E6 polytope: Difference between revisions

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Orthographic projections in the E₆ Coxeter plane
2₂₁	1₂₂

In 6-dimensional geometry, there are 39 uniform polytopes with E₆ symmetry. The two simplest forms are the 2₂₁ and 1₂₂ polytopes, composed of 27 and 72 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the E₆ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 39 polytopes can be made in the E₆, D₅, D₄, D₂, A₅, A₄, A₃ Coxeter planes. A_k has k+1 symmetry, D_k has 2(k-1) symmetry, and E₆ has 12 symmetry.

Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E₆ symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.

#	Coxeter plane graphs						Coxeter diagram Names
#	Aut(E₆) [18/2]	E₆ [12]	D₅ [8]	D₄ / A₂ [6]	A₅ [6]	D₃ / A₃ [4]	Coxeter diagram Names
1							2₂₁ Icosihepta-heptacontidipeton (jak)
2							Rectified 2₂₁ Rectified icosihepta-heptacontidipeton (rojak)
3							Trirectified 2₂₁ Trirectified icosihepta-heptacontidipeton (harjak)
4							Truncated 2₂₁ Truncated icosihepta-heptacontidipeton (tojak)
5							Cantellated 2₂₁ Cantellated icosihepta-heptacontidipeton

#	Coxeter plane graphs							Coxeter diagram Names
#	Aut(E₆) [18]	E₆ [12]	D₅ [8]	D₄ / A₂ [6]	A₅ [6]	D₆ / A₄ [10]	D₃ / A₃ [4]	Coxeter diagram Names
6								1₂₂ Pentacontatetrapeton (mo)
7								Rectified 1₂₂ / Birectified 2₂₁ Rectified pentacontatetrapeton (ram)
8								Birectified 1₂₂ Birectified pentacontatetrapeton (barm)
9								Truncated 1₂₂ Truncated pentacontatetrapeton (tim)

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Klitzing, Richard. "6D uniform polytopes (polypeta)".

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

@@ Line 6: / Line 6: @@
 |[[File:up 1 22 t0 E6.svg|160px]]<BR>[[1 22 polytope|1<sub>22</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
 |}
-In 6-dimensional [[geometry]], there are 39 [[uniform 6-polytope|uniform polytopes]] with E<sub>6</sub> symmetry. The two simplest forms are the [[2_21 polytope|2<sub>21</sub>]] and [[1_22 polytope|1<sub>22</sub>]] polytopes, composed of 27 and 72 [[vertex (geometry)|vertices]] respectively.
+In 6-dimensional [[geometry]], there are 39 [[uniform 6-polytope|uniform polytopes]] with E<sub>6</sub> symmetry. The two simplest forms are the [[2 21 polytope|2<sub>21</sub>]] and [[1 22 polytope|1<sub>22</sub>]] polytopes, composed of 27 and 72 [[vertex (geometry)|vertices]] respectively.
 They can be visualized as symmetric [[orthographic projection]]s in [[Coxeter plane]]s of the E<sub>6</sub> Coxeter group, and other subgroups.
@@ Line 15: / Line 15: @@
 Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E<sub>6</sub> symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.
+{{Clear}}
 {| class="wikitable"
 |-
 !rowspan=2|#
-!colspan=5|[[Coxeter plane]] graphs
+!colspan=6|[[Coxeter plane]] graphs
 !rowspan=2|[[Coxeter diagram]]<br />Names
 |-
-!Alt(E<sub>6</sub>)<BR>[9]
+!Aut(E<sub>6</sub>)<BR>[18/2]
 !E<sub>6</sub><BR>[12]
 !D<sub>5</sub><BR>[8]
@@ Line 43: / Line 43: @@
 |-
 !rowspan=2|#
-!colspan=6|[[Coxeter plane]] graphs
+!colspan=7|[[Coxeter plane]] graphs
 !rowspan=2|[[Coxeter diagram]]<br />Names
 |-
@@ Line 58: / Line 58: @@
 |7||||[[File:up 2_21_t2_E6.svg|80px]]||[[File:up 2_21_t2_D5.svg|80px]]||[[File:up 2_21_t2_D4.svg|80px]]||[[File:up 2_21_t2_A5.svg|80px]]||[[File:up 2_21_t2_A4.svg|80px]]||[[File:up 2_21_t2_D3.svg|80px]]||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}<BR>[[Rectified 1 22|Rectified 1<sub>22</sub>]] /  Birectified 2<sub>21</sub><BR>Rectified pentacontatetrapeton (ram)
 |-  style="text-align:center; background:#e0f0e0;"
-|8||||[[File:up 1_22_t2_E6.svg|80px]]||[[File:up 1_22_t2_D5.svg|80px]]||[[File:up 1_22_t2_D4.svg|80px]]||[[File:up 1_22_t2_A5.svg|80px]]||[[File:up 1_22_t2_A4.svg|80px]]||[[File:up 1_22_t2_D3.svg|80px]]||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}<BR>[[Bicantellated 2 21|Bicantellated 2<sub>21</sub>]] / Birectified 1<sub>22</sub><BR>Birectified pentacontatetrapeton (barm)
+|8||||[[File:up 1_22_t2_E6.svg|80px]]||[[File:up 1_22_t2_D5.svg|80px]]||[[File:up 1_22_t2_D4.svg|80px]]||[[File:up 1_22_t2_A5.svg|80px]]||[[File:up 1_22_t2_A4.svg|80px]]||[[File:up 1_22_t2_D3.svg|80px]]||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}<BR>[[Birectified 1 22|Birectified 1<sub>22</sub>]]<BR>Birectified pentacontatetrapeton (barm)
 |-  style="text-align:center; background:#e0f0e0;"
 |9||||[[File:up 1_22_t01_E6.svg|80px]]||[[File:up 1_22_t01_D5.svg|80px]]||[[File:up 1_22_t01_D4.svg|80px]]||[[File:up 1_22_t01_A5.svg|80px]]||[[File:up 1_22_t01_A4.svg|80px]]||[[File:up 1_22_t01_D3.svg|80px]]||{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}<BR>[[Truncated 1 22|Truncated 1<sub>22</sub>]]<BR>Truncated pentacontatetrapeton (tim)
@@ Line 66: / Line 66: @@
 * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
 ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
-* '''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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter]
+* '''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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter]
-** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
+** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10]
 ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
 ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]