Transcript Lecture 2

Rank-4 Coxeter Groups with
Quaternions
and 4D polytopes
Mehmet Koca
Department of Physics
College of Science
Sultan Qaboos University
Muscat-OMAN
[email protected]
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Outline
2.1. Representations of the group elements of rank-4
Coxeter groups with quaternions
2.2. 4D polytopes with the W (A 4 ) symmetry
2.3. 4D polytopes with the W (B 4 ) symmetry
2.4. 4D polytopes with the W (F4 ) symmetry
2.5. 4D polytopes with the W (H 4 ) symmetry
2.6. Maximal Subgroups of W (H 4 ) and associated
4D polytopes, snub 24-cell and Grandantiprism
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2.1. Representations of the group elements of rank-4
Coxeter groups with quaternions
Correspondence between the rank-4 Coxeter-Weyl groups
and the finite subgroups of quaternions
Five rank-4 Coxeter groups
W (D 4 ), W (A 4 ), W (B 4 ), W (F4 ), W (H 4 ) , I 2 (n)  I 2 (n)
are described with the quaternionic roots and the
quaternionic group elements
The group W (D 4 ) will be discussed in relation with the snub
24-cell
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Let
I 2 ( n)  I 2 ( n )
denotes the Coxeter diagram
n
n


Let p  exp(e1 ) and q  exp(e1

)e2 be two orthogonal unit
n
quaternions generating the dicyclic group of order 4n. Then the
following set of quaternions describes the root system
n
I 2 ( n)  I 2 ( n)  { p , q }; k  1, 2,..., 2 n
k
k
If s , t  (I 2 (n )  I 2 (n )) are arbitrary unit quaternions the group

Aut ( I 2 ( n)  I 2 ( n))  {[ s, t ]  [ s, t ] } of order 4n  4n is
represented by the elements of the dicyclic group
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The Coxeter Dynkin diagram of A4
r1
simple roots scaled by
 1  1,
1
2
2 can
r2
r3
r4
be taken as
 2  (1  e1  e2  e3 ),
 3  e1 ,
1
2
 4  (e1  e2  e3 ) Coxeter-Weyl group
symbolically
W (A 4 )  {[I , cIc ]  [I , cIc ]}
Here I represents the binary icosahedral group with c 
element of the binary icosahedral group
~
I obtained
1
2
(e3  e2 )  T  and ~
p  p (   ) is an
from I by interchanging  and  .
The Dynkin diagram symmetry leads to the group extension, namely,
pc ]  [ p, c~
pc ]}
Aut( A4 )  W ( A4 ) :   {[ p ,  c~
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The F4 diagram with quaternionic reflection
generators
4
1
1
r1  [ (1  e1  e2  e3 ),  (1  e1  e2  e3 )] , r2
2
2
 [e3 , e3 ] ,
1
1

r3  [(e2  e3 ), (e2  e3 )] , r4  [(e1  e2 ), (e1  e2 )]
2
2
leads to the Coxeter-Weyl group
W ( F4 )  A  A
A  [T , T ]  [T , T }  { [Va , Vb ]   [Vi , V j ]} ,
a ,b
i, j
A  [T , T ]  [T , T ]  { [Va ,Vb ]  [Vi ,V j ] } ,

