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Hexapentakis truncated icosahedron

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Hexapentakis truncated icosahedron
Conway notation ktI
Geodesic polyhedron {3,5+}3,0
Faces 180
Edges 270
Vertices 92
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.

Construction

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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)tI Spherical ktI
Image
Form 3-frequency subdivided
icosahedron
1-frequency subdivided
hexakis
truncated icosahedron
1-frequency subdivided
truncated icosahedron
Spherical polyhedron
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Polyhedron Truncated Icosahedron #Pentakis truncated Icosahedron #Hexakis truncated Icosahedron Hexapentakis truncated Icosahedron
Image
Conway tI k5tI k6tI k5k6tI

Pentakis truncated icosahedron

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Pentakis truncated icosahedron
Conway notation k5tI
Faces 132:
60 triangles
20 hexagons
Edges 90
Vertices 72
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

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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.

Hexakis truncated icosahedron

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Hexakis truncated icosahedron
Conway notation k6tI
Faces 132:
120 triangles
12 pentagons
Edges 210
Vertices 80
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

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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.

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

See also

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References

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  • Antony Pugh, Polyhedra: a visual approach, 1976, Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra
  • Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4
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