|
|
| Line 1: |
Line 1: |
| {| align=right class=wikitable width=300
| | This entry was posted in Uncategorized on uly,by wps. Low cost Ray Ban New Launch Sun shades Sale RB . low cost ray bans Leae a reply The H includes a healed situation contained inside a black ceramic situation.<br><br> |
| |+ Graphs of three [[List of regular polytopes#Ten Dimensions|regular]] and related [[uniform polytope]]s.
| |
| |-
| |
| | || || || || || || || || || ||
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:10-simplex t0.svg|100px]]<br />[[10-simplex]]</small>
| |
| |colspan=4|[[File:10-simplex t01.svg|100px]]<br /><small>[[Truncated 10-simplex]]</small>
| |
| |colspan=4|[[File:10-simplex t1.svg|100px]]<br /><small>[[Rectified 10-simplex]]</small>
| |
| |- align=center valign=top
| |
| |colspan=6|[[File:10-simplex t02.svg|150px]]<br /><small>[[Cantellated 10-simplex]]</small>
| |
| |colspan=6|[[File:10-simplex t03.svg|150px]]<br /><small>[[Runcinated 10-simplex]]</small>
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:10-simplex t04.svg|100px]]<br /><small>[[Stericated 10-simplex]]</small>
| |
| |colspan=4|[[File:10-simplex t05.svg|100px]]<br /><small>[[Pentallated 10-simplex]]</small>
| |
| |colspan=4|[[File:10-simplex t06.svg|100px]]<br /><small>[[Hexicated 10-simplex]]</small>
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:10-simplex t07.svg|100px]]<br />[[Heptellated 10-simplex]]
| |
| |colspan=4|[[File:10-simplex t08.svg|100px]]<br />[[Octellated 10-simplex]]
| |
| |colspan=4|[[File:10-simplex t09.svg|100px]]<br />[[Ennecated 10-simplex]]
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:10-orthoplex.svg|100px]]<br />[[10-orthoplex]]
| |
| |colspan=4|[[File:Truncated 10-orthoplex.png|100px]]<br />[[Truncated 10-orthoplex]]
| |
| |colspan=4|[[File:Rectified decacross.png|100px]]<br />[[Rectified 10-orthoplex]]
| |
| |- align=center valign=top
| |
| |colspan=4|[[File:10-cube.svg|100px]]<br />[[10-cube]]
| |
| |colspan=4|[[File:Truncated 10-cube.png|100px]]<br />[[Truncated 10-cube]]
| |
| |colspan=4|[[File:Rectified 10-cube.png|100px]]<br />[[Rectified 10-cube]]
| |
| |- align=center valign=top
| |
| |colspan=6|[[File:10-demicube.svg|150px]]<br />[[10-demicube]]
| |
| |colspan=6|[[File:Truncated 10-demicube.png|150px]]<br />[[Truncated 10-demicube]]
| |
| |}
| |
| In ten-dimensional [[geometry]], a 10-polytope is a 10 dimensional [[polytope]] contained by 9-polytope facets. Each [[8-polytope]] [[Ridge (geometry)|ridge]] being shared by exactly two [[9-polytope]] [[Facet (mathematics)|facets]].
| |
|
| |
|
| A '''uniform 10-polytope''' is one which is [[vertex-transitive]], and constructed from [[uniform 9-polytope|uniform]] [[Facet (geometry)|facets]].
