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The hexapentakis truncated icosahedron is a convex polyhedron constructed as an
It has similar groups of irregular pentagons as the pentagonal hexecontahedron.
Hexatetrakis truncated octahedron
Try "ktI" ( Conway polyhedron notation) Geodesic polyhedra
augmented
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
*Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...
truncated icosahedron. It is geodesic polyhedron
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 polyhed ...
3,0, with pentavalent vertices separated by an edge-direct distance of 3 steps.
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 truegeodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
curved edges on the surface of a sphere. and spherical triangle faces.
Related polyhedra
Pentakis truncated icosahedron
The pentakis truncated icosahedron is a convex polyhedron constructed as anaugmented
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
*Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...
truncated icosahedron, adding pyramids to the 12 pentagonal faces, creating 60 new triangular faces.
It is geometrically similar to the icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
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 adodecahedral prism
In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces: 30 squares and 24 pentagons. It has 80 edges and 40 vertices.
It can be ...
.
:
Hexakis truncated icosahedron
The hexakis truncated icosahedron is a convex polyhedron constructed as anaugmented
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
*Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...
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 (apentakis 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 i ...
), and then its 6-valance vertices truncated.
:
It has similar groups of irregular pentagons as the pentagonal hexecontahedron.
See also
Hexatetrakis truncated octahedron
References
* Antony Pugh, ''Polyhedra: a visual approach'', 1976, Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra * Reprinted by Dover 1999 {{ISBN, 978-0-486-40921-4External links
Try "ktI" ( Conway polyhedron notation) Geodesic polyhedra