a ,b

i, j
a, b  0,  ; i, j  1, 2,3
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The F4 diagram with quaternionic reflection
generators
The automorphism group
Aut ( F4 )  W ( F4 ) :   {[O, O]  [O, O] }
With
O  T T 
1
(e 2  e 3 ), e 2 ] leading to
Dynkin diagram symmetry   [
2
1
1
1
1
1
(1  e1 ), e1 
(1  e1 ), e2 
(e2  e3 ), e3 
(e2  e3 )
2
2
2
2
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The Coxeter-Weyl group W (B 4 ) is a subgroup of
represented by quaternions as
W (F4 )
W (B 4 )  B  C  B   C 
B  {[V0 ,V0 ]  [V ,V ]  [V ,V ]} ,
C  {[V1 ,V1 ]  [V2 ,V2 ]  [V3 ,V3 ]}
The Coxeter Group W (H 4 )
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quaternionic simple roots
1   2e1 ,  2 
2
2
( e1  e 2   e 3 ),  3   2e 2 ,  4 
(  e 2   e 3 )
2
2
W ( H 4 )  {[ I , I ]  [ I , I ] }, I : Binary icosahedral group of order 120
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2.2. 4D polytopes with the W(A4) symmetry
  (a1a2 a3 a4 )  a11  a22  a33  a4 4
i , ( i  1,2,3,4) vectors of the dual basis satisfying ( i ,  j )   ij
For the "highest" weight vector the Dynkin indices are non-negative
integers ai  0
The Cartan matrix defined by ( i ,  j )  C ij and its inverse
(i ,  j )  (C 1 ) ij
 2 1 0 0 



1
2

1
0


,
C 
0  1 2  1


 0 0 1 2 


The dual vectors
i
given by
C 1
4

1 3
 
5 2

1

3 2 1

6 4 2
.
4 6 3

2 3 4 
in terms of quaternions
1
(  5  e 2  e3 ),
10
1
3 
(  5e1   2 e 2   2 e3 ),
10
1 
1
( 2e 2  2e3 ),
10
1
2
4 
( 2e 2  2e3 )  
c.
10
5
2 
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Branching of W (A4 ) under W (A3 )
W ( A4 )  {W ( A3 )  W ( A3 )d  W ( A3 )d 2  W ( A3 )d 3  W ( A3 )d 4 }
d  r1r2 r3r4
(a1a2a3a4 ) A4  (a1a2a3 ) A3 ((a1  2a2  3a3  4a4 ))  (a1a2 (a3  a4 )) A3 ( (a1  2a2  3a3  a4 ))
 (a1 (a2  a3 )a4 ) A3 ( (a1  2a2  2a3  a4 ))  ((a1  a2 )a3a4 ) A3 ( (a1  3a2  2a3  a4 ))
 (a2a3a4 ) A3 (4a1  3a2  2a3  a4 ).
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Examples
5-Cell:
(1000) A4  O (100) A3 (1)  (0) A3 (4)
Voronoi cell of the A*4 lattice
(1111) A4  (111) A3 (10)  O (112) A3 ( 5)  O (121) A3 (0)  O (211) A3 (5)  O (111)A3 (10)
Runcinated 5-cell (Root system of A4 )
(1001) A4  O (100) A3 (5)  (101) A3 (0)  (001) A3 (5)
These two polytopes have a larger symmetry :
Aut( A4 )  W ( A4 ) : 
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Duals of the A4 polytopes
Examples:
Dual of the 5-cell (1000)A
4
Relevant maximal subgroup and cell: W (A3 )  r1 , r2 , r3  , (100)A
3
The center of the tetrahedron (100)A can be represented by the
vector (0001) since it is left invariant by the group W (A3 )  r1, r2 , r3  .
3
Number of cells:
W (A 4 ) 120

