Snub 24-cell

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Snub 24-cell
Schlegel half-solid alternated cantitruncated 16-cell.png
Type Uniform 4-polytope
Schläfli symbol [1] s{3,4,3}
sr{3,3,4}
s{31,1,1}
Coxeter-Dynkin
diagrams

CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel 4a.pngCDel nodea.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png or CDel node h.pngCDel splitsplit1.pngCDel branch3 hh.pngCDel node h.png

Cells14496 3.3.3 (oblique)  Tetrahedron.png
24 3.3.3 Tetrahedron.png
24 3.3.3.3.3 Icosahedron.png
Faces480 {3}
Edges432
Vertices96
Vertex figure Snub 24-cell vertex figure.png
(Tridiminished icosahedron)
Symmetry groups [3+,4,3], 1/2F4, order 576

[(3,3)+,4], 1/2B4, order 192
[31,1,1]+, 1/2D4, order 96

Contents

Dual Dual snub 24-cell
Properties convex
Uniform index 30 31 32
Vertex figure: Tridiminished icosahedron
8 faces:

5 3.3.3 and 3 3.3.3.3.3 Tridiminished icosahedron.png
Vertex figure: Tridiminished icosahedron
8 faces:
Tetrahedron vertfig.png Icosahedron vertfig.png
5 3.3.3 and 3 3.3.3.3.3

In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. 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.

Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids.

Semiregular polytope

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 other two are the rectified 5-cell and rectified 600-cell.)

Alternative names

Geometry

Coordinates

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).

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, in much the same way that 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 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.

Constructions

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-cell600-cell
24-cell h01 F4.svg 600-cell t0 F4.svg

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.

Structure

Snub 24-cell-net.png
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.

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.

Symmetry

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. [6]

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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
s{3,4,3}
Snub 24-cell F4-verf.png 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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
sr{3,3,4}
Snub 24-cell C4-verf.png 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 CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
s{31,1,1}
Snub 24-cell B4-verf.png Three sets of 8 icosahedra (red, green, and blue)
Two sets of tetrahedra: 96 (yellow) and 24 (cyan)

Projections

Orthographic projections

orthographic projections
Coxeter plane F4B4
Graph 24-cell h01 F4.svg 24-cell h01 B4.svg
Dihedral symmetry [12]+[8/2]
Coxeter planeD4 / B3 / A2B2 / A3
Graph 24-cell h01 B3.svg 24-cell h01 B2.svg
Dihedral symmetry[6]+[4]
Ortho solid 969-uniform polychoron 343-snub.png
Orthogonal projection
Centered on hyperplane of one icosahedron.

Perspective projections

Perspective projections
Snub24cell-perspective-cell-first-01.png
Perspective projection centered on an icosahedral cell, with 4D viewpoint placed at a distance of 5 times the vertex-center radius. The nearest icosahedral cell is rendered in solid color, and the other cells are in edge-outline. Cells facing away from the 4D viewpoint are culled, to reduce visual clutter.
Snub24cell-perspective-cell-first-02.png
The same projection, now with 4 of the 8 icosahedral cells surrounding the central cell shown in green.
Snub24cell-perspective-cell-first-03.png
The same projection as above, now with the other 4 icosahedral cells surrounding the central cell shown in magenta. The animated version of this image gives a good view on the layout of these cells.

From this particular viewpoint, one of the gaps containing tetrahedral cells can be seen. Each of these gaps are filled by 5 tetrahedral cells, not shown here.

Snub24cell-perspective-cell-first-04.png
Same projection as above, now with the central tetrahedral cell in the gap filled in. This tetrahedral cell is joined to 4 other tetrahedral cells, two of which fills the two gaps visible in this image. The other two each lies between a green tetrahedral cell, a magenta cell, and the central cell, to the left and right of the yellow tetrahedral cell.

Note that in these images, cells facing away from the 4D viewpoint have been culled; hence there are only a total of 1 + 8 + 6 + 24 = 39 cells accounted for here. The other cells lie on the other side of the snub 24-cell. Part of the edge outline of one of them, an icosahedral cell, can be discerned here, overlying the yellow tetrahedron.

Snub24cell-perspective-cell-first-05.png
In this image, only the nearest icosahedral cell and the 6 yellow tetrahedral cells from the previous image are shown.
Snub24cell-perspective-cell-first-06.png
Now the 12 tetrahedral cells joined to the central icosahedral cell and the 6 yellow tetrahedral cells are shown. Each of these cells is surrounded by the central icosahedron and two of the other icosahedral cells shown earlier.
Snub24cell-perspective-cell-first-07.png
Finally, the other 12 tetrahedral cells joined to the 6 yellow tetrahedral cells are shown here. These cells, together with the 8 icosahedral cells shown earlier, comprise all the cells that share at least 1 vertex with the central cell.

