Snub 24-cell

or or [(3,3)+,4], ⁠1/2⁠B4, order 192 [31,1,1]+, ⁠1/2⁠D4, order 96 In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells.

In total it has 480 triangular faces, 432 edges, and 96 vertices.

One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper.

[2] He called it a tetricosahedric for being made of tetrahedron and icosahedron cells.

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 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 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 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 rectified 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).

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.

[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.

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

24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):

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).

Each of these gaps are filled by 5 tetrahedral cells, not shown here.

Part of the edge outline of one of them, an icosahedral cell, can be discerned here, overlying the yellow tetrahedron.

The Dual snub 24-cell has 144 identical irregular cells.

Each cell has faces of two kinds: 3 kites and 6 isosceles triangles.

The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.

[9] The snub 24-cell can be obtained as a diminishing of the 600-cell at 24 of its vertices, in fact those of a vertex inscribed 24-cell.

There is also a further such bi-diminishing, when the vertices of a second vertex inscribed 24-cell would be diminished as well.

The full snub 24-cell can also be constructed although it is not uniform, being composed of irregular tetrahedra on the alternated vertices.

The snub 24-cell is a part of the F4 symmetry family of uniform 4-polytopes.