| | can not see how he place disconnect all the magic strings, so I can only see him simply stand in front of Shaw butterfly. ABC held its matches at Atlanta's Municipal Auditorium eery Friday night. Lian Si,<br><br>" he says. iane BondareffAPAs Time Warner Ray Ban Sale subscribers in major marketplaces like New York,Los Angeles and allas go devoid of CBS programming for the th day,neither side seems near to crying uncle.<br><br>Though the dowbside to enable thses is normally some people today belieed genuinely plasticy together with fairly flimsy. If you need additional assistance please speak to us. On uly,the Washington Moments posted my Op Ed: CUTLER: Tricky questions about immigration 'reform' Who will uncover the answers though there is nevertheless time to preent a catastrophe.<br><br>ray ban Menu Skip to content material HomeSample Site on neer had the experience of its ray ban sunglasses Could on neer hearts amazed,low cost ray ban sunglasses concealed weapon can be mentioned without the need of untrue hair, neer experienced it not hit scenarios. Sae: off Reiews There are at the moment no product reiews.<br><br>The people assembled there are not radicals,but liberals: "superior burghers respectable suburbanites. There are means you can make sure [http://tinyurl.com/ode2fpd http://www.accessrentacar.com.au/rayban.asp] that the product or service will be deliered. Very good jogging sneakers are Ray Ban Sunglasses Outlet,<br><br>associates Comment Likes Social Cycle Group associates Comments Likes See All Films rier and Bicycle owner Security Recommendations Added by Andrew Abbess Responses Likes yr old on Rostreor MTB trails Additional by Andrew Abbess Responses Likes Edinburgh Trams Security Cycling with the Trams Additional by Andrew Abbess Responses Likes Increase Videos See All Discussion board My reaction to the Town Centre Session Started out by Murray Thomson ul , . Replies Likes Here's my response to the Town Centre Session becoming carried out by Stirling Council which I will be e mailing to Alistair awson. Be sure to discover down below my reaction to your consultation on changes ofContinue Cycle parking at Stirling station Started out by Philippa Cochrane.<br><br>aid Plouffe,the campaign supervisor for Obama's work and a best adiser for the duration of his first time period,tweeted: "Not an Angeleno but will neer ignore Eric Garcetti bundled up in the course of frigid very last days of Iowa,canassing relentlessly for then Sen Obama.<br><br>It will not just take Sigmund Freud to see in William's action a direct obstacle to Prince Charles's failure to take action in opposition to the photographers who hounded his mom to her death. I suggest,I can name seeral mass taking pictures eents due to the fact August by yourself due to the fact Aurora. Charles would hae preferred the spouse and children to dismiss the images,<br><br>concise fashion and very good top quality, no question grow to be the ray ban this brand name as a result of almost decades after enduring nonetheless critical factors. This entry was posted in faux ray bans and tagged pretend ray ban sun shades on February,by admin.<br><br>agreed. Maps Work For the Neighborhood Flexibility of Facts Act Foundations Basic Facts Get hold of us bhcSae Off Low-priced Ray Ban Sun shades Outlet Online. You can observe any responses to this entry by way of RSS . Announcements Campus bookstores closed for inentory une (QC),<br><br>this is my career. For Homer the hero was a battler who deals in loss of life and seeks immortality through alor. for Pindar the hero was poised in glory amongst gods and guys. But this lady cannot wait around for food stuff.<br><br>we'e had some sort of outside the house pressure. Van der Sloot admitted to killing Flores,but blamed posttraumatic anxiety problem for being accused of Holloway's disappearance for building him eliminate the Peruian. This year,<br><br>getting rid of weight way too immediately can be perilous. ear when the countryside arrived to a seere listening to impairment, modest point oice what they listen to not, but see every single other lips flaring,<br><br>all lenses are equipped to block harmful ultraiolet (u) and one particular hundred percent and screening of infrared mild and other destructive rays. (Andrew HarrerBloomberg,ia Getty)What's Within America's Banks. Experimenting with his wife and anny went horribly incorrect after him shedding some layers of flesh on his nipple.<br><br>so the presence of individuals amazing. To observe bubble undesirable four piano aged gentleman instructed reporters that he was brought up like swimming, piano, observing anyone who floats on the h2o for a extended time to get the Guinness E-book of Documents,<br><br>reproduction sunglasses united kingdom, Uncategorized affordable Ray Ban Wayfarer Pure White With Brand Beside Sunglasses wholesale united kingdom cheap ray ban wayfarers coolest ray bans faux Ray Ban unior Gray Black Frame With Brand Besid Sunglasses wholesale united kingdom wholesale Ray Ban Orange on the internet retail store uk Apr buy rayban wayfarer acquire rayban wayfarer Invest in rayban wayfarer sportswear product sales hae been to the wonderful flood situation, obtain rayban wayfarer eye catching brand sportswear has grow to be a silly (Cha) their uniforms, in purchase to reerse the image of the substantially necessary injection of trend things.<br><br>If you have any type of inquiries regarding where and how you can use [http://tinyurl.com/ode2fpd http://www.accessrentacar.com.au/rayban.asp], you could call us at the web page. |
| | |
| A [[5-polytope#A note on generality of terms for n-polytopes and elements|proposed]] name for 10-polytope is polyxennon (plural: polyxenna), created from poly- ''xenna'' (a variation on [[ennea]] meaning nine) and -on.