Dual

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. [7]

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. Accordingly, this one is known as the bi-24-diminished 600-cell.

D4 uniform polychora
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
CDel node 1.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
CDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
CDel node 1.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
CDel node h.pngCDel splitsplit1.pngCDel branch3 hh.pngCDel node h.png
4-demicube t0 D4.svg 4-cube t1 B3.svg 4-demicube t01 D4.svg 4-cube t12 B3.svg 4-demicube t1 D4.svg 24-cell t2 B3.svg 24-cell t23 B3.svg 24-cell h01 B3.svg
{3,31,1}
h{4,3,3}
2r{3,31,1}
h3{4,3,3}
t{3,31,1}
h2{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
r{3,31,1}
{31,1,1}={3,4,3}
rr{3,31,1}
r{31,1,1}=r{3,4,3}
tr{3,31,1}
t{31,1,1}=t{3,4,3}
sr{3,31,1}
s{31,1,1}=s{3,4,3}

The snub 24-cell is also called a semi-snub 24-cell because it is not a true snub (alternation of an omnitruncated 24-cell). 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 the largest facet of the 4-dimensional honeycomb, the snub 24-cell honeycomb.

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

24-cell family polytopes
Name 24-cell truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell
Schläfli
symbol
{3,4,3}t0,1{3,4,3}
t{3,4,3}
s{3,4,3}t1{3,4,3}
r{3,4,3}
t0,2{3,4,3}
rr{3,4,3}
t1,2{3,4,3}
2t{3,4,3}
t0,1,2{3,4,3}
tr{3,4,3}
t0,3{3,4,3}t0,1,3{3,4,3}t0,1,2,3{3,4,3}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel
diagram
Schlegel wireframe 24-cell.png Schlegel half-solid truncated 24-cell.png Schlegel half-solid alternated cantitruncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Cantel 24cell1.png Bitruncated 24-cell Schlegel halfsolid.png Cantitruncated 24-cell schlegel halfsolid.png Runcinated 24-cell Schlegel halfsolid.png Runcitruncated 24-cell.png Omnitruncated 24-cell.png
F4 24-cell t0 F4.svg 24-cell t01 F4.svg 24-cell h01 F4.svg 24-cell t1 F4.svg 24-cell t02 F4.svg 24-cell t12 F4.svg 24-cell t012 F4.svg 24-cell t03 F4.svg 24-cell t013 F4.svg 24-cell t0123 F4.svg
B4 24-cell t0 B4.svg 4-cube t123.svg 24-cell h01 B4.svg 24-cell t1 B4.svg 24-cell t02 B4.svg 24-cell t12 B4.svg 24-cell t012 B4.svg 24-cell t03 B4.svg 24-cell t013 B4.svg 24-cell t0123 B4.svg
B3(a) 4-cube t0 B3.svg 24-cell t01 B3.svg 24-cell h01 B3.svg 24-cell t1 B3.svg 24-cell t02 B3.svg 24-cell t12 B3.svg 24-cell t012 B3.svg 24-cell t03 B3.svg 24-cell t013 B3.svg 24-cell t0123 B3.svg
B3(b) 24-cell t3 B3.svg 24-cell t23 B3.svg 24-cell t2 B3.svg 24-cell t13 B3.svg 24-cell t123 B3.svg 24-cell t023 B3.svg
B2 24-cell t0 B2.svg 24-cell t01 B2.svg 24-cell h01 B2.svg 24-cell t1 B2.svg 24-cell t02 B2.svg 24-cell t12 B2.svg 24-cell t012 B2.svg 24-cell t03 B2.svg 24-cell t013 B2.svg 24-cell t0123 B2.svg

See also

Citations

  1. Klitzing.
  2. 1 2 Gosset 1900.
  3. Conway, Burgiel & Goodman-Strass 2008, p. 401, 26. Gosset's semi-snub polyoctahedron.
  4. 1 2 Coxeter 1973, pp. 151–153, §8.4. The snub {3,4,3}.
  5. Coxeter 1973, pp. 50–52, §3.7.
  6. Koca, Ozdes Koca & Al-Barwani 2012.
  7. Koca, Al-Ajmi & Ozdes Koca 2011.

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References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds