Editing 600-cell (section)

==== Golden chords ====
[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]]
The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]].
The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc.
The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
{{see also|24-cell#Hypercubic chords}}
The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏<sup>−1</sup>) beginning with a vertex; see column ''a''}} at eight different [[Chord (geometry)|chord]] lengths from each other.
These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}}
In ascending order of length, they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=The fractional square roots are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <math>\tfrac{1}{\phi} = \phi^{-1}</math>
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽<sup>2</sup> ≈ 0.382<br>
For example:
{{indent|7}}𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618|name=fractional square roots|group=}}

Notice that the four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons.
The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[golden ratio]] <big>φ</big> {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to <big>𝜋</big>:<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=John Carlos Baez|access-date=10 October 2022}}</ref><br>
: {{sfrac|𝜋|5}} = arccos ({{sfrac|φ|2}})
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing <big>φ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br>
: <big>φ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the <big>φ</big> = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is <big>φ</big>, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the <big>φ</big> chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <big>φ</big>{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}