Editing 5-cell (section)

== Construction ==
=== As a Boerdijk–Coxeter helix ===
[[File:5-cell 5-ring net.png|480px|thumb|The 5-cell Boerdijk&ndash;Coxeter helix]]
A 5-cell can be constructed as a [[Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}}{{Failed verification|date=March 2025|reason=Banchoff 2013 describes the decomposition of the 8-cell and 24-cell into tori, but neither discusses the 5-cell nor anything relating to Boerdijk–Coxeter helices.}} The 10 triangle faces can be seen in a 2D net within a [[triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[pentagon#Regular pentagons|regular pentagon]] which is the [[Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].

=== Net ===
[[File:5-cell net.png|thumb|right|[[Net (polyhedron)|Net of five tetrahedra (one hidden)]]]]
When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges, and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. This makes the six tetrahedron as its [[Cell (geometry)|cell]].{{sfn|Akiyama|Hitotumatu|Sato|2012}}

===Coordinates===
The simplest set of [[Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates:

:<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi}  \right)/(\tfrac{1}{\phi}-2)</math>

The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space:

:<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math>
:<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left( 0, 0, 0,\frac{4}\sqrt{5}  \right)</math>

Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates:

:<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5},  -1 \right)/4</math>
:<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math>
:<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math>
:<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math>
:<math>\left( 0, 0, 0, 1 \right)</math>

Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are:

:<math>\left( \frac{1}{\sqrt{10}},\  \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left( \frac{1}{\sqrt{10}},\  \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)</math>
:<math>\left( \frac{1}{\sqrt{10}},\  -\sqrt{\frac{3}{2}},\ 0,\                   0   \right)</math>
:<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\                   0,\                   0   \right)</math>

Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives:
:<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math>
:<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math>
:<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math>
:<math>\left(-1, 0, 0, 0 \right)</math>

The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[facet (geometry)|facet]] of, respectively, the [[5-orthoplex]] or the [[rectified penteract]].