Root system of a semi-simple Lie algebra: Difference between revisions

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|- align=center valign=top
|[[File:6-cube t0.svg|120px]]<BR>[[6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}
|[[File:6-cube t01.svg|120px]]<BR>'''Truncated 6-cube'''<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node|4|node}}
|[[File:6-cube t12.svg|120px]]<BR>'''Bitruncated 6-cube'''<BR>{{CDD|node|4|node_1|3|node_1|3|node|3|node|4|node}}
|rowspan=2 valign=center|[[File:6-cube t23.svg|120px]]<BR>'''Tritruncated 6-cube'''<BR>{{CDD|node|4|node|3|node_1|3|node_1|3|node|4|node}}
|- align=center valign=top
|[[File:6-cube t5.svg|120px]]<BR>[[6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node|4|node_1}}
|[[File:6-cube t45.svg|120px]]<BR>[[Truncated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node_1|4|node_1}}
|[[File:6-cube t34.svg|120px]]<BR>[[Bitruncated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node_1|3|node_1|4|node}}
|-
!colspan=4|[[Orthogonal projection]]s in BC<sub>6</sub> [[Coxeter plane]]
|}
In six-dimensional [[geometry]], a '''truncated 6-cube''' (or '''truncated hexeract''') is a convex [[uniform 6-polytope]], being a [[Truncation (geometry)|truncation]] of the regular [[6-cube]].


There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the [[cube|cubic]] cells of the 6-cube.  
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== Truncated 6-cube ==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Truncated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform polypeton]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t{4,3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||76
|-
|bgcolor=#e7dcc3|4-faces||464
|-
|bgcolor=#e7dcc3|Cells||1120
|-
|bgcolor=#e7dcc3|Faces||1520
|-
|bgcolor=#e7dcc3|Edges||1152
|-
|bgcolor=#e7dcc3|Vertices||384
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||Elongated 5-cell pyramid
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
=== Alternate names===
* Truncated hexeract (Acronym: tox) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3x4x - tox)</ref>
 
===Construction and coordinates===
The truncated 6-cube may be constructed by [[Truncation (geometry)|truncating]] the vertices of the [[6-cube]] at <math>1/(\sqrt{2}+2)</math> of the edge length. A regular [[5-simplex]] replaces each original vertex.
 
The [[Cartesian coordinate]]s of the vertices of a ''truncated 6-cube'' having edge length&nbsp;2 are the permutations of:
 
:<math>\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)</math>
 
=== Images ===
{{6-cube Coxeter plane graphs|t01|150}}
 
=== Related polytopes ===
The ''[[Truncation (geometry)|truncated]] [[6-cube]]'', is fifth in a sequence of truncated [[hypercube]]s:
{{Truncated hypercube polytopes}}
 
== Bitruncated 6-cube ==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Bitruncated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform polypeton]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2t{4,3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
=== Alternate names===
* Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)<ref>Klitzing, (o3o3o3x3x4o - botox)</ref>
 
===Construction and coordinates===
The [[Cartesian coordinate]]s of the vertices of a ''bitruncated 6-cube'' having edge length&nbsp;2 are the permutations of:
:<math>\left(0,\ \pm1,\ \pm2,\ \pm2,\ \pm2,\ \pm2 \right)</math>
 
=== Images ===
{{6-cube Coxeter plane graphs|t12|150}}
 
=== Related polytopes ===
The ''[[bitruncation|bitruncated]] [[6-cube]]'' is fourth in a sequence of bitruncated [[hypercube]]s:
{{Bitruncated hypercube polytopes}}
 
== Tritruncated 6-cube ==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Tritruncated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform polypeton]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 3t{4,3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
=== Alternate names===
* Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)<ref>Klitzing, (o3o3x3x3o4o - xog)</ref>
 
===Construction and coordinates===
The [[Cartesian coordinate]]s of the vertices of a ''tritruncated 6-cube'' having edge length&nbsp;2 are the permutations of:
:<math>\left(0,\ 0,\ \pm1,\ \pm2,\ \pm2,\ \pm2 \right)</math>
 
=== Images ===
{{6-cube Coxeter plane graphs|t23|150}}
 
=== Related polytopes==
{{2-isotopic_uniform_hypercube_polytopes}}
 
== Related polytopes==
 
These polytopes are from a set of 63 [[Uniform polypeton|uniform polypeta]] generated from the B<sub>6</sub> [[Coxeter plane]], including the regular [[6-cube]] or [[6-orthoplex]].
 
{{Hexeract family}}
 
== Notes==
{{reflist}}
 
== References==
 
{{Empty section|date=March 2013}}
 
== References==
* [[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''', editied 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]
*** (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]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
 
== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
*{{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
 
{{Polytopes}}
 
[[Category:6-polytopes]]

Latest revision as of 21:30, 20 November 2014

Making sure you have the correct size is very important; a poor fitting helmet can be uncomfortable and unsafe. Horn: A horn can let the other driver know that you are there. If you go on a regular mountain biking, you will develop a healthier body and disposition. Music sounds better with a background like the Continental Divide, and you simply may find out once and for all, as there are music vacations and concerts happening just about each weekend in the summertime. Just like coasting, you'll also spend a great deal of time pedaling while standing.

It'll have you on the edge of your seat and holding your breath for Zink to land this record-breaking backflip. in summary let me give some procuations you bshould take when ridding it , ridding a bike at high speed can lead to accidents just maintain a minimum and also follow the traffic lights , if it is for sport just make enough practice and enjoy the game. Fitting is logical, the instructions are clear few tools are required. - Get a bike that is lightweight and made out of carbon fiber materials. Then sense the direct route it will take to go from your nose to the target location and then back out through the nose.

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When discount sales are on, you need to be sure you understand the terms and conditions of your purchase, including warranty information. There are also tires with detailed treads which are very prominent and these are very useful for the biker because of the grip they provide. It is primarily design of Billet Motorola clutch kit that replaces your stock clutch within your GPX 150, 160 and additionally YX 140, 150, 160. The dedication and solid experience of the experts, it is not difficult to accomplish such goal. To help you choose which one you to buy, here are the list of bikes and their use.

In the recent article section, you can find the list of biking related articles with a detailed description present on the official website. If you have any queries with regards to exactly where and how to use Popular mountain bike sizing., you can get hold of us at our site. The bike should be examined to check it is working correctly. You can come across cutting edge package deals being incorporated all of the time, together with our normal products which might be our preferred sellers. Whether it's a testament to South Dakota's commitment to recycling, or just whimsy, Bone Collector embodies the features of the best bike trails in the world: Sedona, Moab, Flagstaff, Whistler and Tahoe, without the long drive or expensive plane ticket. Part of Park City's distinction as the International Mountain Bike Association's first and only gold-level Ride Center is its abundance of beginner trails.

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