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Dual Snub 24-cell
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 convex uniform 4-polytope composed of 96 regular 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.

3D Visualization of the hull of the dual snub 24-Cell, with vertices colored by overlap count:
The (42) yellow have no overlaps.
The (51) orange have 2 overlaps.
The (18) tetrahedral hull surfaces are uniquely colored.

Semiregular polytope

It was discovered by Koca et al. in a 2011 paper.

The vertices of a dual snub 24-cell are obtained through non-commutative multiplication of the simple roots (T') used in the quaternion base generation of the 600 vertices of the 120-cell. The following orbits of 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

Constructions

One can build it from the subsets of the 120-cell, namely the 24 vertices of T=24-cell, 24 vertices of the alternate T'=D₄ 24-cell, and 96 vertices of the alternate snub 24-cell S'=T' ^n=1-4 using the quaternion construction of the 120-cell and non-commutative multiplication.

2D Orthogonal projection

Dual Snub 24-cell
2D projection of the dual snub 24-cell with color coded vertex overlaps.

Dual

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

See also

• Snub 24-cell honeycomb

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H. S. M. Coxeter (1973). Regular Polytopes. New York: Dover Publications Inc. pp. 151–152, 156–157.
• Snub icositetrachoron - Data and images
• 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 31, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) s3s4o3o - sadi".
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4) [1], Mehmet Koca, Nazife Ozdes Koca, Muataz Al-Barwani (2012);Int. J. Geom. Methods Mod. Phys. 09, 1250068 (2012)
• Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system, Mehmet Koca, Mudhahir Al-Ajmi, Nazife Ozdes Koca (2011);Linear Algebra and its Applications,Volume 434, Issue 4 (2011),Pages 977-989,ISSN 0024-3795

Category:4-polytopes

DualSnub24Cell

This is a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points along the critical line $\zeta (1/2+it)$ for real values of $t$ running from 0 to 50. The consecutive zeros have 50 red plot points between each with zeros identified by magenta concentric rings (scaled to show the relative distance between their values of t). Gram's law states that the curve usually crosses the real axis once between zeros.

The first failure of Gram's law occurs at the 127'th zero and the Gram point g₁₂₆, which are in the "wrong" order.

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