| |
| | |
| == Regular 10-polytopes ==
| |
| | |
| Regular 10-polytopes can be represented by the [[Schläfli symbol]] {p,q,r,s,t,u,v,w,x}, with '''x''' {p,q,r,s,t,u,v,w} 9-polytope [[Facet (mathematics)|facets]] around each [[Peak (geometry)|peak]].
| |
| | |
| There are exactly three such [[List of regular polytopes#Convex 4|convex regular 10-polytopes]]:
| |
| # {3,3,3,3,3,3,3,3,3} - [[10-simplex]]
| |
| # {4,3,3,3,3,3,3,3,3} - [[10-cube]]
| |
| # {3,3,3,3,3,3,3,3,4} - [[10-orthoplex]]
| |
| | |
| There are no nonconvex regular 10-polytopes.
| |
| | |
| == Euler characteristic ==
| |
| | |
| The [[Euler characteristic]] for 10-polytopes that are topological [[9-sphere]]s (including all convex 10-polytopes) is zero. χ=V-E+F-C+f<sub>4</sub>-f<sub>5</sub>+f<sub>6</sub>-f<sub>7</sub>+f<sub>8</sub>-f<sub>9</sub>=0.
| |
| | |
| == Uniform 10-polytopes by fundamental Coxeter groups ==
| |
| | |
| Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the [[Coxeter-Dynkin diagram]]s:
| |
| | |
| {| class=wikitable
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| |-
| |
| |1||A<sub>10</sub>|| [3<sup>9</sup>]||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| |-
| |
| |2||B<sub>10</sub>||[4,3<sup>8</sup>]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| |-
| |
| |3||D<sub>10</sub>||[3<sup>7,1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| |}
| |
| | |
| Selected regular and uniform 10-polytopes from each family include:
| |
| # [[Simplex]] family: A<sub>10</sub> [3<sup>9</sup>] - {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| #* 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
| |
| #*# {3<sup>9</sup>} - '''[[10-simplex]]''' - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| # [[Hypercube]]/[[orthoplex]] family: B<sub>10</sub> [4,3<sup>8</sup>] - {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| #* 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
| |
| #*# {4,3<sup>8</sup>} - '''[[10-cube]]''' or '''dekeract''' - {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| #*# {3<sup>8</sup>,4} - '''[[10-orthoplex]]''' or '''decacross''' - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
| |
| #*# h{4,3<sup>8</sup>} - '''[[10-demicube]]''' {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}.
| |
| # [[Demihypercube]] D<sub>10</sub> family: [3<sup>7,1,1</sup>] - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| #* 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
| |
| #*# '''1<sub>7,1</sub>''' - '''[[10-demicube]]''' or '''demidekeract''' - {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
| |
| #*# '''7<sub>1,1</sub>''' - '''[[10-orthoplex]]''' - {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
| |
| | |
| == The A<sub>10</sub> family ==
| |
| | |
| The A<sub>10</sub> family has symmetry of order 39,916,800 (11 [[factorial]]).
