Dual snub 24-cell explained

bgcolor=#e7dcc3 align=center colspan=3	Dual snub 24-cell
bgcolor=#ffffff align=center colspan=3	Orthogonal projection
Type	4-polytope
Cells	96
Faces	432	144 kites 288 Isosceles triangle
Edges	480
Vertices	144
Dual	Snub 24-cell
Properties	convex

In geometry, the dual snub 24-cell is a 144 vertex convex 4-polytope composed of 96 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.

Geometry

The dual snub 24-cell, first described by Koca et al. in 2011, is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.

Construction

The vertices of a dual snub 24-cell are obtained using quaternion simple roots (T') in the generation of the 600 vertices of the 120-cell. The following describe

T and T' 24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):
O(0100) : T =
O(1000) : V1
O(0010) : V2
O(0001) : V3

With quaternions

(p,q) where \bar p is the conjugate of p and [p,q]:r → r' = prq and [p,q] ^{*:r →} r'' = p\bar rq , then the Coxeter group W(H _{4)=\lbrace[p,\bar} p] ⊕ [p,\bar p] ^{*\rbrace} is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given

p \in T such that \bar p = \pm p ^{4,} \bar p ^{2=\pm} p ^{3,} \bar p ^{3=\pm} p ^{2,} \bar p ^{4=\pm} p and p ^{\dagger} as an exchange of -1 /\phi \leftrightarrow \phi within p where \phi = (1+ \sqrt{ 5)/() } is the golden ratio, we can construct:S = \sum ^{4 ⊕} _{i =1} p ^{i} T I = T + S = \sum ^{4 ⊕} _{i =0} p ^{i} T J = \sum ^{4 ⊕} _{i,j =0} p ^{i\bar} p ^{\dagger} T'

• the alternate snub 24-cell

S' = \sum ^{4 ⊕} _{i =1} p ^{i\bar} p ^{\dagger} T'

and finally the dual snub 24-cell can then be defined as the orbits of

T ⊕ T' ⊕ S' .

Dual

The dual polytope of this polytope is the Snub 24-cell.

See also

• Snub 24-cell honeycomb

References

• Thorold. Gosset. Thorold Gosset. On the Regular and Semi-Regular Figures in Space of n Dimensions. Messenger of Mathematics. Macmillan. 1900.
• Book: Coxeter, H.S.M. . Harold Scott MacDonald Coxeter . 1973 . 1948 . Regular Polytopes . Dover . New York . 3rd . Regular Polytopes (book) .
• Book: John Horton Conway. John. Conway. Heidi. Burgiel. Chaim. Goodman-Strauss. The Symmetries of Things. 2008. 978-1-56881-220-5.
• Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4). Mehmet. Koca. Nazife. Ozdes Koca. Muataz. Al-Barwani. 2012. Int. J. Geom. Methods Mod. Phys.. 09. 8 . 10.1142/S0219887812500685 . 1106.3433 . 119288632 .
• Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system. Mehmet. Koca. Mudhahir. Al-Ajmi. Nazife. Ozdes Koca. Linear Algebra and Its Applications. 434. 4. 2011. 977–989. 10.1016/j.laa.2010.10.005 . 18278359 . 0024-3795. free. 0906.2109.