Polytope commun 4 – Wikipedia Géometrie sacrée

Tesseract est l’un des 6 polytopes communs convexes

En mathématiques, un 4-polytope régulier est un polytope commun à quatre dimensions. Ce sont les analogues à quatre dimensions de polyèdres ordinaires en trois dimensions et les polygones communs en deux dimensions.

Le mathématicien suisse Ludwig Schläfli a décrit pour la première fois les 4-polytopes au milieu des années 1800, bien que l'ensemble n'ait été découvert que plus tard.

Il y a six polytopes réguliers convexes et dix étoiles normaux, pour un total de seize.

histoire(éditer)

Les 4 polytopes ordinaires convexes ont été décrits pour la première fois par le mathématicien suisse Ludwig Schläfli au milieu du XIXe siècle. Il a découvert qu'il n'y en avait que six.

Schläfli a également trouvé quatre polytopes 4 étoiles communes: la grande 120 cellules, la grande 120 cellules numérotées, la grande 600 cellules et la grande cellule 120 cellules. Il a sauté les six autres parce qu'il n'autorisait pas les formes ayant échoué avec la caractéristique d'Euler des cellules ou des figures de vertex (pour les tori à trou nul: faE + V = 2). Elle exclut les cellules et les en-têtes tels que 5.5 / 2 et 5 / 2.5.

Edmund Hess (1843-1903) a publié la liste complète dans son livre allemand de 1883 Exercez l’apprentissage de la salle de bal avec une exposition spéciale à l’utilisation de la théorie du carnage et du polyèdre gai.

construction(éditer)

L'existence d'un 4-polytope régulier

p,q,rstyle p, q, r

est limité par l'existence du polyèdre commun

p,q,q,rstyle p, q, q, r}

formant leurs cellules et un limitation d'angle dièdre

pour s'assurer que les cellules se rencontrent pour former une surface 3 fermée.

Les six polytopes convexes et les dix étoiles décrits sont les seules solutions à ces limitations.

Il y a quatre symboles non-convexes-Schläfli p, q, r qui ont des cellules valides p, q et des en-têtes q, r et envoient le test diédral, mais ne produisent pas les nombres finaux: 3.5 / 2, 3, 4.3.5 / 2, 5 / 2.3.4, 5 / 2.3.5 / 2.

4-polytopes convexes réguliers(éditer)

Les 4-polytopes convexes communs sont les analogues à quatre dimensions des solides platoniques tridimensionnels et les polygones réguliers convexes à deux dimensions.

Cinq d'entre eux peuvent être considérés comme analogues aux solides platoniques. Une figure supplémentaire, la cellule à 24 cellules, n’a pas d’équivalent presque tridimensionnel.

Chaque polytope commun convexe est limité par un ensemble de trois dimensions cellules qui sont tous des solides platoniques du même type et de la même taille. Ceux-ci sont montés ensemble le long de leurs faces respectives régulièrement.

propriétés(éditer)

Les tableaux suivants montrent certaines propriétés des six polytopes 4 communs convexes. Les groupes de symétrie de ces 4 polytopes sont tous des groupes coxeter et sont donnés dans la notation décrite dans cet article. Le numéro qui suit le nom du groupe correspond à l'ordre du groupe.

noms image famille Schläfli
Coxeter
V E fa C Vert.
Fig.
double Groupe de symétrie
5-cell
pentachoron
pentatope
4-simplex
4-simplex t0.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/105px-4-simplex_t0.svg.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/158px-4-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/210px-4-simplex_t0.svg.png 2x "data file width =" 1600 "hauteur du fichier de données =" 1600 n-simplex
(ENn famille)
3,3,3
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
5 10 10
3
5
3,3
3,3 (Autoduaux) FR4
(3,3,3)
120
8 cellules
octachoron
Tesseract
4 cube
4-kube t0.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/105px-4-cube_t0.svg.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/158px-4-cube_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/210px-4-cube_t0.svg.png 2x "fichier de données width =" 1600 "fichier de données height =" 1600 n-cube
(Bn famille)
4,3,3
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
16 32 24
4
8
4,3
3,3 16 Cell- B4
(4,3,3)
384
16 Cell-
hexadecachoron
4 orthoplex
4-cube t3.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/105px-4-cube_t3.svg.png "décodage =" asynchrone "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/158px-4-cube_t3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/210px-4-cube_t3.svg.png 2x "fichier de données width =" 1600 "fichier de données height =" 1600 n-orthoplex
(Bn famille)
3,3,4
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
8 24 32
3
16
3,3
3,4 8 cellules B4
(4,3,3)
384
24 Cell-
icositetrachoron
octaplex
poly-octaèdre (pO)
24 cellules t0 F4.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/105px-24-cell_t0_F4.svg.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/158px-24-cell_t0_F4.svg.png 1.5x , //upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/210px-24-cell_t0_F4.svg.png 2x "data-file-width =" 1600 "hauteur du fichier de données =" 1600 fan famille 3,4,3
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
24 96 96
3
24
3,4
4,3 (Autoduaux) fa4
(3,4,3)
1152
120 cellules
hecatonicosachoron
dodecacontachoron
dodecaplex
polydodécaèdre (pD)
Graphique à 120 cellules H4.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/105px-120-cell_graph_H4.svg.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/158px-120-cell_graph_H4.svg.png 1.5x , //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/210px-120-cell_graph_H4.svg.png 2x "fichier-largeur-de-données =" 800 "hauteur du fichier de données =" 800 polytope n-pentagonal
(Hn famille)
5,3,3
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 5.png "src =" http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png "decoding =" async "width =" 7 "height =" 23 "fichier de données width =" 7 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
600 1200 720
5
120
5,3
3,3 600 cellules H4
(5,3,3)
14400
600 cellules
hexacosichoron
tetraplex
polytétraèdre (pT)
Graphique à 600 cellules H4.svg "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/105px-600-cell_graph_H4.svg.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/158px-600-cell_graph_H4.svg.png 1.5x , //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell-graph_H4.svg/210px-600-cell-graph_H4.svg.png 2x "fichier-largeur-de-données =" 800 "hauteur du fichier de données =" 800 polytope n-pentagonal
(Hn famille)
3,3,5
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 5.png "src =" http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png "decoding =" async "width =" 7 "height =" 23 "fichier de données width =" 7 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
120 720 1200
3
600
3,3
3,5 120 cellules H4
(5,3,3)
14400

John Conway préconise les noms simplex, orthoplex, tesseract, octaplex ou polyoctaèdre (pO), dodécaplex ou polydodécaèdre (pD) et tétraplex ou polytétraèdre (pT).(1)

Norman Johnson préconise les noms n-cell ou pentacoron, tesseract ou octacoron, hexadecachoron, icositetrachoron, hecatonicosachoron (ou dodecacontachoron) et hexacoccinone, qui forgent le terme polychoron être une analogie 4D avec un polyèdre 3D et un polygone 2D, exprimée à partir de racines grecques poly ("plusieurs") et Choros ("salle" ou "espace").(2)(3)

La caractéristique d'Euler de tous les 4-polytopes est zéro, nous avons l'analogue à 4 dimensions de la formule polyédrique d'Euler:

Nk indique le nombre k-faces dans le polytope (un sommet est une face 0, une arête est une face, etc.).

La topologie d'un 4-polytope donné est définie par ses nombres de Betti et ses coefficients de torsion.(4)

En tant que configurations(éditer)

Un 4-polytope commun peut être décrit comme une matrice de configuration contenant un compteur d'éléments constitutifs. Les rainures et les colonnes correspondent aux coins, aux arêtes, aux faces et aux cellules. Les nombres diagonaux (en haut à gauche et en bas à droite) indiquent le nombre d'éléments de chaque élément dans le 4-polytope. Les nombres non diagonaux indiquent combien d'éléments de la colonne apparaissent dans ou sur l'élément de la ligne. Par exemple, il y a 2 coins en chaque bord (chaque bord a 2 coins) et 2 cellules se rencontrent sur chaque visage (chaque visage appartient 2 cellules), dans n'importe quel 4-polytope commun. Notez que la configuration du double polytope peut être obtenue en faisant pivoter la matrice de 180 degrés.(5)(6)

