Editing 600-cell (section)

==== Characteristic orthoscheme ====
{| class="wikitable floatright"
!colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}}
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!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
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!align=right|𝒍
|align=center|<small><math>\tfrac{1}{\phi} \approx 0.618</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
|align=center|<small>164°29′</small>
|align=center|<small><math>\pi-2\text{𝟁}</math></small>
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!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
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!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
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!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
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!align=right|<small><math>_0R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
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!align=right|<small><math>_1R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
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!align=right|<small><math>_2R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
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!align=right|<small><math>_0R^4/l</math></small>
|align=center|<small><math>1</math></small>
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!align=right|<small><math>_1R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}} \approx 0.951</math></small>
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!align=right|<small><math>_2R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</math></small>
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!align=right|<small><math>_3R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math></small>
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!align=right|<small><math>\text{𝜼}</math></small>
|align=center|
|align=center|<small>37°44′40″</small>
|align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small>
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Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[orthoscheme]] is a [[chiral]] irregular [[simplex]] with [[right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[facet (geometry)|facet]]s (its ''mirror walls'').
Every regular polytope can be dissected radially into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center.
The characteristic orthoscheme has the shape described by the same [[Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}
The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[Coxeter-Dynkin diagram]] {{CDD|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets.
It is an irregular [[Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]].
The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius.
The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center.
Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme.
The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length <small><math>\text{𝒍} = \tfrac{1}{\phi} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell).
The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.