Editing 600-cell (section)

===== Decagons =====
[[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six]] configuration 30<sub>2</sub>12<sub>5</sub> characteristic of the H<sub>4</sub> polytopes.{{^|Efn|name=Schläfli double six}} The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]]
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.

Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}}
Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations.
The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}}

The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[Hopf fibration]]s make the 600-cell a [[Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30<sub>2</sub>12<sub>5</sub>.{{^|Efn|name=Schläfli double six}}
It is called the [[Schläfli double six]] configuration after [[Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3<sup>2, 2, 1</sup>] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}