| |
| | |
| There are 512+16-1=527 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing. | |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Graph
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name
| |
| !colspan=10|Element counts
| |
| |-
| |
| || 9-faces|| 8-faces|| 7-faces|| 6-faces|| 5-faces|| 4-faces|| Cells|| Faces|| Edges|| Vertices
| |
| |-
| |
| |- align=center
| |
| !1
| |
| |[[File:10-simplex t0.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0</sub>{3,3,3,3,3,3,3,3,3}<br />[[10-simplex]] (ux)
| |
| |11||55||165||330||462||462||330||165||55||11
| |
| |- align=center
| |
| !2
| |
| |[[File:10-simplex t1.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub>1</sub>{3,3,3,3,3,3,3,3,3}<br />[[Rectified 10-simplex]] (ru)
| |
| || || || || || || || || ||495 ||55
| |
| |- align=center
| |
| !3
| |
| |[[File:10-simplex t2.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t<sub>2</sub>{3,3,3,3,3,3,3,3,3}<br />[[Birectified 10-simplex]] (bru)
| |
| || || || || || || || || ||1980 ||165
| |
| |- align=center
| |
| !4
| |
| |[[File:10-simplex t3.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t<sub>3</sub>{3,3,3,3,3,3,3,3,3}<br />[[Trirectified 10-simplex]] (tru)
| |
| || || || || || || || || ||4620 ||330
| |
| |- align=center
| |
| !5
| |
| |[[File:10-simplex t4.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<br />t<sub>4</sub>{3,3,3,3,3,3,3,3,3}<br />[[Quadrirectified 10-simplex]] (teru)
| |
| || || || || || || || || ||6930 ||462
| |
| |- align=center
| |
| !6
| |
| |[[File:10-simplex t01.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}<br />t<sub>0,1</sub>{3,3,3,3,3,3,3,3,3}<br />[[Truncated 10-simplex]] (tu)
| |
| || || || || || || || || ||550 ||110
| |
| |- align=center
| |
| !7
| |
| |[[File:10-simplex t02.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}<br />t<sub>0,2</sub>{3,3,3,3,3,3,3,3,3}<br />[[Cantellated 10-simplex]]
| |
| || || || || || || || || ||4455 ||495
| |
| |- align=center
| |
| !8
| |
| |[[File:10-simplex t12.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}<br />t<sub>1,2</sub>{3,3,3,3,3,3,3,3,3}<br />[[Bitruncated 10-simplex]]
| |
| || || || || || || || || ||2475 ||495
| |
| |- align=center
| |
| !9
| |
| |[[File:10-simplex t03.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}<br />t<sub>0,3</sub>{3,3,3,3,3,3,3,3,3}<br />[[Runcinated 10-simplex]]
| |
| || || || || || || || || ||15840 ||1320
| |
| |- align=center
| |
| !10
| |
| |[[File:10-simplex t13.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}<br />t<sub>1,3</sub>{3,3,3,3,3,3,3,3,3}<br />[[Bicantellated 10-simplex]]
| |
| || || || || || || || || ||17820 ||1980
| |
| |- align=center
| |
| !11
| |
| |[[File:10-simplex t23.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}<br />t<sub>2,3</sub>{3,3,3,3,3,3,3,3,3}<br />[[Tritruncated 10-simplex]]
| |
| || || || || || || || || ||6600 ||1320
| |
| |- align=center
| |
| !12
| |
| |[[File:10-simplex t04.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,4</sub>{3,3,3,3,3,3,3,3,3}<br />[[Stericated 10-simplex]]
| |
| || || || || || || || || ||32340 ||2310
| |
| |- align=center
| |
| !13
| |
| |[[File:10-simplex t14.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}<br />t<sub>1,4</sub>{3,3,3,3,3,3,3,3,3}<br />[[Biruncinated 10-simplex]]
| |
| || || || || || || || || ||55440 ||4620
| |
| |- align=center
| |
| !14
| |
| |[[File:10-simplex t24.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}<br />t<sub>2,4</sub>{3,3,3,3,3,3,3,3,3}<br />[[Tricantellated 10-simplex]]
| |
| || || || || || || || || ||41580 ||4620
| |
| |- align=center
| |
| !15
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}<br />t<sub>3,4</sub>{3,3,3,3,3,3,3,3,3}<br />[[Quadritruncated 10-simplex]]
| |
| || || || || || || || || ||11550 ||2310