5-cell
3,3,3
16 Cell-
3,3,4
Tesseract
4,3,3
24 Cell-
3,4,3
600 cellules
3,3,5
120 cellules
5,3,3

visualisation(éditer)

Le tableau suivant montre quelques projections bidimensionnelles de ces 4 polytopes. Divers autres visualisations sont disponibles dans les liens externes ci-dessous. Les graphiques Coxeter-Dynkin sont également fournis sous le symbole Schläfli.

FR4 = (3,3,3) BC4 = (4.3.3) fa4 = (3,4,3) H4 = (5.3.3)
5-cell 8 cellules 16 Cell- 24 Cell- 120 cellules 600 cellules
3,3,3 4,3,3 3,3,4 3,4,3 5,3,3 3,3,5
CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23 CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23 CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23 CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 4.png "src =" http://upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23 CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 5.png "src =" http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png "decoding =" async "width =" 7 "height =" 23 "fichier de données width =" 7 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23 CDel node 1.png "src =" http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png "decoding =" async "width =" 9 "height =" 23 "fichier de données- width = "9" data-file-height = "23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 3.png "src =" http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png "decoding =" async "width =" 6 "height =" 23 "fichier de données width =" 6 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23CDel 5.png "src =" http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png "decoding =" async "width =" 7 "height =" 23 "fichier de données width =" 7 "data-file-height =" 23CDel node.png "src =" http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png "decoding =" async "width =" 5 "height =" 23 "fichier de données width =" 5 "data-file-height =" 23
Projections orthographiques 3D solides
Tetrahedron.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/105px-Tetrahedron.png "decoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/158px-Tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2 /25/Tetrahedron.png/210px-Tetrahedron.png 2x "fichier de données width =" 1000 "fichier de données height =" 1000
tétraèdre
enveloppe
(Cellule / sommet centré)
Hexahedron.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hexahedron.png/105px-Hexahedron.png "décodage =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hexahedron.png/158px-Hexahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3 /33/Hexahedron.png/210px-Hexahedron.png 2x "fichier de données width =" 1000 "fichier de données height =" 1000
Enveloppe cubique
(Cell-centrée)
Ortho-16 cellules cellulaires.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/16-cell_ortho_cell-centered.png/105px-16-cell_ortho_cell-centered.png "dekoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/c/ca/16-cell_ortho_cell-centered.png/158px-16-cell_ortho_cell -centered.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/16-cell_ortho_cell-centered.png/210px-16-cell_ortho_cell-centered.png 2x "data-file-width = "1000" hauteur du fichier de données = "1000
enveloppe cubique
(Cell-centrée)
Ortho solid 24-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Ortho_solid_24-cell.png/105px-Ortho_solid_24-cell.png "decoding =" async " width = "105" height = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Ortho_solid_24-cell.png/158px-Ortho_solid_24-cell.png 1.5x, // upload ".wikimedia.org / wikipedia / commons / thumb / d / d9 / Ortho_solid_24-cell.png / 210px-Ortho_solid_24-cell.png 2x" fichier de données width = "1000" fichier de données height = "1000
Cuboctahedral
enveloppe
(Cell-centrée)
Ortho solid 120-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Ortho_solid_120-cell.png/105px-Ortho_solid_120-cell.png "decoding =" async " width = "105" height = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Ortho_solid_120-cell.png/158px-Ortho_solid_120-cell.png 1.5x, // télécharger .wikimedia.org / wikipedia / commons / thumb / 7 / 7a / Ortho_solid_120-cell.png / 210px-Ortho_solid_120-cell.png 2x "fichier de données width =" 1000 "fichier de données height =" 1000
Rhombique raccourci
triacontaèdre
enveloppe
(Cell-centrée)
Ortho solid 600-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/4/47/Ortho_solid_600-cell.png/105px-Ortho_solid_600-cell.png "decoding =" async " width = "105" height = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/4/47/Ortho_solid_600-cell.png/158px-Ortho_solid_600-cell.png 1.5x, // upload ".wikimedia.org / wikipedia / commons / thumb / 4/47 / Ortho_solid_600-cell.png / 210px-Ortho_solid_600-cell.png 2x" fichier de données width = "1000" fichier de données height = "1000
pentakis icosidodécaédrique
enveloppe
(Vertex centré)
Schémas Schlegel en mode filaire (projection en perspective)
Schlegel wireframe 5-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/105px-Schlegel_wireframe_5-cell.png "decoding =" async " width = "105" height = "101" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/158px-Schlegel_wireframe_5-cell.png 1.5x, // télécharger .wikimedia.org / wikipedia / commons / thumb / 5 / 5a / Schlegel_wireframe_5-cell.png / 210px-Schlegel_wireframe_5-cell.png 2x "fichier de données width =" 817 "data-file-height =" 785
Cell-centrée
