Snub 24-cell

The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking even permutations of

(0, ±1, ±φ, ±φ²)

where φ = 1+√5/2 ≈ 1.618 is the golden ratio.

The unit-radius coordinates of the snub 24-cell, with edges of length φ⁻¹ ≈ 0.618, are the even permutations of

(±φ/2, ±1/2, ±φ⁻¹/2, 0)

These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell in the golden ratio in a consistent manner dimensionally analogous to the way the 12 vertices of an icosahedron or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This can be done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector.[4] This is equivalent to the snub truncation construction of the 24-cell described below.

The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the 600-cell.

The snub 24-cell is derived from the 24-cell by a special form of truncation.

Truncations remove vertices by cutting through the edges incident to the vertex; forms of truncation differ by where on the edge the cut is made. The common truncations of the 24-cell include the recitified 24-cell (which cuts each edge at its midpoint, producing a polytope bounded by 24 cubes and 24 cuboctahedra), and the truncated 24-cell (which cuts each edge one-third of its length from the vertex, producing a polytope bounded by 24 cubes and 24 truncated octahedra). In these truncations a cube is produced in place of the removed vertex, because the vertex figure of the 24-cell is a cube and the cuts are equidistant from the vertex.

The snub truncation of the 24-cell[4] cuts each edge into two golden sections (such that the larger section is in the golden ratio ~1.618 to the smaller section, and the original edge is in the golden ratio to the larger section). The cut must be made in alternate directions on alternate edges incident to each vertex, in order to have a coherent result. The edges incident to a vertex in the 24-cell are the 8 radii of its cubical vertex figure. The only way to choose alternate radii of a cube is to choose the four radii of a tetrahedron (inscribed in the cube) to be cut at the smaller section of their length from the vertex, and the opposite four radii (of the other tetrahedron that can be inscribed in the cube) to be cut at the larger section of their length from the vertex. There are of course two ways to do this; both produce a cluster of five regular tetrahedra in place of the removed vertex, rather than a cube.

This construction has an analogy in 3 dimensions: the construction of the icosahedron (the "snub octahedron") from the octahedron, by the same method.[5] That is how the snub-24 cell's icosahedra are produced from the 24-cell's octahedra during truncation.

The snub 24-cell is related to the truncated 24-cell by an alternation operation. Half the vertices are deleted, the 24 truncated octahedron cells become 24 icosahedron cells, the 24 cubes become 24 tetrahedron cells, and the 96 deleted vertex voids create 96 new tetrahedron cells.

Orthogonal projection, F₄ Coxeter plane

Snub 24-cell	600-cell

The snub 24-cell may also be constructed by a particular diminishing of the 600-cell: by removing 24 vertices from the 600-cell corresponding to those of an inscribed 24-cell, and then taking the convex hull of the remaining vertices. This is equivalent to removing 24 icosahedral pyramids from the 600-cell.

Conversely, the 600-cell may be constructed from the snub 24-cell by augmenting it with 24 icosahedral pyramids.

Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits ${\displaystyle O(\Lambda )=W(H_{4})=I}$ of order 120.[6] The following describe ${\displaystyle T}$ and ${\displaystyle T'}$ 24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):

O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}

O(1000) : V1

O(0010) : V2

O(0001) : V3

With quaternions ${\displaystyle (p,q)}$ where ${\displaystyle {\bar {p}}}$ is the conjugate of ${\displaystyle p}$ and ${\displaystyle [p,q]:r\rightarrow r'=prq}$ and ${\displaystyle [p,q]^{*}:r\rightarrow r''=p{\bar {r}}q}$, then the Coxeter group ${\displaystyle W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace }$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given ${\displaystyle p\in T}$ such that ${\displaystyle {\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p}$ and ${\displaystyle p^{\dagger }}$ as an exchange of ${\displaystyle -1/\varphi \leftrightarrow \varphi }$ within ${\displaystyle p}$, we can construct the snub 24-cell as ${\displaystyle S=\sum _{i=1}^{4}\oplus p^{i}T}$

A net of the snub 24-cell with blue icosahedra, and red and yellow tetrahedra.

The icosahedral cells fit together face-to-face leaving voids between them filled by clusters of five tetrahedral cells.[7]

Each icosahedral cell is joined to 8 other icosahedral cells at 8 triangular faces in the positions corresponding to an inscribing octahedron. The remaining triangular faces are joined to tetrahedral cells, which occur in pairs that share an edge on the icosahedral cell.

The tetrahedral cells may be divided into two groups, of 96 yellow cells and 24 red cells respectively (as colored in the net illustration). Each yellow tetrahedral cell is joined via its triangular faces to 3 blue icosahedral cells and one red tetrahedral cell, while each red tetrahedral cell is joined to 4 yellow tetrahedra. Thus, the tetrahedral cells occur in clusters of five (four yellow cells face-bonded around a red central one, each red/yellow pair lying in a different hyperplane). The red central tetrahedron of the five shares each of its six edges with a different icosahedral cell, and with the pair of yellow tetrahedral cells which shares that edge on the icosahedral cell.

The snub 24-cell has three vertex-transitive colorings based on a Wythoff construction on a Coxeter group from which it is alternated: F₄ defines 24 interchangeable icosahedra, while the B₄ group defines two groups of icosahedra in an 8:16 counts, and finally the D₄ group has 3 groups of icosahedra with 8:8:8 counts.[8]

Symmetry (order)	Constructive name	Coxeter-Dynkin diagram Extended Schläfli symbol	Vertex figure (Tridiminished icosahedron)	Cells (Colored as faces in vertex figures)
1/2F4 [3+,4,3] (576)	Snub 24-cell	s{3,4,3}		One set of 24 icosahedra (blue) Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1/2B4 [(3,3)+,4] (192)	Snub rectified 16-cell	sr{3,3,4}		Two sets icosahedra: 8, 16 each (red and blue) Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1/2D4 [31,1,1]+ (96)	Omnisnub demitesseract	s{31,1,1}		Three sets of 8 icosahedra (red, green, and blue) Two sets of tetrahedra: 96 (yellow) and 24 (cyan)