120-cell: Difference between revisions

The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords and the regularform polygons they formthat lie in planes of four kinds: great squaresquares ☐ planes [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagonhexagons and great triangletriangles △ planes [[24-cell#Hexagons|characteristic of the 24-cell]], great decagondecagons and great pentagonpentagons 𝜙 planes [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and pentagramskew pentagrams ✩ faceor planesdecagrams which[[5-cell#Geodesics doand notrotations|characteristic lieof inthe any5-cell]] onewhich central plane, but on a skeware Petrie polygonpolygons that circlescircle through a set of △ central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes.; Rotationsrotations of this type are an expression of the [[Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[24-cell#Hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] and the 120-cell{{Efn|name=120-cell characteristic rotation}} are skew, and do not lie in a single central plane; they describe a skew pentagram ✩ (or compound pentagramdecagram) and only form polygons which occupy face planes butpolygons, not central planes.{{Efn|name=irregularpolygons; greatrotations dodecagon}}of Thethis 120-cell's pentagram ✩ rotationstype are expressions of all four symmetry groups includingthe [[Tetrahedral symmetry|<math>A_4</math>]], [[Hyperoctahedralsymmetry group|<math>B_4</math>]], [[F4 (mathematics)|<math>F_4</math>]] and [[H4 polytope|<math>H_4</math>]].|name=edge rotation planes}}

|rowspan=2|#1<br><big>✩</big>△

|rowspan=2|#9<br><big>𝜙</big><big>✩</big>

|rowspan=2|#12<br><big>𝜙</big><big>✩</big>