Schfrel filaire 8-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/105px-Schlegel_wireframe_8-cell.png "decoding = async" width = "105" height = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/158px-Schlegel_wireframe_8-cell.png 1.5x, // télécharger .wikimedia.org / wikipedia / commons / thumb / a / a2 / Schlegel_wireframe_8-cell.png / 210px-Schlegel_wireframe_8-cell.png 2x "fichier de données width =" 1000 "data-file-height =" 1000
Cell-centrée
Schfrel filaire 16-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/105px-Schlegel_wireframe_16-cell.png "decoding =" async " width = "105" height = "107" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/158px-Schlegel_wireframe_16-cell.png 1.5x, // télécharger .wikimedia.org / wikipedia / commons / thumb / 9/90 / Schlegel_wireframe_16-cell.png / 210px-Schlegel_wireframe_16-cell.png 2x "fichier de données width =" 768 "data-file-height =" 786
Cell-centrée
Schlegel wireframe 24-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/105px-Schlegel_wireframe_24-cell.png "decoding =" async " width = "105" height = "99" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/158px-Schlegel_wireframe_24-cell.png 1.5x, // télécharger .wikimedia.org / wikipedia / commons / thumb / 7/77 / Schlegel_wireframe_24-cell.png / 210px-Schlegel_wireframe_24-cell.png 2x "fichier de données width =" 862 "data-file-height =" 816
Cell-centrée
Schlegel wireframe 120-cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Schlegel_wireframe_120-cell.png/105px-Schlegel_wireframe_120-cell.png "dekoding =" async " width = "105" height = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Schlegel_wireframe_120-cell.png/158px-Schlegel_wireframe_120-cell.png 1.5x, // opplasting .wikimedia.org / wikipedia / commons / thumb / 8 / 8e / Schlegel_wireframe_120-cell.png / 210px-Schlegel_wireframe_120-cell.png 2x "data-filbredde =" 1000 "data-fil-høyde =" 1000
Celle-sentrert
Schlegel wireframe 600-celle vertex-centered.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/105px-Schlegel_wireframe_600-cell_vertex-centered. png "dekoding =" async "width =" 105 "height =" 105 "srcset =" // upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/158px-Schlegel_wireframe_600- cell_vertex-centered.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/210px-Schlegel_wireframe_600-cell_vertex-centered.png 2x "datafilbredde = "1000" data-fil-høyde = "1000
Vertex-sentrert
Wireframe stereografiske fremspring (3-sfæren)
Stereografisk polytope 5cell.png "src =" http://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Stereographic_polytope_5cell.png/105px-Stereographic_polytope_5cell.png "dekoding =" async "width =" 105 "høyde = "105" srcset = "// upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Stereographic_polytope_5cell.png/158px-Stereographic_polytope_5cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb /f/f5/Stereographic_polytope_5cell.png/210px-Stereographic_polytope_5cell.png 2x "data-filbredde =" 1000 "data-fil-høyde =" 1000 Stereographic polytope 8cell.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Stereographic_polytope_8cell.png/105px-Stereographic_polytope_8cell.png" decoding="async" width="105" height="105" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Stereographic_polytope_8cell.png/158px-Stereographic_polytope_8cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Stereographic_polytope_8cell.png/210px-Stereographic_polytope_8cell.png 2x" data-file-width="1000" data-file-height="1000 Stereographic polytope 16cell.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/105px-Stereographic_polytope_16cell.png" decoding="async" width="105" height="95" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/158px-Stereographic_polytope_16cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/210px-Stereographic_polytope_16cell.png 2x" data-file-width="1100" data-file-height="1000 Stereographic polytope 24cell.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_polytope_24cell.png/105px-Stereographic_polytope_24cell.png" decoding="async" width="105" height="95" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_polytope_24cell.png/158px-Stereographic_polytope_24cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_polytope_24cell.png/210px-Stereographic_polytope_24cell.png 2x" data-file-width="1100" data-file-height="1000 Stereographic polytope 120cell.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/db/Stereographic_polytope_120cell.png/105px-Stereographic_polytope_120cell.png" decoding="async" width="105" height="105" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Stereographic_polytope_120cell.png/158px-Stereographic_polytope_120cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Stereographic_polytope_120cell.png/210px-Stereographic_polytope_120cell.png 2x" data-file-width="1000" data-file-height="1000 Stereographic polytope 600cell.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Stereographic_polytope_600cell.png/105px-Stereographic_polytope_600cell.png" decoding="async" width="105" height="105" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Stereographic_polytope_600cell.png/158px-Stereographic_polytope_600cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Stereographic_polytope_600cell.png/210px-Stereographic_polytope_600cell.png 2x" data-file-width="1100" data-file-height="1100

