedron), and then its 6-valance vertices truncated.

Dk6tI.png

It has similar groups of irregular pentagons as the pentagonal hexecontahedron.

See also

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Regular icosahedron Platonic solid

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

Kepler–Poinsot polyhedron

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

Rhombicosidodecahedron Archimedean solid

In geometry, the rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

Truncated icosidodecahedron

In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

Triakis icosahedron

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

Pentakis dodecahedron

In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan Solid, meaning that it is a dual of an Archimedean Solid, in this case, the Truncated Icosahedron.

Snub disphenoid

In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra and one of the 92 Johnson solids. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

Small stellated dodecahedron a Kepler-Poinsot polyhedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {52,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.

Chamfered dodecahedron

The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

Conway polyhedron notation

In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

Tetradecahedron Polyhedron with 14 faces

A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.

Pentakis icosidodecahedron

The pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron.

Small hexagonal hexecontahedron

In geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.

Goldberg polyhedron

In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.

Chamfer (geometry)

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

Geodesic polyhedron Polyhedron made from triangles that approximates a sphere

A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra with mostly hexagonal faces.

Order-5 truncated pentagonal hexecontahedron

The order-5 truncated pentagonal hexecontahedron is a convex polyhedron with 72 faces: 60 hexagons and 12 pentagons triangular, with 210 edges, and 140 vertices. Its dual is the pentakis snub dodecahedron.

References