Table of kanichora (Meta, 11)

From Hi.gher. Space

(Difference between revisions)
m (correct names of dual-invariant figures)
m (ontology)
 
(8 intermediate revisions not shown)
Line 1: Line 1:
 +
<[#ontology [kind meta] [cats 4D Uniform Polytope]]>
This table shows the four-dimensional [[kanitope]]s arranged by their [[polytope family|family]] and [[Dx number]].
This table shows the four-dimensional [[kanitope]]s arranged by their [[polytope family|family]] and [[Dx number]].
-
Forms enclosed in parentheses are non-canonical forms, which duplicate other cells of the table. In 4D, this occurs in the pentachoric and icositetrachoric families due to them being [[self-dual]], and in the hexadecachoric family due to the hexadecachoron being equivalent to the [[Demihypercube|demitesseract]].
+
Forms enclosed in parentheses are non-canonical forms, which duplicate other cells of the table. In 4D, this occurs in the pyrochoric and xylochoric families due to them being [[self-dual]], and in the aerochoric family due to the aerochoron being equivalent to the [[Demihypercube|demitesseract]].
 +
 
 +
Variants enclosed in parentheses indicate it is a variant of the dual, rather than a variant of the parent.
{|cellpadding="2" cellspacing="1"
{|cellpadding="2" cellspacing="1"
-
!bgcolor="#E0E0E0" rowspan="2" width="30"|[[Dx]]
+
!bgcolor="#E0E0E0" rowspan="2" width="5%"|[[Dx]]
-
!bgcolor="#E0E0E0" rowspan="2" width="80"|[[Coxeter-Dynkin|Coxeter-<br>Dynkin]]
+
!bgcolor="#E0E0E0" rowspan="2" width="10%"|[[Coxeter-Dynkin|Coxeter-<br>Dynkin]]
-
!bgcolor="#E0E0E0" colspan="4" width="40%"|[[Variance group]]
+
!bgcolor="#E0E0E0" rowspan="2" width="15%"|[[Variant]]
-
!bgcolor="#E0E0E0" colspan="4" width="40%"|[[Polychoric group]]
+
!bgcolor="#E0E0E0" colspan="4" width="70%"|[[Symmetry group]]
|-
|-
-
!bgcolor="#E0E0E0"|Standard
+
!bgcolor="#E0E0E0"|Pyrochoric
-
!bgcolor="#E0E0E0"|Krieger
+
!bgcolor="#E0E0E0"|Staurochoric
-
!bgcolor="#E0E0E0"|Conventional
+
!bgcolor="#E0E0E0"|Xylochoric
-
!bgcolor="#E0E0E0"|Alt. conv.
+
!bgcolor="#E0E0E0"|Rhodochoric
-
!bgcolor="#E0E0E0"|Pentachoric
+
-
!bgcolor="#E0E0E0"|Hexadecachoric
+
-
!bgcolor="#E0E0E0"|Icositetrachoric
+
-
!bgcolor="#E0E0E0"|Hexacosichoric
+
|-
|-
-
|1||xooo||Parent||Parent||Parent||&nbsp;||[[Pyrochoron|Kp1]]||[[Geochoron|Ke1]]||[[Xylochoron|Kk1]]||[[Cosmochoron|Ks1]]
+
|1||xooo||Parent||[[Pyrochoron|x3o3o3o]]||[[Geochoron|x4o3o3o]]||[[Xylochoron|x3o4o3o]]||[[Cosmochoron|x5o3o3o]]
|-
|-
-
|2||oxoo||Rectate||Rectate||Hemicate||&nbsp;||[[3-pyrotomochoron|Kp2]]||[[3-geotomochoron|Ke2]]||[[3-xylotomochoron|Kk2]]||[[3-cosmotomochoron|Ks2]]
+
|2||oxoo||[[Rectate]]||[[Pyrorectichoron|o3x3o3o]]||[[Georectichoron|o4x3o3o]]||[[Xylorectichoron|o3x4o3o]]||[[Cosmorectichoron|o5x3o3o]]
|-
|-
-
|3||xxoo||Truncate||Truncate||Truncate||&nbsp;||[[2-pyrotomochoron|Kp3]]||[[2-geotomochoron|Ke3]]||[[2-xylotomochoron|Kk3]]||[[2-cosmotomochoron|Ks3]]
+
|3||xxoo||[[Truncate]]||[[Pyrotomochoron|x3x3o3o]]||[[Geotomochoron|x4x3x3o]]||[[Xylotomochoron|x3x4o3o]]||[[Cosmotomochoron|x5x3o3o]]
|-
|-
-
|4||ooxo||---||---||Dual,hemic||&nbsp;||[[3-pyrotomochoron|(Kp2)]]||[[Xylochoron|''(Kk1)'']]||[[3-xylotomochoron|(Kk2)]]||[[3-hydrotomochoron|Ks4]]
+
|4||ooxo||(Rectate)||[[Pyrorectichoron|(o3x3o3o)]]||[[Xylochoron|''(x3o4o3o)'']]||[[Xylorectichoron|(o3x4o3o)]]||[[Hydrorectichoron|o3x3o5o]]
|-
|-
-
|5||xoxo||Cantellate||sm Rhombi-||Cantellate||&nbsp;||[[Cantellated 5-cell|Kp5]]||[[Cantellated 8-cell|Ke5]]||[[Cantellated 24-cell|Kk5]]||[[Cantellated 120-cell|Ks5]]