| |
| |- align=center
| |
| !16
| |
| |[[File:10-simplex t05.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,5</sub>{3,3,3,3,3,3,3,3,3}<br />[[Pentellated 10-simplex]]
| |
| || || || || || || || || ||41580 ||2772
| |
| |- align=center
| |
| !17
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub>1,5</sub>{3,3,3,3,3,3,3,3,3}<br />[[Bistericated 10-simplex]]
| |
| || || || || || || || || ||97020 ||6930
| |
| |- align=center
| |
| !18
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}<br />t<sub>2,5</sub>{3,3,3,3,3,3,3,3,3}<br />[[Triruncinated 10-simplex]]
| |
| || || || || || || || || ||110880 ||9240
| |
| |- align=center
| |
| !19
| |
| |[[File:10-simplex t35.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}<br />t<sub>3,5</sub>{3,3,3,3,3,3,3,3,3}<br />[[Quadricantellated 10-simplex]]
| |
| || || || || || || || || ||62370 ||6930
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !20
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}<br />t<sub>4,5</sub>{3,3,3,3,3,3,3,3,3}<br />[[Quintitruncated 10-simplex]]
| |
| || || || || || || || || ||13860 ||2772
| |
| |- align=center
| |
| !21
| |
| |[[File:10-simplex t06.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,6</sub>{3,3,3,3,3,3,3,3,3}<br />[[Hexicated 10-simplex]]
| |
| || || || || || || || || ||34650 ||2310
| |
| |- align=center
| |
| !22
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub>1,6</sub>{3,3,3,3,3,3,3,3,3}<br />[[Bipentellated 10-simplex]]
| |
| || || || || || || || || ||103950 ||6930
| |
| |- align=center
| |
| !23
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t<sub>2,6</sub>{3,3,3,3,3,3,3,3,3}<br />[[Tristericated 10-simplex]]
| |
| || || || || || || || || ||161700 ||11550
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !24
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t<sub>3,6</sub>{3,3,3,3,3,3,3,3,3}<br />[[Quadriruncinated 10-simplex]]
| |
| || || || || || || || || ||138600 ||11550
| |
| |- align=center
| |
| !25
| |
| |[[File:10-simplex t07.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,7</sub>{3,3,3,3,3,3,3,3,3}<br />[[Heptellated 10-simplex]]
| |
| || || || || || || || || ||18480 ||1320
| |
| |- align=center
| |
| !26
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub>1,7</sub>{3,3,3,3,3,3,3,3,3}<br />[[Bihexicated 10-simplex]]
| |
| || || || || || || || || ||69300 ||4620
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !27
| |
| |
| |
| |
| |
| {{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t<sub>2,7</sub>{3,3,3,3,3,3,3,3,3}<br />[[Tripentellated 10-simplex]]
| |
| || || || || || || || || ||138600 ||9240
| |
| |- align=center
| |
| !28
| |
| |[[File:10-simplex t08.svg|60px]]
| |
| |
| |
| {{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,8</sub>{3,3,3,3,3,3,3,3,3}<br />[[Octellated 10-simplex]]
| |
| || || || || || || || || ||5940 ||495
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !29
| |
| |
| |
| |
| |
| {{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub>1,8</sub>{3,3,3,3,3,3,3,3,3}<br />[[Biheptellated 10-simplex]]
| |
| || || || || || || || || ||27720 ||1980
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !30
| |
| |[[File:10-simplex t09.svg|60px]]
| |
| |
| |
| {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub>0,9</sub>{3,3,3,3,3,3,3,3,3}<br />[[Ennecated 10-simplex]]
| |
| || || || || || || || || ||990 ||110
| |
| |- align=center
| |
| !31
| |
| |
| |
| | {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<br />t<sub><small>0,1,2,3,4,5,6,7,8,9</small></sub>{3,3,3,3,3,3,3,3,3}<br />[[Omnitruncated 10-simplex]]
| |
| || || || || || || || || ||199584000||39916800
| |
| |}
| |
| | |
| == The B<sub>10</sub> family ==
| |
| | |
| There are 1023 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings.
