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Omnitruncated polyhedron

In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.

All omnitruncated polyhedra are considered as zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r.

More generally, an omnitruncated polyhedron is a bevel operator in Conway polyhedron notation.

List of convex omnitruncated polyhedra

There are three convex forms. These forms can be seen as red faces of one regular polyhedron, yellow or green faces of the dual polyhedron, and blue faces at the truncated vertices of the quasiregular polyhedron.

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Wythoff
symbol
p q r {{pipe}}

! Omnitruncated polyhedron

! Regular/quasiregular polyhedra

align=center

! 3 3 2 {{pipe}}

| 100px
Truncated octahedron
{{CDD|node_1|3|node_1|3|node_1}}

| 100px 100px 100px
Tetrahedron/Octahedron/Tetrahedron

align=center

! 4 3 2 {{pipe}}

| 100px
Truncated cuboctahedron
{{CDD|node_1|4|node_1|3|node_1}}

| 100px100px100px
Cube/Cuboctahedron/Octahedron

align=center

! 5 3 2 {{pipe}}

| 100px
Truncated icosidodecahedron
{{CDD|node_1|5|node_1|3|node_1}}

| 100px100px100px
Dodecahedron/Icosidodecahedron/Icosahedron

List of nonconvex omnitruncated polyhedra

There are 5 nonconvex uniform omnitruncated polyhedra.

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Wythoff
symbol
p q r {{pipe}}

! Omnitruncated star polyhedron

!Wythoff
symbol
p q r {{pipe}}

! Omnitruncated star polyhedron

colspan=2|Right triangle domains (r=2)

!colspan=2|General triangle domains

align=center

! 3 4/3 2 {{pipe}}

| 100px
Great truncated cuboctahedron
{{CDD|node_1|4|rat|d3|node_1|3|node_1}}

! 4 4/3 3 {{pipe}}

| 100px
Cubitruncated cuboctahedron

align=center

! 3 5/3 2 {{pipe}}

| 100px
Great truncated icosidodecahedron
{{CDD|node_1|5|rat|d3|node_1|3|node_1}}

! 5 5/3 3 {{pipe}}

| 100px
Icositruncated dodecadodecahedron

align=center

! 5 5/3 2 {{pipe}}

| 100px
Truncated dodecadodecahedron
{{CDD|node_1|5|rat|d3|node_1|5|node_1}}

= Other even-sided nonconvex polyhedra =

There are 8 nonconvex forms with mixed Wythoff symbols p q (r s) |, and bow-tie shaped vertex figures, 2p.2q.-2q.-2p. They are not true omnitruncated polyhedra. Instead, the true omnitruncates p q r | or p q s | have coinciding 2r-gonal or 2s-gonal faces that must be removed respectively to form a proper polyhedron. All these polyhedra are one-sided, i.e. non-orientable. The p q r | degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols.

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Omnitruncated polyhedron

! Image

! Wythoff symbol

Cubohemioctahedron

| 100px

| 3/2 2 3 {{pipe}}
2 3 (3/2 3/2) {{pipe}}

Small rhombihexahedron

| 100px

| 3/2 2 4 {{pipe}}
2 4 (3/2 4/2) {{pipe}}

Great rhombihexahedron

| 100px

| 4/3 3/2 2 {{pipe}}
2 4/3 (3/2 4/2) {{pipe}}

Small rhombidodecahedron

| 100px

| 2 5/2 5 {{pipe}}
2 5 (3/2 5/2) {{pipe}}

Small dodecicosahedron

| 100px

| 3/2 3 5 {{pipe}}
3 5 (3/2 5/4) {{pipe}}

Rhombicosahedron

| 100px

| 2 5/2 3 {{pipe}}
2 3 (5/4 5/2) {{pipe}}

Great dodecicosahedron

| 100px

| 5/2 5/3 3 {{pipe}}
3 5/3 (3/2 5/2) {{pipe}}

Great rhombidodecahedron

| 100px

| 3/2 5/3 2 {{pipe}}
2 5/3 (3/2 5/4) {{pipe}}

General omnitruncations (bevel)

Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra:

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!Coxeter

!trrC

!trrD

!trtT

!trtC

!trtO

!trtI

Conway

!baO

!baD

!btT

!btC

!btO

!btI

Image

|100px

|100px

|100px

|100px

|100px

|100px

See also

References

  • {{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 | issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003| bibcode=1954RSPTA.246..401C | s2cid=202575183 }}
  • {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }}
  • {{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 }}
  • Har'El, Z. [https://web.archive.org/web/20110927223146/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/20110930090207/http://www.math.technion.ac.il/~rl/ Zvi Har'El], [https://web.archive.org/web/20110927225147/http://www.math.technion.ac.il/~rl/kaleido/ Kaleido software], [https://www.math.technion.ac.il/~rl/kaleido/poly.html Images], [https://www.math.technion.ac.il/~rl/kaleido/dual.html dual images]
  • [http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.] Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.

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Category:Polyhedra