5
W (A3 )
24
The vertices of the dual cell : (0001)A which is another 5-cell.
4
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Dual of the polytope (1001)A  (1  4 )A (runcinated 5-cell)
The generators r2 and r3 fixing the vector 1  4 form a
dihedral group of order 6 indicating that the polytope has 20
vertices. Numbers of edges, faces and cells are determined
similarly:
4
4
N 0  20, N 1  60 , N 2  70 , N 3  30
30 cells=10 tetrahedra+20 triangular prisms.
The centers of two tetrahedra :
1 and 4
Centers of six triangular prisms :
2 , r2 r32 , (r2 r3 ) 22 ; 3 , r2 r33 , (r2 r3 ) 23
These vectors constitute the hyperplane orthogonal to the vector
1  4
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In a suitable quaternionic basis these 8 vectors form a
rhombohedron with a dihedral symmetry D3
1  ( 1,1, 1) ,
ω4  ( 1, 1,1)
ω3  ( τ, σ, 2 ) ,
r2 r3ω3  ( σ, 2,  τ ) ,
r3 r2 ω3  ( 2, τ,  σ ),
ω2  ( 2,  τ, σ ) ,
r2 r3ω2  (  σ, 2, τ ) , r3r2ω2  (  τ,  σ, 2 ) .
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2.3. 4D polytopes with the W (B 4 ) symmetry
Examples
16-Cell(tetrahedron):hyperoctahedron with 8
vertices
(1000)B 4  (100)B 3 (0)  (0)B 3 (1)  (0)B 3 (1)
N 0  8, N 1  24 , N 2  32 , N 3  16
Quaternionic vertices
V 0  {1, e1 , e 2 , e 3}
8-Cell(cube):hypercube with 16 vertices
1
1
(0001) B 4  (001) B 3 ( )  (001) B 3 (  )
2
2
Quaternionic vertices
1
V  V   {1  e1  e 2  e 3 }
2
It is dual to the hyperoctahedron
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2.4. 4D polytopes with the W (F4 ) symmetry:
all roots of norm 2
C F4
1
2

1
2

0  2

 0
0
0
 2
2
1
 2

 3
(C F4 ) 1  
2 2

 2
0

0
,

1
2 
3
2 2
6
4 2
4 2
6
2 2
3
2

2 2

3 

2 
The maximal Coxeter subgroups:
W (B 4 ) , W (B ) W (A )
3
1
Two subgroups W (B 3L )  r1 ,r2 ,r3 , W (B 3R )  r2 ,r3 ,r4 
not conjugate in the group W (F4 ) but conjugate in
Aut (F4 )
The octahedral group W (B ) plays an important role when an
arbitrary polytope of the group W (F4 ) is projected into 3D.
3
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The quaternionic representations of the groups W (B 3 ) C 2 :
W (B 3R ) C 2  {[T , T ]  [T , T ]  [T , T ]  [T , T ]}
 ]  [T , qTq ]  [T , qT q ]}, q 
W (B 3L ) C 2  {[T , qTq ]  [T , qT q
1  e1
2
The group W (B 3 ) C 2 of order 96 can be embedded in the group
different ways,
W (F4 )
in 12
In each embedding, a quaternion with  sign left invariant
W (B 3R ) C 2 leaves 1 invariant
1  e1
W (B 3L ) C 2 leaves  (
)
2
invariant
the conjugates of W (B ) C leave the vectors of the set T invariant
the conjugates of W (B ) C leave the vectors from T  invariant.
3R
2
3L
2
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Branching W (F ) polytopes under the group W (B 4 )
4
3
W (F4 )  W (B 4 ) g i ; g 1  [1,1], g 2  [0 ,1], g 2  [0 2 ,1], 0 
i 1
1
(1  e1  e 2  e 3 ).
2
(a1 , a2 , a3 , a4 )F4  {( 2a1  2a2  a3 ), a4 , a3 , a2 }B 4  {a3 , a4 , ( 2a2  a3 ), a1}B 4 
{( 2a2  a3 ), a4 , a3 , (a1  a2 )}B 4 .
24-cell:
(1, 0, 0, 0) F4  ( 2, 0, 0, 0) B 4  (0, 0, 0,1) B 4
T V 0 V  , (1, 0, 0, 0)  1(unit of quaternion)
24-cell consists of 24 octahedral cells
every vertex is shared by 6 octahedra
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Let the vertex represented by the quaternion 1.
The following quaternions represent vertices of an octahedron
1, e1 ,
1
(1  e1  e 2  e 3 )
2
The center of this octahedron is represented, up to a scale factor,
1  e1
by the quaternion 2 T 
1  e1
, pi  e i p 0 , i  1, 2,3
2
Vertices of above octahedron p0  pi , i  1, 2,3
Introduce the orthogonal vectors p 0 
shifted in the 4th dimension p 0
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The dual polytope 24-cell (0, 0, 0,1) F decomposes under
the subgroup W (B 4 ) as
4
(0, 0, 0,1) F4  (0,1, 0, 0) B 4
vertices, edges, faces and cells respectively decompose
as
24=24, 96=96, 96=32+64, 24=16+8
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Branching W (F ) polytopes and under W (B
4
The right coset decomposition of
octahedral group W (B )
W (F4 )
3R
)
under the
3R
24
W (F4 )  W (B 3R ) g i , g i  [T ,1]
i 1
[T ,1] , [1,T ] invariant
subgroups of W (F4 )
3a2 2a3  a4

