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.
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.
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
{\displaystyle T}
and
T
′
{\displaystyle T'}
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
(
p
,
q
)
{\displaystyle (p,q)}
where
p
¯
{\displaystyle {\bar {p}}}
is the conjugate of
p
{\displaystyle p}
and
p
,
q
:
r
→
r
′
=
p
r
q
{\displaystyle [p,q]:r\rightarrow r'=prq}
and
p
,
q
∗
:
r
→
r
″
=
p
r
¯
q
{\displaystyle [p,q]^{*}:r\rightarrow r''=p{\bar {r}}q}
, then the
Coxeter group
W
(
H
4
)
=
{
p
,
p
¯
⊕
p
,
p
¯
∗
}
{\displaystyle W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace }
is the symmetry group of the
600-cell and the
120-cell of order 14400.
Given
p
∈
T
{\displaystyle p\in T}
such that
p
¯
=
±
p
4
,
p
¯
2
=
±
p
3
,
p
¯
3
=
±
p
2
,
p
¯
4
=
±
p
{\displaystyle {\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p}
and
p
†
{\displaystyle p^{\dagger }}
as an exchange of
−
1
/
ϕ
↔
ϕ
{\displaystyle -1/\phi \leftrightarrow \phi }
within
p
{\displaystyle p}
where
ϕ
=
1
+
5
2
{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}
is the
golden ratio , we can construct:
the
snub 24-cell
S
=
∑
i
=
1
4
⊕
p
i
T
{\displaystyle S=\sum _{i=1}^{4}\oplus p^{i}T}
the
600-cell
I
=
T
+
S
=
∑
i
=
0
4
⊕
p
i
T
{\displaystyle I=T+S=\sum _{i=0}^{4}\oplus p^{i}T}
the
120-cell
J
=
∑
i
,
j
=
0
4
⊕
p
i
p
¯
†
j
T
′
{\displaystyle J=\sum _{i,j=0}^{4}\oplus p^{i}{\bar {p}}^{\dagger j}T'}
the alternate snub 24-cell
S
′
=
∑
i
=
1
4
⊕
p
i
p
¯
†
i
T
′
{\displaystyle S'=\sum _{i=1}^{4}\oplus p^{i}{\bar {p}}^{\dagger i}T'}
and finally the dual snub 24-cell can then be defined as the orbits of
T
⊕
T
′
⊕
S
′
{\displaystyle T\oplus T'\oplus S'}
.
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
±
{\displaystyle \pm }
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
3
2
=
.866
{\displaystyle {\tfrac {\sqrt {3}}{2}}=.866}
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
The dual polytope of this polytope is the
Snub 24-cell .
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 .