Indicwiki: 1 revision imported from :alpha:Template:Regular_convex_4-polytopes

2023-08-18T02:06:31Z

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alpha>Indicwiki: Created page with "{| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes {{#if:{{{radius|..."

2023-07-26T10:28:44Z

Created page with "{| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes {{#if:{{{radius|..."

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{| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;"
!colspan=8|[[Regular 4-polytopes|Regular convex 4-polytopes]] {{#if:{{{radius|}}}|of radius {{{radius|}}}|}}
|-
!align=right|[[Coxeter_group|Symmetry group]]
|align=center|[[Tetrahedral symmetry|A4]]
|align=center colspan=2|[[Hyperoctahedral_group|B4]]
|align=center|[[F4_(mathematics)|F4]]
|align=center colspan=2|[[H4_polytope|H4]]
|-
!valign=top align=right|Name
|valign=top align=center|[[5-cell]] 
Hyper-[[tetrahedron]] 
5-point
|valign=top align=center|[[16-cell]] 
Hyper-[[octahedron]] 
8-point
|valign=top align=center|[[8-cell]] 
Hyper-[[cube]] 
16-point
|valign=top align=center|[[24-cell]] 
 24-point
|valign=top align=center|[[600-cell]] 
Hyper-[[Regular icosahedron|icosahedron]] 
120-point
|valign=top align=center|[[120-cell]] 
Hyper-[[Regular dodecahedron|dodecahedron]] 
600-point
|-
!align=right|[[Schläfli symbol]]
|align=center|{3, 3, 3}
|align=center|{3, 3, 4}
|align=center|{4, 3, 3}
|align=center|{3, 4, 3}
|align=center|{3, 3, 5}
|align=center|{5, 3, 3}
|-
!align=right|[[Coxeter diagram|Coxeter mirrors]]
|align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}
|align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}}
|align=center|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}}
|align=center|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
|align=center|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}
|align=center|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}
|-
!align=right|Mirror dihedrals
|align=center|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|align=center|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|align=center|{{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|align=center|{{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|align=center|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|5}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|align=center|{{sfrac|𝝅|5}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|-
!valign=top align=right|Graph
|align=center|[[Image:4-simplex t0.svg|120px]]
|align=center|[[Image:4-cube t3.svg|120px]]
|align=center|[[Image:4-cube t0.svg|120px]]
|align=center|[[Image:24-cell t0 F4.svg|120px]]
|align=center|[[Image:600-cell graph H4.svg|120px]]
|align=center|[[Image:120-cell graph H4.svg|120px]]
|-
!align=right|Vertices
|align=center|5 tetrahedral
|align=center|8 octahedral
|align=center|16 tetrahedral
|align=center|24 cubical
|align=center|120 icosahedral
|align=center|600 tetrahedral
|-
!valign=top align=right|[[120-cell#Chords|Edges]]
|align=center|10 triangular
|align=center|24 square
|align=center|32 triangular
|align=center|96 triangular
|align=center|720 pentagonal
|align=center|1200 triangular
|-
!valign=top align=right|Faces
|align=center|10 triangles
|align=center|32 triangles
|align=center|24 squares
|align=center|96 triangles
|align=center|1200 triangles
|align=center|720 pentagons
|-
!valign=top align=right|Cells
|align=center|5 tetrahedra
|align=center|16 tetrahedra
|align=center|8 cubes
|align=center|24 octahedra
|align=center|600 tetrahedra
|align=center|120 dodecahedra
|-
!valign=top align=right|[[600-cell#Clifford parallel cell rings|Tori]]
|align=center|1 [[5-cell#Boerdijk–Coxeter helix|5-tetrahedron]]
|align=center|2 [[16-cell#Helical construction|8-tetrahedron]]
|align=center|2 [[8-cell#Construction|4-cube]]
|align=center|4 [[24-cell#Cell rings|6-octahedron]]
|align=center|20 [[600-cell#Boerdijk–Coxeter helix rings|30-tetrahedron]]
|align=center|12 [[120-cell#Intertwining rings|10-dodecahedron]]
|-
!valign=top align=right|Inscribed
|align=center|120 in 120-cell
|align=center|675 in 120-cell
|align=center|2 16-cells
|align=center|3 8-cells
|align=center|25 24-cells
|align=center|10 600-cells
|-
!valign=top align=right|[[Great circle|Great polygons]]
|align=center|
|align=center|2 [[16-cell#Coordinates|squares]] x 3
|align=center|4 rectangles x 4
|align=center|4 [[24-cell#Hexagons|hexagons]] x 4
|align=center|12 [[600-cell#Geodesics|decagons]] x 6
|align=center|100 [[120-cell#Chords|irregular hexagons]] x 4
|-
!valign=top align=right|[[Petrie polygon]]s
|align=center|1 [[5-cell#Boerdijk–Coxeter helix|pentagon]]
|align=center|1 [[16-cell#Helical construction|octagon]]
|align=center|2 [[Octagon#Skew octagon|octagon]]s
|align=center|2 [[Dodecagon#Skew dodecagon|dodecagon]]s
|align=center|4 [[30-gon#Petrie polygons|30-gon]]s
|align=center|20 [[30-gon#Petrie polygons|30-gon]]s
|-
!valign=top align=right|{{#ifeq:{{{radius|}}}|1|[[24-cell#Hexagons|Long radius]]|{{#ifeq:{{{radius|}}}|{{radic|2}}|[[24-cell#Squares|Long radius]]|Long radius}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2}</math>}}}}
|-
!valign=top align=right|Edge length