Regular star (Schläfli–Hess) 4-polytopes(éditer)

This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.(7)

A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations a,g,s are commutable, defining a cubic framework. There are 7 densities seen in vertical positioning, with 2 dual forms having the same density.

ils Schläfli–Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes).(8) They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol p,q,r in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra, which are in turn analogous to the pentagram.

Names(éditer)

Their names given here were given by John Conway, extending Cayley's names for the Kepler–Poinsot polyhedra: along with stellated et grand, he adds a grand modifier. Conway offered these operational definitions:

  1. stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram)
  2. greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
  3. aggrandizement – replaces the cells by large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell)

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron 3,3,5 (a tetrahedral 600-cell), pI=polyicoshedron 3,5,5/2 (an icosahedral 120-cell), and pD=polydodecahedron 5,3,3 (a dodecahedral 120-cell), with prefix modifiers: g, unet s for great, (ag)grand, and stellated. The final stellation, the great grand stellated polydodecahedron contains them all as gaspD.

Symmetry(éditer)

All ten polychora have (3,3,5) (H4) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: (3,5,5/2), (5,5/2,5), (5,3,5/2), (5/2,5,5/2), (5,5/2,3), and (3,3,5/2).

Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

propriétés(éditer)

Remarque:

The cells (polyhedra), their faces (polygons), the polygonal edge figures et polyhedral vertex figures are identified by their Schläfli symbols.

