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.
Semiregular polytope edit
It was discovered by Koca et al. in a 2011 paper.[1]
Coordinates edit
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
Constructions edit
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.
Dual Snub 24-cell |
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Dual edit
The dual polytope of this polytope is the Snub 24-cell.
See also edit
Notes edit
References edit
- 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
DualSnub24Cell
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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