+
|5||xoxo||[[Cantellate]]||[[Pyrocantichoron|x3o3x3o]]||[[Geocantichoron|x4o3x3o]]||[[Xylocantichoron|x3o4x3o]]||[[Cosmocantichoron|x5o3x3o]]
|-
|-
-
|6||oxxo||Bitruncate||---||Rectate||&nbsp;||[[Pyromesochoron|Kp6]]||[[Stauromesochoron|Ke6]]||[[Xylomesochoron|Kk6]]||[[Rhodomesochoron|Ks6]]
+
|6||oxxo||[[Mesotope]]||[[Pyromesochoron|o3x3x3o]]||[[Stauromesochoron|o4x3x3o]]||[[Xylomesochoron|o3x4x3o]]||[[Rhodomesochoron|o5x3x3o]]
|-
|-
-
|7||xxxo||Cantitruncate||gr Rhombi-||Cantitruncate||&nbsp;||[[Cantitruncated 5-cell|Kp7]]||[[Cantitruncated 8-cell|Ke7]]||[[Cantitruncated 24-cell|Kk7]]||[[Cantitruncated 120-cell|Ks7]]
+
|7||xxxo||[[Cantitruncate]]||[[Pyrocantitomochoron|x3x3x3o]]||[[Geocantitomochoron|x4x3x3o]]||[[Xylocantitomochoron|x3x4x3o]]||[[Cosmocantitomochoron|x5x3x3o]]
|-
|-
-
|8||ooox||---||---||Dual||&nbsp;||[[Pyrochoron|(Kp1)]]||[[Aerochoron|Ke8]]||[[Xylochoron|(Kk1)]]||[[Hydrochoron|Ks8]]
+
|8||ooox||[[Dual]]||[[Pyrochoron|(x3o3o3o)]]||[[Aerochoron|x3o3o4x]]||[[Xylochoron|(x3o4o3o)]]||[[Hydrochoron|x3o3o5o]]
|-
|-
-
|9||xoox||Runcinate||---||Runcinate||Rect,rect||[[Runcinated 5-cell|Kp9]]||[[Runcinated 8-cell|Ke9]]||[[Runcinated 24-cell|Kk9]]||[[Runcinated 120-cell|Ks9]]
+
|9||xoox||[[Peritope]]||[[Pyroperichoron|x3o3o3x]]||[[Stauroperichoron|x4o3o3x]]||[[Xyloperichoron|x3o4o3x]]||[[Rhodoperichoron|x5o3o3x]]
|-
|-
-
|10||oxox||---||---||Dual,cantel||&nbsp;||[[Cantellated 5-cell|(Kp5)]]||[[3-xylotomochoron|''(Kk2)'']]||[[Cantellated 24-cell|(Kk5)]]||[[Cantellated 600-cell|Ks10]]
+
|10||oxox||(Cantellate)||[[Pyrocantichoron|(x3o3x3o)]]||[[Xylorectichoron|''(o3x4o3o)'']]||[[Xylocantichoron|(x3o4x3o)]]||[[Hydrocantichoron|x3o3x5o]]
|-
|-
-
|11||xxox||Runcitruncate||---||Runcitruncate||&nbsp;||[[Runcitruncated 5-cell|Kp11]]||[[Runcitruncated 8-cell|Ke11]]||[[Runcitruncated 24-cell|Kk11]]||[[Runcitruncated 120-cell|Ks11]]
+
|11||xxox||[[Runcitruncate]]||[[Pyroruncitomochoron|x3x3o3x]]||[[Georuncitomochoron|x4x3o3x]]||[[Xyloruncitomochoron|x3x4o3x]]||[[Cosmoruncitomochoron|x5x3o3x]]
|-
|-
-
|12||ooxx||---||---||Dual,trunc||&nbsp;||[[2-pyrotomochoron|(Kp3)]]||[[2-aerotomochoron|Ke12]]||[[2-xylotomochoron|(Kk3)]]||[[2-hydrotomochoron|Ks12]]
+
|12||ooxx||(Truncate)||[[Pyrotomochoron|(x3x3o3o)]]||[[Aerotomochoron|x3x3o4o]]||[[Xylotomochoron|(x3x4o3o)]]||[[Hydrotomochoron|x3x3o5o]]
|-
|-
-
|13||xoxx||---||Prismatorh||Dual,runcic||&nbsp;||[[Runcitruncated 5-cell|(Kp11)]]||[[Runcitruncated 16-cell|Ke13]]||[[Runcitruncated 24-cell|(Kk11)]]||[[Runcitruncated 600-cell|Ks13]]
+
|13||xoxx||(Runcitruncate)||[[Pyroruncitomochoron|(x3x3o3x)]]||[[Aeroruncitomochoron|x3x3o4x]]||[[Xyloruncitomochoron|(x3x4o3x)]]||[[Hydroruncitomochoron|x3x3o5x]]
|-
|-
-
|14||oxxx||---||---||Dual,cantic||&nbsp;||[[Cantitruncated 5-cell|(Kp7)]]||[[2-xylotomochoron|''(Kk3)'']]||[[Cantitruncated 24-cell|(Kk7)]]||[[Cantitruncated 600-cell|Ks14]]
+
|14||oxxx||(Cantitruncate)||[[Pyrocantitomochoron|(x3x3x3o)]]||[[Xylocantitomochoron|''(x3x4o3o)'']]||[[Xylocantitomochoron|(x3x4x3o)]]||[[Hydrocantitomochoron|x3x3x5o]]
|-
|-
-
|15||xxxx||Omnitruncate||---||Omnitruncate||Rect,trunc||[[Pyropantomochoron|Kp15]]||[[Stauropantomochoron|Ke15]]||[[Xylopantomochoron|Kk15]]||[[Rhodopantomochoron|Ks15]]
+
|15||xxxx||[[Pantome]]||[[Pyropantochoron|x3x3x3x]]||[[Stauropantochoron|x4x3x3x]]||[[Xylopantochoron|x3x4x3x]]||[[Rhodopantochoron|x5x3x3x]]
-
|-
+
-
|0||oooo||Snub||Snub||Snub||&nbsp;||
+
|}
|}
-
 