nom
Conway (abbrev.)
Orthogonal
projection
Schläfli
Coxeter
C
p, q
fa
p
E
r
V
q, r
Dens. χ
Icosahedral 120-cell
polyicosahedron (pI)
Ortho solid 007-uniform polychoron 35p-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ortho_solid_007-uniform_polychoron_35p-t0.png/75px-Ortho_solid_007-uniform_polychoron_35p-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ortho_solid_007-uniform_polychoron_35p-t0.png/113px-Ortho_solid_007-uniform_polychoron_35p-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Ortho_solid_007-uniform_polychoron_35p-t0.png/150px-Ortho_solid_007-uniform_polychoron_35p-t0.png 2x" data-file-width="1000" data-file-height="1000 3,5,5/2
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
120
3,5
Icosahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/25px-Icosahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/38px-Icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/50px-Icosahedron.png 2x" data-file-width="911" data-file-height="952
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
120
5,5/2
Great dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/25px-Great_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/38px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/50px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958
4 480
Small stellated 120-cell
stellated polydodecahedron (spD)
Ortho solid 010-uniform polychoron p53-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/54/Ortho_solid_010-uniform_polychoron_p53-t0.png/75px-Ortho_solid_010-uniform_polychoron_p53-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Ortho_solid_010-uniform_polychoron_p53-t0.png/113px-Ortho_solid_010-uniform_polychoron_p53-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Ortho_solid_010-uniform_polychoron_p53-t0.png/150px-Ortho_solid_010-uniform_polychoron_p53-t0.png 2x" data-file-width="1000" data-file-height="1000 5/2,5,3
CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23
120
5/2,5
Small stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/25px-Small_stellated_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/38px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/50px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
120
5,3
Dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/25px-Dodecahedron.png" decoding="async" width="25" height="24" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/38px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/50px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951
4 −480
Great 120-cell
great polydodecahedron (gpD)
Ortho solid 008-uniform polychoron 5p5-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ortho_solid_008-uniform_polychoron_5p5-t0.png/75px-Ortho_solid_008-uniform_polychoron_5p5-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ortho_solid_008-uniform_polychoron_5p5-t0.png/113px-Ortho_solid_008-uniform_polychoron_5p5-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Ortho_solid_008-uniform_polychoron_5p5-t0.png/150px-Ortho_solid_008-uniform_polychoron_5p5-t0.png 2x" data-file-width="1000" data-file-height="1000 5,5/2,5
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
120
5,5/2
Great dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/25px-Great_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/38px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/50px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
120
5/2,5
Small stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/25px-Small_stellated_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/38px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/50px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953
6 0
Grand 120-cell
grand polydodecahedron (apD)
Ortho solid 009-uniform polychoron 53p-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/be/Ortho_solid_009-uniform_polychoron_53p-t0.png/75px-Ortho_solid_009-uniform_polychoron_53p-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Ortho_solid_009-uniform_polychoron_53p-t0.png/113px-Ortho_solid_009-uniform_polychoron_53p-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Ortho_solid_009-uniform_polychoron_53p-t0.png/150px-Ortho_solid_009-uniform_polychoron_53p-t0.png 2x" data-file-width="1000" data-file-height="1000 5,3,5/2
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
120
5,3
Dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/25px-Dodecahedron.png" decoding="async" width="25" height="24" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/38px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/50px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
120
3,5/2
Great icosahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/25px-Great_icosahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/38px-Great_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/50px-Great_icosahedron.png 2x" data-file-width="908" data-file-height="953
20 0
Great stellated 120-cell
great stellated polydodecahedron (gspD)
Ortho solid 012-uniform polychoron p35-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Ortho_solid_012-uniform_polychoron_p35-t0.png/75px-Ortho_solid_012-uniform_polychoron_p35-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Ortho_solid_012-uniform_polychoron_p35-t0.png/113px-Ortho_solid_012-uniform_polychoron_p35-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Ortho_solid_012-uniform_polychoron_p35-t0.png/150px-Ortho_solid_012-uniform_polychoron_p35-t0.png 2x" data-file-width="1000" data-file-height="1000 5/2,3,5
CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23
120
5/2,3
Great stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/25px-Great_stellated_dodecahedron.png" decoding="async" width="25" height="24" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/38px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/50px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
120
3,5
Icosahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/25px-Icosahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/38px-Icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/50px-Icosahedron.png 2x" data-file-width="911" data-file-height="952
20 0
Grand stellated 120-cell
grand stellated polydodecahedron (aspD)
Ortho solid 013-uniform polychoron p5p-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ortho_solid_013-uniform_polychoron_p5p-t0.png/75px-Ortho_solid_013-uniform_polychoron_p5p-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ortho_solid_013-uniform_polychoron_p5p-t0.png/113px-Ortho_solid_013-uniform_polychoron_p5p-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ortho_solid_013-uniform_polychoron_p5p-t0.png/150px-Ortho_solid_013-uniform_polychoron_p5p-t0.png 2x" data-file-width="1000" data-file-height="1000 5/2,5,5/2
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
120
5/2,5
Small stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/25px-Small_stellated_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/38px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/50px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
120
5,5/2
