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 144 vertex convex 4-polytope composed of 96 irregular cells. Each cell has faces of two kinds: 3 kites and 6 isosceles triangles.[1] The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.

Geometry

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The dual snub 24-cell, first described by Koca et al. in 2011,[2] is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.[3]

Construction

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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.[4] The following describe   and   24-cells as quaternion orbit 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

 

With quaternions   where   is the conjugate of   and   and  , then the Coxeter group   is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given   such that   and   as an exchange of   within   where   is the golden ratio, we can construct:

  • the snub 24-cell  
  • the 600-cell  
  • the 120-cell  
  • the alternate snub 24-cell  

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

Projections

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3D Orthogonal projections
 
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) sets of tetrahedral surfaces are uniquely colored.
 
3D overlay of the dual snub 24-cell with the orthogonal projection of the 120-cell which forms an outer hull of a unit circumradius chamfered dodecahedron. Of the 600 vertices in the 120-cell (J), 120 of the dual snub 24-cell (T'+S') are a subset of J and 24 (the T 24-cell) are not. Some of those 24 can be seen projecting outside the convex 3D hull of the 120-cell. As itemized in the hull data of this diagram, the 8 16-cell vertices of T have 6 with unit norm and can be seen projecting outside the center of 6 hexagon faces, while 2 with a  1 in the 4th dimension get projected to the origin in 3D. The 16 other vertices are the 8-cell Tesseract which project to norm   inside the 120-cell 3D hull. Please note: the face and cell count data, along with the area and volume, within this image are from Mathematica automated tetrahedral cell analysis and are not based on the 96 kite cells of the dual snub 24-cell.
2D Orthogonal projections
 
2D projection of the dual snub 24-cell with color coded vertex overlaps
 
2D Projections to selected Coxeter Planes

Dual

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The dual polytope of this polytope is the Snub 24-cell.[5]

See also

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Citations

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  1. ^ Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–987, Fig. 4.
  2. ^ Koca, Al-Ajmi & Ozdes Koca 2011.
  3. ^ Gosset 1900.
  4. ^ Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.
  5. ^ Coxeter 1973, pp. 151–153, §8.4. The snub {3,4,3}.

References

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  • Gosset, Thorold (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions". Messenger of Mathematics. Macmillan.
  • Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
  • Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. ISBN 978-1-56881-220-5.
  • Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)". Int. J. Geom. Methods Mod. Phys. 09 (8). arXiv:1106.3433. doi:10.1142/S0219887812500685. S2CID 119288632.
  • Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
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
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