| |
| | |
| Twelve cases are shown below: ten single-ring ([[Rectification (geometry)|rectified]]) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Graph
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name
| |
| !colspan=10|Element counts
| |
| |-
| |
| ! 9-faces
| |
| ! 8-faces
| |
| ! 7-faces
| |
| ! 6-faces
| |
| ! 5-faces
| |
| ! 4-faces
| |
| ! Cells
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- align=center
| |
| !1
| |
| |[[File:10-cube t0.svg|60px]]
| |
| | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>0</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[10-cube]] (deker)
| |
| |20||180||960||3360||8064||13440||15360||11520||5120||1024
| |
| |- align=center
| |
| !2
| |
| |[[File:Truncated 10-cube.png|60px]]
| |
| | {{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>0,1</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[Truncated 10-cube]] (tade)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |51200
| |
| |10240
| |
| |- align=center
| |
| !3
| |
| |[[File:10-cube t1.svg|60px]]
| |
| | {{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>1</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[Rectified 10-cube]] (rade)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |46080
| |
| |5120
| |
| |- align=center
| |
| !4
| |
| |[[File:10-cube t2.svg|60px]]
| |
| | {{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>2</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[Birectified 10-cube]] (brade)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |184320
| |
| |11520
| |
| |- align=center
| |
| !5
| |
| |[[File:10-cube t3.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>3</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[Trirectified 10-cube]] (trade)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |322560
| |
| |15360
| |
| |- align=center
| |
| !6
| |
| |[[File:10-cube t4.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}<br />t<sub><small>4</small></sub>{4,3,3,3,3,3,3,3,3}<br />[[Quadrirectified 10-cube]] (terade)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |322560
| |
| |13440
| |
| |- align=center
| |
| !7
| |
| |[[File:10-cube t5.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}<br />t<sub><small>4</small></sub>{3,3,3,3,3,3,3,3,4}<br />[[Quadrirectified 10-orthoplex]] (terake)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |201600
| |
| |8064
| |
| |- align=center
| |
| !8
| |
| |[[File:10-cube t6.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}<br />t<sub><small>3</small></sub>{3,3,3,3,3,3,3,4}<br />[[Trirectified 10-orthoplex]] (trake)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |80640
| |
| |3360
| |
| |- align=center
| |
| !9
| |
| |[[File:10-cube t7.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}<br />t<sub><small>2</small></sub>{3,3,3,3,3,3,3,3,4}<br />[[Birectified 10-orthoplex]] (brake)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |20160
| |
| |960
| |
| |- align=center
| |
| !10
| |
| |[[File:10-cube t8.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}<br />t<sub><small>1</small></sub>{3,3,3,3,3,3,3,3,4}<br />[[Rectified 10-orthoplex]] (rake)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |2880
| |
| |180
| |
| |- align=center
| |
| !11
| |
| |[[File:Truncated 10-orthoplex.png|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}<br />t<sub><small>0,1</small></sub>{3,3,3,3,3,3,3,3,4}<br />[[Truncated 10-orthoplex]] (take)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |3060
| |
| |360
| |
| |- align=center
| |
| !12
| |
| |[[File:10-cube t9.svg|60px]]
| |
| | {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}<br />t<sub><small>0</small></sub>{3,3,3,3,3,3,3,3,4}<br />[[10-orthoplex]] (ka)
| |
| |1024||5120||11520||15360||13440||8064||3360||960||180||20
| |
| |}
| |
| | |
| == The D<sub>10</sub> family ==
| |
| | |
| The D<sub>10</sub> family has symmetry of order 1,857,945,600 (10 [[factorial]] x 2<sup>9</sup>).