)}  {(a2 , a3 , ( 2a1  2a2  a3  a4 )) B 3
2
2
a
a a a
a
 ( 2  3 4 )}  {(a2 , (a3  a4 ), ( 2a1  2a2  a3 )) B 3  ( 2  3 )} 
2
2
2
2
a2
a
{((a2  2a3 ), a4 , ( 2a1  2a2  a3 )) B 3  ( } +{((a1  2a2  2a3 ), a4 , a3 ) B 3  ( 1 )}+
2
2
1
{((a1  a2 ), a3 , a4 ) B 3  ( (a1  3a2  2 2a3  2a4 )}  {((a1  a2 ), a3 , ( 2a2  a3  a4 )) B 3 
2
1
1
( (a1  a2  2a3  2a4 )}  {((a1  a2 ), (a3  a4 ), ( 2a2  a3 )) B 3  (a1  a2  2a3 )} 
2
2
1
1
{( (a1  a2  2a3 ), a4 , ( 2a2  a3 )) B 3  (a1  a2 )}  {(a1 , ( 2a2  a3 ), a4 ) B 3 
2
2
2a3  a4
a3
a
a
( 1  a2 
)}  {(a1 , ( 2a2  a3  a4 ), a3 ) B 3  ( 1  a2 
)} 
2
2
2
2
1
{((a1 , ( 2a2  a3 ), (a3  a4 )) B 3  (a1  2a2  2a3  2a4 )}.
2
(a1 , a2 , a3 , a4 ) F4  {(a2 , a3 , a4 ) B 3  (a1 
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Branching of 24-cell under W (B 3R )
1
1
1
1
(1, 0, 0, 0) F4  {(0, 0, 0) B 3 
}  {(0, 0,1) B 3 (0)}  { (1, 0, 0) B 3  (
)}.
2
2
2
2 2
1
1
T  {1}  {e1 , e 2 , e 3}  (1  e1  e 2  e 3 )  (1  e1  e 2  e 3 )
2
2
The dual 24-cell
1
(0, 0, 0,1) F4  T  decomposes
2
as
1
1
1
1
(0, 0, 0,1) F4  { (0, 0,1) B 3  }  { (0,1, 0) B 3 (0)}
2
2
2
2
with quaternions
T  {
1
1
1
1
1
1
(1  e1 ),
(1  e 2 ),
(1  e 3 )}  { (e1  e 2 ),
(e 2  e 3 ),
(e 3  e1 )}.
2
2
2
2
2
2
The first set represents two octahedra oppositely placed
with respect to the origin along the fourth direction
Second set represents a cuboctahedra with 12 vertices around the center
of the sphere
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Dual polytope of the polytope (1, 0, 0,1) F4 W (F4 )(1  4 )
The polytope has
N 0  144 vertices, N 1  576 edges, N 2  672 faces, and N 3  240 cells.
Faces consist of 192 triangles, 288 squares, and 192 triangles.
The cells consist of 24+24 octahedra, 96+96 triangular prisms.
The vectors representing the centers of the cells at the vertex (1  4 )
1 , 4 , 2 , r2 r32 , (r2 r3 ) 2 2 , (r2 r3 )3 2 , 3 , r2 r33 , (r2 r3 ) 2 3 , (r2 r3 ) 33

