Dual Snub 24-cell

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 ,qr\rightarrow r'=prq and ,q*:r\rightarrow r''=p\bar rq, then the Coxeter group W(H_4)=\lbrace ,\bar p\oplus ,\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 $\phi=\frac$ is the golden ratio, we can construct:

* the snub 24-cell $S=\sum_^4\oplus p^i T$
* the 600-cell $I=T+S=\sum_^4\oplus p^i T$
* the 120-cell $J=\sum_^4\oplus p^i\bar p^T'$
* the alternate snub 24-cell $S'=\sum_^4\oplus p^i\bar p^T'$

and finally the dual snub 24-cell can then be defined as the orbits of $T \oplus T' \oplus S'$.

Projections

Dual

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

See also

* Snub 24-cell honeycomb

Citations

References

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{{DEFAULTSORT:Dual snub 24-Cell
4-polytopes