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Order-5 truncated pentagonal hexecontahedron

{{Short description|Convex polyhedron with 72 faces}}

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!bgcolor=#e7dcc3 colspan=2|Order-5 truncated pentagonal hexecontahedron

align=center colspan=2|250px
bgcolor=#e7dcc3|Conwayt5gD or [https://levskaya.github.io/polyhedronisme/?recipe=A10wD wD]
bgcolor=#e7dcc3|Goldberg{5+,3}2,1
bgcolor=#e7dcc3|FullereneC140
bgcolor=#e7dcc3|Faces72:
60 hexagons
12 pentagons
bgcolor=#e7dcc3|Edges210
bgcolor=#e7dcc3|Vertices140
bgcolor=#e7dcc3|Symmetry groupIcosahedral (I)
bgcolor=#e7dcc3|Dual polyhedronPentakis snub dodecahedron
bgcolor=#e7dcc3|Propertiesconvex, chiral
bgcolor=#e7dcc3|Net160px

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.

It is Goldberg polyhedron {5+,3}2,1 in the icosahedral family, with chiral symmetry. The relationship between pentagons steps into 2 hexagons away, and then a turn with one more step.

It is a Fullerene C140.{{cite journal|doi=10.1002/anie.201505516 | volume=54 | title=Giant Spherical Cluster with I-C140 Fullerene Topology | year=2015 | journal=Angewandte Chemie International Edition | pages=13431–13435 | last1 = Heinl | first1 = Sebastian| issue=45 | pmc=4691335 | pmid=26411255}}

Construction

It is explicitly called a pentatruncated pentagonal hexecontahedron since only the valence-5 vertices of the pentagonal hexecontahedron are truncated.Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, 2013, Chapter 9 Goldberg polyhedra [https://books.google.com/books?id=kZtCAAAAQBAJ&dq=%22truncated+pentagonal+hexecontahedron%22&pg=PA128]

:240px

Its topology can be constructed in Conway polyhedron notation as t5gD and more simply wD as a whirled dodecahedron, reducing original pentagonal faces and adding 5 distorted hexagons around each, in clockwise or counter-clockwise forms. This picture shows its flat construction before the geometry is adjusted into a more spherical form. The snub can create a (5,3) geodesic polyhedron by k5k6.

: 240px

Related polyhedra

The whirled dodecahedron creates more polyhedra by basic Conway polyhedron notation. The zip whirled dodecahedron makes a chamfered truncated icosahedron, and Goldberg (4,1). Whirl applied twice produces Goldberg (5,3), and applied twice with reverse orientations produces goldberg (7,0).

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|+ Whirled dodecahedron polyhedra

!"seed"

ambotruncatezipexpandbevelsnubchamferwhirlwhirl-reverse
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|80px
wD = G(2,1)
[https://levskaya.github.io/polyhedronisme/?recipe=A10wD wD]

|80px
awD
[https://levskaya.github.io/polyhedronisme/?recipe=A10awD awD]

|80px
twD
[https://levskaya.github.io/polyhedronisme/?recipe=A10twD twD]

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zwD = G(4,1)
[https://levskaya.github.io/polyhedronisme/?recipe=A10tdwD zwD]

|80px
ewD
[https://levskaya.github.io/polyhedronisme/?recipe=A10ewD ewD]

|80px
bwD
[https://levskaya.github.io/polyhedronisme/?recipe=A10bwD bwD]

|80px
swD
[https://levskaya.github.io/polyhedronisme/?recipe=A10wD swD]

|80px
cwD = G(4,2)
[https://levskaya.github.io/polyhedronisme/?recipe=A10wcD cwD]

|80px
wwD = G(5,3)
[https://levskaya.github.io/polyhedronisme/?recipe=A10wwD wwD]

|80px
wrwD = G(7,0)
[https://levskaya.github.io/polyhedronisme/?recipe=A10wrwD wrwD]

dual||join||needle||kis||ortho||medial||gyro||dual chamfer||dual whirl||dual whirl-reverse
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|80px
dwD
[https://levskaya.github.io/polyhedronisme/?recipe=C100dwD dwD]

|80px
jwD
[https://levskaya.github.io/polyhedronisme/?recipe=C100jwD jwD]

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nwD
[https://levskaya.github.io/polyhedronisme/?recipe=C100kdwD nwD]

|80px
kwD
[https://levskaya.github.io/polyhedronisme/?recipe=C100kwD kwD]

|80px
owD
[https://levskaya.github.io/polyhedronisme/?recipe=C1000owD owD]

|80px
mwD
[https://levskaya.github.io/polyhedronisme/?recipe=A100mwD mwD]

|80px
gwD
[https://levskaya.github.io/polyhedronisme/?recipe=A100gwD gwD]

|80px
dcwD
[https://levskaya.github.io/polyhedronisme/?recipe=A100dcwD dcwD]

|80px
dwwD
[https://levskaya.github.io/polyhedronisme/?recipe=A100dwwD dwwD]

|80px
dwrwD
[https://levskaya.github.io/polyhedronisme/?recipe=A100dwrwD dwrwD]

See also

References

{{reflist}}

  • {{cite journal | title=A class of multi-symmetric polyhedra | first=Michael | last=Goldberg | journal= Tohoku Mathematical Journal | year=1937 | volume=43 | pages=104–108 |url=https://www.jstage.jst.go.jp/article/tmj1911/43/0/43_0_104/_article}}
  • {{cite book|first=George |last=Hart | authorlink = George W. Hart | chapter=Goldberg Polyhedra | title=Shaping Space |url=https://archive.org/details/shapingspaceexpl00sene |url-access=limited | edition= 2nd | editor-first=Marjorie | editor-last=Senechal | editor-link = Marjorie Senechal | pages=[https://archive.org/details/shapingspaceexpl00sene/page/n126 125]–138 | publisher=Springer | year=2012 | doi=10.1007/978-0-387-92714-5_9 |isbn=978-0-387-92713-8 }}
  • {{cite web | title=Mathematical Impressions: Goldberg Polyhedra | first=George | last=Hart | authorlink = George W. Hart | date=June 18, 2013 | url=https://www.simonsfoundation.org/multimedia/mathematical-impressions-goldberg-polyhedra/ | publisher= Simons Science News }}
  • [http://www.pnas.org/content/early/2014/02/04/1310939111 Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses], Stan Schein and James Maurice Gaye, PNAS, Early Edition doi: 10.1073/pnas.1310939111

External links

  • [http://www.georgehart.com/virtual-polyhedra/conway_notation.html VRML polyhedral generator] Try "t5gI" (Conway polyhedron notation)

Category:Goldberg polyhedra

Category:Pentagonal tilings

Category:Snub tilings

Category:Fullerenes

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