2 2
2
This cell of the dual polytope is tetragonal trapezohedron with
10 vertices with the symmetry D 4 :C 2 of order 16
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The dihedral group D generated by D 4   r2 , r3  and C 2
is the group generated by the diagram symmetry
1  4
p

and pi  e i p 0 , i  1, 2,3
0
Define
4
1  4
1   p1 , 4  p1
2  (2 2  3) p1  ( 2  1) p 2  p 3
r2 r32  (2 2  3) p1  p 2  ( 2  1) p 3
(r2 r3 ) 2 2  (2 2  3) p1  ( 2  1) p 2  p 3
(r2 r3 )3 2  (2 2  3) p1  p 2  ( 2  1) p 3
3  (3  2 2) p1  p 2  ( 2  1) p 3
r2 r33  (3  2 2) p1  ( 2  1) p 2  p 3
(r2 r3 ) 2 3  (3  2 2) p1  p 2  ( 2  1) p 3
(r2 r3 )3 3  (3  2 2) p1  ( 2  1) p 2  p 3
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Eight faces consist of kites with sides
16  10 2  1.363 and 80  56 2  0.897
the area is equal to 0.934
The dual polytope is the union of the orbits
 (1, 0, 0, 0) F   (0, 0, 0,1) F  (0,1, 0, 0) F  (0, 0,1, 0) F
4
4
4
4
The vertices of the dual polytope is on two concentric radii of the spheres
S 3 with radii r1  r4  2.414 and r2  r3  2.449.
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Dual polytope of the polytope (0,1,1, 0) F4 W ( F4 )(2  3 )
The polytope has
N 0  288 vertices, N 1  576 edges, N 2  336 faces, and N 3  48 cells
Faces consist of 96 triangles, 144 squares, and 96 triangles
The cells consist of 24+24 truncated cubes
The vectors representing the centers of the cells at the vertex (2  3 )
1 , r11 , 4 , r44
It is a solid with four vertices which has a symmetry D 2 : C 2
D 2   r1 , r4  and C 2 is generated by the diagram symmetry
D  [
1
(e 2  e 3 ), e 2 ]
.
2
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The cell of the dual polytope consist of four faces of isosceles triangles
of two sides 4  2 2 and one side 2 .
Define
p0 
2  3
and pi  e i p 0 , i  1, 2,3
2  3
1  (1  2) p1  (1  2) p 2  p 3 , r11  (1  2) p1  p 2  (1  2) p 3
4  (1  2) p1  p 2  (1  2) p 3 , r44  (1  2) p1  (1  2) p 2  p 3
The dual polytope is the union of the vertices determined
by the polytopes (1, 0, 0, 0)F  (0, 0, 0,1)F .
3
The 48 vertices of the dual polytope are on the same S sphere
4
4
with radius 2
It is cell and vertex transitive under the group
Bangalore conference, 16-22 December, 2012
Aut (F4 )
28
2.5. 4D polytopes with the H4 symmetry
5
1   2e1 ,  2 
C H4
 2



 0

 0
2
2
( e1  e 2   e 3 ),  3   2e 2 ,  4 
(  e 2   e 3 )
2
2
 0 0 
4


2 1 0 
1
4  3
, (C ) H 4  
 2
1 2 1


0 1 2 

3
6
4
2
2
 

4
2 
2(2   ) 2    .