Great dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/25px-Great_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/38px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/50px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958
66 0
Great grand 120-cell
great grand polydodecahedron (gapD)
Ortho solid 011-uniform polychoron 53p-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Ortho_solid_011-uniform_polychoron_53p-t0.png/75px-Ortho_solid_011-uniform_polychoron_53p-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Ortho_solid_011-uniform_polychoron_53p-t0.png/113px-Ortho_solid_011-uniform_polychoron_53p-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Ortho_solid_011-uniform_polychoron_53p-t0.png/150px-Ortho_solid_011-uniform_polychoron_53p-t0.png 2x" data-file-width="1000" data-file-height="1000 5,5/2,3
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
120
5,5/2
Great dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/25px-Great_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/38px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/50px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
120
5/2,3
Great stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/25px-Great_stellated_dodecahedron.png" decoding="async" width="25" height="24" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/38px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/50px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950
76 −480
Great icosahedral 120-cell
great polyicosahedron (gpI)
Ortho solid 014-uniform polychoron 3p5-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ortho_solid_014-uniform_polychoron_3p5-t0.png/75px-Ortho_solid_014-uniform_polychoron_3p5-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ortho_solid_014-uniform_polychoron_3p5-t0.png/113px-Ortho_solid_014-uniform_polychoron_3p5-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Ortho_solid_014-uniform_polychoron_3p5-t0.png/150px-Ortho_solid_014-uniform_polychoron_3p5-t0.png 2x" data-file-width="1000" data-file-height="1000 3,5/2,5
CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23
120
3,5/2
Great icosahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/25px-Great_icosahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/38px-Great_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/50px-Great_icosahedron.png 2x" data-file-width="908" data-file-height="953
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
720
5
Regular pentagon.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/25px-Regular_pentagon.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/38px-Regular_pentagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Regular_pentagon.svg/50px-Regular_pentagon.svg.png 2x" data-file-width="400" data-file-height="400
120
5/2,5
Small stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/25px-Small_stellated_dodecahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/38px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/50px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953
76 480
Grand 600-cell
grand polytetrahedron (apT)
Ortho solid 015-uniform polychoron 33p-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ortho_solid_015-uniform_polychoron_33p-t0.png/75px-Ortho_solid_015-uniform_polychoron_33p-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ortho_solid_015-uniform_polychoron_33p-t0.png/113px-Ortho_solid_015-uniform_polychoron_33p-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ortho_solid_015-uniform_polychoron_33p-t0.png/150px-Ortho_solid_015-uniform_polychoron_33p-t0.png 2x" data-file-width="1000" data-file-height="1000 3,3,5/2
CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23
600
3,3
Tetrahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/25px-Tetrahedron.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/38px-Tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/50px-Tetrahedron.png 2x" data-file-width="1000" data-file-height="1000
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
120
3,5/2
Great icosahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/25px-Great_icosahedron.png" decoding="async" width="25" height="26" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/38px-Great_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/50px-Great_icosahedron.png 2x" data-file-width="908" data-file-height="953
191 0
Great grand stellated 120-cell
great grand stellated polydodecahedron (gaspD)
Ortho solid 016-uniform polychoron p33-t0.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Ortho_solid_016-uniform_polychoron_p33-t0.png/75px-Ortho_solid_016-uniform_polychoron_p33-t0.png" decoding="async" width="75" height="75" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Ortho_solid_016-uniform_polychoron_p33-t0.png/113px-Ortho_solid_016-uniform_polychoron_p33-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Ortho_solid_016-uniform_polychoron_p33-t0.png/150px-Ortho_solid_016-uniform_polychoron_p33-t0.png 2x" data-file-width="1000" data-file-height="1000 5/2,3,3
CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 3.png" src="http://upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" data-file-width="6" data-file-height="23CDel node.png" src="http://upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" data-file-width="5" data-file-height="23CDel 5.png" src="http://upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" data-file-width="7" data-file-height="23CDel rat.png" src="http://upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" data-file-width="3" data-file-height="23CDel d2.png" src="http://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_d2.png" decoding="async" width="4" height="23" data-file-width="4" data-file-height="23CDel node 1.png" src="http://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" data-file-width="9" data-file-height="23
120
5/2,3
Great stellated dodecahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/25px-Great_stellated_dodecahedron.png" decoding="async" width="25" height="24" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/38px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/50px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950
720
5/2
Star polygon 5-2.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/25px-Star_polygon_5-2.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/38px-Star_polygon_5-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Star_polygon_5-2.svg/50px-Star_polygon_5-2.svg.png 2x" data-file-width="250" data-file-height="250
1200
3
Regular triangle.svg" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/25px-Regular_triangle.svg.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/38px-Regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Regular_triangle.svg/50px-Regular_triangle.svg.png 2x" data-file-width="400" data-file-height="400
600
3,3
Tetrahedron.png" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/25px-Tetrahedron.png" decoding="async" width="25" height="25" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/38px-Tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Tetrahedron.png/50px-Tetrahedron.png 2x" data-file-width="1000" data-file-height="1000
191 0

