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Ditrigonal polyhedron

In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal.

Ditrigonal vertex figures

There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993

The three uniform star polyhedron with Wythoff symbol of the form 3 | p q or {{sfrac|3|2}} | p q are ditrigonal, at least if p and q are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form p.q.p.q.p.q or (p.q)3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ditrigonal means "having two sets of 3 angles").[http://mathworld.wolfram.com/UniformPolyhedron.html Uniform Polyhedron], Mathworld (retrieved 10 June 2016)

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Type

!Small ditrigonal icosidodecahedron

!Ditrigonal dodecadodecahedron

!Great ditrigonal icosidodecahedron

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!Image

|125px

|125px

|125px

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!Vertex figure

|125px

|125px

|125px

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!Vertex configuration

|3.{{frac|5|2}}.3.{{frac|5|2}}.3.{{frac|5|2}}

|5.{{frac|5|3}}.5.{{frac|5|3}}.5.{{frac|5|3}}

|(3.5.3.5.3.5)/2

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!Faces

|32
20 {3}, 12 { {{frac|5|2}} }

|24
12 {5}, 12 { {{frac|5|2}} }

|32
20 {3}, 12 {5}

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!Wythoff symbol

|3 {{pipe}} 5/2 3

|3 {{pipe}} 5/3 5

|3 {{pipe}} 3/2 5

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!Coxeter diagram

|File:Small ditrigonal icosidodecahedron cd.png

|File:Ditrigonal dodecadodecahedron cd.png

|File:Great ditrigonal icosidodecahedron cd.png

Other uniform ditrigonal polyhedra

The small ditrigonal dodecicosidodecahedron and the great ditrigonal dodecicosidodecahedron are also uniform.

Their duals are respectively the small ditrigonal dodecacronic hexecontahedron and great ditrigonal dodecacronic hexecontahedron.

See also

References

=Notes=

{{reflist}}

=Bibliography=

  • Coxeter, H.S.M., M.S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.
  • Har'El, Z. [https://web.archive.org/web/20090715034226/http://www.math.technion.ac.il/~rl/docs/uniform.pdf Uniform Solution for Uniform Polyhedra.], Geometriae Dedicata 47, 57–110, 1993. [https://web.archive.org/web/20090727182130/http://www.math.technion.ac.il/~rl Zvi Har'El], [https://web.archive.org/web/20110520092545/http://www.math.technion.ac.il/~rl/kaleido/ Kaleido software], [https://web.archive.org/web/20110520080303/http://www.math.technion.ac.il/~rl/kaleido/poly.html Images], [https://web.archive.org/web/20110520080425/http://www.math.technion.ac.il/~rl/kaleido/dual.html dual images]

Further reading

  • Johnson, N.; The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 [https://web.archive.org/web/20131014175200/https://getinfo.de/app/The-theory-of-uniform-polytopes-and-honeycombs/id/TIBKAT%3A22693604X]
  • {{Citation | last1=Skilling | first1=J. | title=The complete set of uniform polyhedra | jstor=74475 | mr=0365333 | year=1975 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=278 | issue=1278 | pages=111–135 | doi=10.1098/rsta.1975.0022| bibcode=1975RSPTA.278..111S | s2cid=122634260 }}

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Category:Polyhedra