2 
2 2 
Quaternionic representations of the dual vectors
1 4
(  e1   2e 3 ), 2   2( 3   e 3 ),
2
1
3  
( (  2)  e 2   e 3 ), 4   2
2
1  
W (H 4 )  [I , I ]  [I , I ] , W (H 3 )  [I , I ]  [I , I ]  A 5  C2
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Maximal subgroups
W (H 3 ) W (A1 )  [I ,  I ]  [I ,  I ]  A 5  C 2  C 2
All conjugates
{W (H 3 ) W (A1 )}q  [I , qIq ]  [I , qIq ]
r1
r2
r3
,
q I
r4
Coxeter diagram of A4
One possible choice of the simple roots the set I
1   2,
Let c 
2 
2
(1  e1  e 2  e 3 ),
2
 3   2e1 ,
4 
2
(e1   e 2   e 3 )
2
1
(e3  e2 )  T  be a fixed quaternion
2
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30
The Dynkin diagram symmetry
represented by the quaternions
 : 1   4 ,  2  3
can be
1
1
  [a, b ] with a  ( e1  e 2   e 3 ) and b  ( e1   e 2  e 3 )
2
2
leading to the
Aut (A 4 ) W (A 4 ) : C 2  {[I , cIc ]  [I , cIc ]}  (A 5 : C 2 ) : C 2
W (H 4 )  [I , I ]  [I , I ]
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The W (H 4 ) polytopes and branching under the group W (H 3 )
The orbit W (H 4 )   H 4 decomposes as
120
 H 4   W (H 3 ) g i  ; g
i 1
The 600-cell
i
 [I ,1]
,
i  1, 2,...,120

(0, 0, 0,1) H 4  (0, 0, 0) H 3 ( 1)  (0, 0,1) H 3 (  ) 
2
1

(0, 0, ) H 3 (  )  (1, 0, 0) H 3 (  )  (0,1, 0) H 3 (0)
2
2
The 120-cell
It is the dual of the 600-cell and its cells are the dodecahedra, 4 of which is
meeting at one point
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32
(1, 0, 0 , 0) H 4  (1, 0, 0) H 3 ( 
3
2
)  ( , 0, 0) H 3 ( 

 2
2
)
2
( , 0, 0) H 3 ( )  (0, , 0) H 3 ( )  (0,1, ) H 3 (  ) 
2
2

1
(0, ,1) H 3 ( )  ( , 0, ) H 3 (  )  (1, 0, 2 ) H 3 (0)
2
2
2
All vectors can be converted to unit quaternions by multiplying each vector

with 2 then the 600 vertices are represented by the set of quaternions
J
4

 p iT p j
i , j 0
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The 720  cell
W (H 4 )(0,0,1,0) has 720 vertices. It consists of 600 octahedra and
120 icosahedra.
 2
 
(0, 0,1, 0) H 4  (0, 0,1) H 3 ( 
)  (0, 0,  2 ) H 3 ( 
)
2
2

(0,1, 0) H 3 (  )  (0, 0, 2 ) H 3 (0)  (0,1,  ) H 3 (  ) 
2
(1, 0,1) H 3 (
2

1
)  (1, 0, ) H 3 (  )  (1,1, 0) H 3 (  )  ( , 0,1) H 3 (0)  (0,  , 0)H 3 (1)
2
2
2
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34
Dual polytopes of the uniform polytopes of the
Coxeter-Weyl group W (H 4 )
Examples

Dual polytope of the 600-cell (0001) H 4 W (H 4 )4  2 I
N0 
N2 
W (H 4 )
W (H 3 )
 120, N 1 
W (H 4 )
 r3 , r4    r1 
W (H 4 )
 r4    r1 , r2 
 1200, N 3 
 720,
W (H 4 )
 r1 , r2 , r3 
 600; N 0  N 1  N 2  N 3  0
The subgroup W (A3 )   r2 , r3 , r4  acting on the vector (0, 0, 0,1) generates a tetrahedron whose
center can be represented by the vector 1
The number of tedrahedra joining to the vertex 4 is given by the formula
W (H 3 ) 120

 20
 r2 , r3  )
6
whose centers constitute the vertices of a dodecahedron W (H 3 )1
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35
Dual polytope (120- cell): W (H 4 )1  (1, 0, 0, 0)H
4
W (A 3 ) 24
At each vertex there are  r , r  )  6  4 dodecahedra
2 3
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Dual polytope of the polytope 720'-cell
(0, 0,1, 0) H 4 W (H 4 )(0, 0,1, 0)
N0 
W (H 4 )
 r1 , r2    r4 
N3 
W (H 4 )
 r1 , r2 , r3 