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\sqrt{\tfrac{5}{2}} \approx 1.581</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{5} \approx 2.236</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\sqrt{2} \approx 1.414</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>2</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2} \approx 1.414</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{2} \approx 1.414</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{1}{\phi} \approx 0.618</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\sqrt{2}}{\phi} \approx 0.874</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>2-\phi \approx 0.382</math>}}}}
|-
!valign=top align=right|Short radius
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{1}{4}</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\sqrt{2}}{4} \approx 0.354</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{1}{2}</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\sqrt{2}}{2} \approx 0.707</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{1}{2}</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\sqrt{2}}{2} \approx 0.707</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\sqrt{\tfrac{1}{2}} \approx 0.707</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>1</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{\tfrac{\phi^4}{4}} \approx 1.309</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\sqrt{\tfrac{\phi^4}{4}} \approx 1.309</math>}}}}
|-
!valign=top align=right|Area
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>10\left(\tfrac{5\sqrt{3}}{8}\right) \approx 10.825</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>10\left(\tfrac{5\sqrt{3}}{4}\right) \approx 21.651</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>32\left(\sqrt{\tfrac{3}{4}}\right) \approx 27.713</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>32\left(\sqrt{3}\right) \approx 55.425</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>24</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>48</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>96\left(\sqrt{\tfrac{3}{16}}\right) \approx 41.569</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>96\left(\sqrt{\tfrac{3}{4}}\right) \approx 83.138</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1200\left(\tfrac{\sqrt{3}}{4\phi^2}\right) \approx 198.48</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>1200\left(\tfrac{2\sqrt{3}}{4\phi^2}\right) \approx 396.95</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{8\phi^4}\right) \approx 90.366</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{4\phi^4}\right) \approx 180.73</math>}}}}
|-
!valign=top align=right|Volume
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>5\left(\tfrac{5\sqrt{5}}{24}\right) \approx 2.329</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>5\left(\tfrac{5\sqrt{10}}{12}\right) \approx 6.588</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>16\left(\tfrac{1}{3}\right) \approx 5.333</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>16\left(\tfrac{2\sqrt{2}}{3}\right) \approx 15.085</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>8</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>8\sqrt{8} \approx 22.627</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>24\left(\tfrac{\sqrt{2}}{3}\right) \approx 11.314</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>24\left(\tfrac{4}{3}\right) = 32</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>600\left(\tfrac{\sqrt{2}}{12\phi^3}\right) \approx 16.693</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>600\left(\tfrac{4}{12\phi^3}\right) \approx 47.214</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6\sqrt{8}}\right) \approx 18.118</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6}\right) \approx 51.246</math>}}}}
|-
!valign=top align=right|4-Content
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{\sqrt{5}}{24}\left(\tfrac{\sqrt{5}}{2}\right)^4 \approx 0.146</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\sqrt{5}}{24}\left(\sqrt{5}\right)^4 \approx 2.329</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{2}{3} \approx 0.667</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{8}{3} \approx 2.666</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>1</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>4</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>2</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>8</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{\text{Short}\times\text{Vol}}{4} \approx 3.863</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\text{Short}\times\text{Vol}}{4} \approx 15.451</math>}}}}
|align=center|{{#ifeq:{{{radius|1}}}|1|<math>\tfrac{\text{Short}\times\text{Vol}}{4} \approx 4.193</math>|{{#ifeq:{{{radius}}}|{{radic|2}}|<math>\tfrac{\text{Short}\times\text{Vol}}{4} \approx 16.770</math>}}}}
|}<noinclude>
[[Category:Geometry templates]]
</noinclude>

Template:Regular convex 4-polytopes - Revision history

Indicwiki: 1 revision imported from :alpha:Template:Regular_convex_4-polytopes

alpha>Indicwiki: Created page with "{| class="wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}" style="white-space:nowrap;" !colspan=8|Regular convex 4-polytopes {{#if:{{{radius|..."