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Hendecagrammic prism

File:HendecagramTypes.png

In geometry, a hendecagrammic prism is a star polyhedron made from two identical regular hendecagrams connected by squares. The related hendecagrammic antiprisms are made from two identical regular hendecagrams connected by equilateral triangles.

Hendecagrammic prisms and bipyramids

There are 4 hendecagrammic uniform prisms, and 6 hendecagrammic uniform antiprisms. The prisms are constructed by 4.4.11/q vertex figures, {{CDD|node_1|2|node|11|rat|q|node_1}} Coxeter diagram. The hendecagrammic bipyramids, duals to the hendecagrammic prisms are also given.

class=wikitable
Symmetry

!colspan=4|Prisms

align=center| D11h
[2,11]
(*2.2.11)

| 64px
4.4.11/2
{{CDD|node_1|2|node|11|rat|d2|node_1}}

| 64px
4.4.11/3
{{CDD|node_1|2|node|11|rat|d3|node_1}}

| 64px
4.4.11/4
{{CDD|node_1|2|node|11|rat|d4|node_1}}

| 64px
4.4.11/5
{{CDD|node_1|2|node|11|rat|d5|node_1}}

valign=top align=center

! D11h
[2,11]
(*2.2.11)

|60px
{{CDD|node_f1|2|node|11|rat|2x|node_f1}}

|60px
{{CDD|node_f1|2|node|11|rat|3x|node_f1}}

|60px
{{CDD|node_f1|2|node|11|rat|4|node_f1}}

|60px
{{CDD|node_f1|2|node|11|rat|5|node_f1}}

Hendecagrammic antiprisms

The antiprisms with 3.3.3.3.11/q vertex figures, {{CDD|node_h|2|node_h|11|rat|q|node_h}}. Uniform antiprisms exist for p/q>3/2,{{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=79|pages=447–457|year=1976|doi=10.1017/S0305004100052440|mr=0397554|issue=3|bibcode=1976MPCPS..79..447S }}.

and are called crossed for p/q<2. For hendecagonal antiprism, two crossed antiprisms can not be constructed as uniform (with equilateral triangles): 11/8, and 11/9.

class=wikitable
Symmetry

!colspan=2|Antiprisms

!colspan=2|Crossed- antiprisms

valign=top align=center

! D11h
[2,11]
(*2.2.11)

| 64px
3.3.3.11/2
 
{{CDD|node_h|2|node_h|11|rat|d2|node_h}}

| 64px
3.3.3.11/4
 
{{CDD|node_h|2|node_h|11|rat|d4|node_h}}

| 64px
3.3.3.11/6
3.3.3.-11/5
{{CDD|node_h|2|node_h|11|rat|6|node_h}}

| Nonuniform
3.3.3.11/8
3.3.3.-11/3

valign=top

!D11d
[2+,11]
(2*11)

| 64px
3.3.3.11/3
 
{{CDD|node_h|2|node_h|11|rat|d3|node_h}}

| 64px
3.3.3.11/5
 
{{CDD|node_h|2|node_h|11|rat|d5|node_h}}

| 64px
3.3.3.11/7
3.3.3.-11/4
{{CDD|node_h|2|node_h|11|rat|d7|node_h}}

| Nonuniform
3.3.3.11/9
3.3.3.-11/2

Hendecagrammic trapezohedra

The hendecagrammic trapezohedra are duals to the hendecagrammic antiprisms.

class=wikitable

!Symmetry

!colspan=3|Trapezohedra

valign=top

!D11h
[2,11]
(*2.2.11)

|60px
{{CDD|node_fh|2|node_fh|11|rat|2x|node_fh}}

|60px
{{CDD|node_fh|2|node_fh|11|rat|4|node_fh}}

|60px
{{CDD|node_fh|2|node_fh|11|rat|6|node_fh}}

valign=top

!D11d
[2+,11]
(2*11)

|60px
{{CDD|node_fh|2|node_fh|11|rat|3x|node_fh}}

|60px
{{CDD|node_fh|2|node_fh|11|rat|5|node_fh}}

|60px
{{CDD|node_fh|2|node_fh|11|rat|7|node_fh}}

See also

References

{{reflist}}

Further reading

  • {{Cite journal | 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 | publisher=The Royal Society | doi=10.1098/rsta.1954.0003 | bibcode=1954RSPTA.246..401C | s2cid=202575183 }}

External links

  • {{Mathworld |urlname=Polygram |title=Polygram}}

Category:Polyhedra