Voir aussi(éditer)

  • Regular polytope
  • List of regular polytopes
  • Infinite regular 4-polytopes:
    • One regular Euclidean honeycomb: 4,3,4
    • Four compact regular hyperbolic honeycombs: 3,5,3, 4,3,5, 5,3,4, 5,3,5
    • Eleven paracompact regular hyperbolic honeycombs: 3,3,6, 6,3,3, 3,4,4, 4,4,3, 3,6,3, 4,3,6, 6,3,4, 4,4,4, 5,3,6, 6,3,5, and 6,3,6.
  • Abstract regular 4-polytopes:
  • Uniform 4-polytope uniform 4-polytope families constructed from these 6 regular forms.
  • Platonisk fast stoff
  • Kepler-Poinsot polyhedra – regular star polyhedron
  • Star polygon – regular star polygons

références(éditer)

Citations(éditer)

  1. ^ Conway, 2008, Chapter 26, Higher Still
  2. ^ "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
  3. ^ Johnson (2015), Chapter 11, Section 11.5 Spherical Coxeter groups
  4. ^ Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  5. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  6. ^ Coxeter, Complex Regular Polytopes, p.117
  7. ^ The Symmetries of Things, John Conway, (2008), p. 406, Fig 26.2
  8. ^ Coxeter, Star polytopes and the Schläfli function f{α,β,γ) s. 122 2. The Schläfli-Hess polytopes

Bibliography(éditer)

  • H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. ISBN 0-471-50458-0.
  • H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
  • D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
  • Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1).
  • Edmund Hess Uber die regulären Polytope höherer Art, Sitzungsber Gesells Beförderung gesammten Naturwiss Marburg, 1885, 31-57
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 (2)
    • (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) (Elemente der Mathematik 44 (2) (1989) 25–36)
  • H. S. M. Coxeter, Regular Complex Polytopes, 2nd. ed., Cambridge University Press 1991. ISBN 978-0-521-39490-1. (3)
  • Peter McMullen and Egon Schulte, Abstract Regular Polytopes, 2002, PDF

Liens externes(éditer)


Un solide de polyèdre doit avoir toutes les faces planes ( par exemple, des solides de Platon, des prismes et des pyramides ), tandis qu’un solide non polyèdre a au minimum une de ses surfaces qui n’est pas plate ( par exemple, barillet, sphère ou tube ) Régulier sous-entend que tous les angles sont de la même mesure, toutes les faces sont de formes congruentes ou semblables dans tous les aspects, et tous les bords sont de la même taille 3D sous-entend que la forme a la largeur, la profondeur et la hauteur. Un polygone est une forme verrouillée dans une figure plane avec au moins cinq bords droits. Un duel est un solide de Platon qui s’adapte à l’intérieur d’un autre solide de Platon et se connecte au point médian de chaque face

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