-
[[Category:Uniform polychora|$]]
 

Latest revision as of 08:11, 11 February 2014

This table shows the four-dimensional kanitopes arranged by their family and Dx number.

Forms enclosed in parentheses are non-canonical forms, which duplicate other cells of the table. In 4D, this occurs in the pyrochoric and xylochoric families due to them being self-dual, and in the aerochoric family due to the aerochoron being equivalent to the demitesseract.

Variants enclosed in parentheses indicate it is a variant of the dual, rather than a variant of the parent.

Dx Coxeter-
Dynkin
Variant Symmetry group
Pyrochoric Staurochoric Xylochoric Rhodochoric
1xoooParentx3o3o3ox4o3o3ox3o4o3ox5o3o3o
2oxooRectateo3x3o3oo4x3o3oo3x4o3oo5x3o3o
3xxooTruncatex3x3o3ox4x3x3ox3x4o3ox5x3o3o
4ooxo(Rectate)(o3x3o3o)(x3o4o3o)(o3x4o3o)o3x3o5o
5xoxoCantellatex3o3x3ox4o3x3ox3o4x3ox5o3x3o
6oxxoMesotopeo3x3x3oo4x3x3oo3x4x3oo5x3x3o
7xxxoCantitruncatex3x3x3ox4x3x3ox3x4x3ox5x3x3o
8oooxDual(x3o3o3o)x3o3o4x(x3o4o3o)x3o3o5o
9xooxPeritopex3o3o3xx4o3o3xx3o4o3xx5o3o3x
10oxox(Cantellate)(x3o3x3o)(o3x4o3o)(x3o4x3o)x3o3x5o
11xxoxRuncitruncatex3x3o3xx4x3o3xx3x4o3xx5x3o3x
12ooxx(Truncate)(x3x3o3o)x3x3o4o(x3x4o3o)x3x3o5o
13xoxx(Runcitruncate)(x3x3o3x)x3x3o4x(x3x4o3x)x3x3o5x
14oxxx(Cantitruncate)(x3x3x3o)(x3x4o3o)(x3x4x3o)x3x3x5o
15xxxxPantomex3x3x3xx4x3x3xx3x4x3xx5x3x3x