| |
| | |
| This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D<sub>10</sub> [[Coxeter-Dynkin diagram]]. Of these, 511 (2×256−1) are repeated from the B<sub>10</sub> family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|Graph
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]<br />Name
| |
| !colspan=10|Element counts
| |
| |-
| |
| ! 9-faces
| |
| ! 8-faces
| |
| ! 7-faces
| |
| ! 6-faces
| |
| ! 5-faces
| |
| ! 4-faces
| |
| ! Cells
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- align=center
| |
| |1||[[File:10-demicube.svg|60px]]||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />[[10-demicube]] (hede)
| |
| |532||5300||24000||64800||115584||142464||122880||61440||11520||512
| |
| |- align=center
| |
| |2||[[File:Truncated 10-demicube.png|60px]]||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}<br />[[Truncated 10-demicube]] (thede)
| |
| | || || || || || || || ||195840 ||23040
| |
| |}
| |
| | |
| == Regular and uniform honeycombs ==
| |
| | |
| There are four fundamental affine [[Coxeter groups]] that generate regular and uniform tessellations in 9-space:
| |
| {| class=wikitable
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| |-
| |
| |1||<math>{\tilde{A}}_9</math>||[3<sup>[10]</sup>]||{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
| |
| |-
| |
| |2||<math>{\tilde{B}}_9</math>||[4,3<sup>7</sup>,4]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
| |
| |-
| |
| |3||<math>{\tilde{C}}_9</math>||h[4,3<sup>7</sup>,4]<br />[4,3<sup>6</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
| |
| |-
| |
| |4||<math>{\tilde{D}}_9</math>||q[4,3<sup>7</sup>,4]<br />[3<sup>1,1</sup>,3<sup>5</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}
| |
| |}
| |
| | |
| Regular and uniform tessellations include:
| |
| * [[List of regular polytopes#Higher dimensions 3|Regular]] [[Hypercubic honeycomb|9-hypercubic honeycomb]], with symbols {4,3<sup>7</sup>,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
| |
| * Uniform [[Alternated hypercubic honeycomb|alternated 9-hypercubic honeycomb]] with symbols h{4,3<sup>7</sup>,4}, {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
| |
| | |
| === Regular and uniform hyperbolic honeycombs ===
| |
| | |
| There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite [[vertex figure]]. However there are [[Coxeter-Dynkin diagram#Rank 4 to 10|3 noncompact hyperbolic Coxeter groups]] of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.
| |
| | |
| {| class=wikitable
| |
| |align=right|<math>{\bar{Q}}_9</math> = [3<sup>1,1</sup>,3<sup>4</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
| |
| |align=right|<math>{\bar{S}}_9</math> = [4,3<sup>5</sup>,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|4a|nodea}}
| |
| |align=right|<math>E_{10}</math> or <math>{\bar{T}}_9</math> = [3<sup>6,2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
| |
| |}
| |
| | |
| Three honeycombs from the <math>E_{10}</math> family, generated by end-ringed Coxeter diagrams are:
| |
| * [[6 21 honeycomb|6<sub>21</sub> honeycomb]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
| |
| * [[2 61 honeycomb|2<sub>61</sub> honeycomb]]: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
| |
| * [[1 62 honeycomb|1<sub>62</sub> honeycomb]]: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
| |
| | |
| == References ==
| |
| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
| |
| * [[Alicia Boole Stott|A. Boole Stott]]: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
| |
| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
| |
| ** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
| |
| ** 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]
| |
| ** (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)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
| |
| * {{KlitzingPolytopes|polyxenna.htm|10D|uniform polytopes (polyxenna)}}
| |
| | |
| == External links ==
| |
| * [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
| |
| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
| |
| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
| |
| * {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
| |
| {{Polytopes}}
| |
| | |
| [[Category:10-polytopes]]
| |