 720, N 1 
W (H 4 )
 r2 , r3 , r4 
W (H 4 )
 r3    r1 
 3600, N 2 
W (H 4 )
 r3 , r4    r1 

W (H 4 )
 r2 , r3 
 3600,
 720; N 0  N 1  N 2  N 3  0
The number of cells sharing the vector 3 as a vertex consists of 2 icosahedra and 5
octahedra. The centers of the cells
2
3
4

,
r
r

,
(
r
r
)

,
(
r
r
)

,
(
r
r
)
1
1 2
1
1 2
1
1 2
Centers of octahedra 1 1 2 1
Centers of icosahedra 4 , r44 up to a scale factor
Scale factor is determined from the relation
(4  1 ).3  0   
2
2 
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37
These seven vectors describe a dipyramid with pentagonal base
with edge length 2  1.41 and the other edges of length
4  1 
8
 2.05
5
Dual polytope consist of 720 vertices represented by the union
of two orbits
(1, 0, 0, 0) H 4 
2
(0, 0, 0,1) H 4
2 
These two orbits define two concentric spheres S 3 with the ratio
of radii
R4
R1
 1.023

3
Define p 0   , p1  e1 p 0 , p 2  e 2 p 0 , p3  e 3 p 0
3

3
after deleting the common components p0  
3
vertices of the dipyramid
(0,1, ), (0, 1,  ),
5 2
( , 1,  ),
2
5
(1,  ,  2 ),
2
5 2
5
( ,1,  ),
( 2 , 0, 0),
2
2
5
(1,  ,  2 ).
2
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(a)
(a)
(b)
2
A pentagonal dipyramid, a typical cell of the polytope O (1, 0, 0, 0)  3   O (0, 0, 0,1)
(b) top view.
.
This polytope is the projection of the Voronoi cell of W ( E 8 ) with 19,440 vertices into
4D space with the residual symmetry W (H 4 )
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39
Dual polytope of the polytope
(1, 0, 0,1) H 4 W (H 4 )(1  4 )
N0 
W (H 4 )
N2 
W (H 4 )
N3 
W (H 4 )
 r2 , r3 
 r1 , r2 
 2400, N 1 

 r1 , r2 , r3 
W (H 4 )

 r3 , r4 

W (H 4 )
 r2 , r3 , r4 
W (H 4 )
 r1    r3 
W (H 4 )
 r1 , r4 


W (H 4 )
 r4    r2 
 7200,
 7440,
W (H 4 )
 r1 , r2 , r4 

W (H 4 )
 r1 , r3 , r4 
 2640
N 0 N1 N 2 N 3  0
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40
Number of cells sharing the same vertex and the vectors representing their
centers
Dodecahedral cell : 1 center represented by
4
Tetrahedral cell : 1 center represented by 1
Triangular prisms : 3 centers represented by
2 , (r2 r3 )2 , (r2 r3 ) 2 2
Pentagonal prisms : 3 centers represented by
3 , (r2 r3 )3 , (r2 r3 ) 2 3
These vectors are defined up to some scale vectors
3 , (r2 r3 )3 , (r2 r3 ) 2 3 , 1 and 4
4
4
4

, 
and  
 3
 4
 3
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The 2640 vertices of the dual polytope
 (1, 0, 0, 0) H  (0,1, 0, 0) H   (0, 0,1, 0) H   (0, 0, 0,1) H
4
4
4
4
3
These vertices lie on the concentric four S spheres with the ratio of radii
R 2 : R 3 : R 4 : R1  2.45 : 2.47 : 2.52 : 2.97.
The dual polytope has 2400 cells each of which is a solid with 8 vertices.
These 8 vertices can be expressed in the basis of unit quaternions defined by
p0 
1  4
, p e p , p e p , p e p
1  4 1 1 0 2 2 0 3 3 0
Excluding the common components
in the direction of p 0 they can be determined in 3 dimensions as follows:
1  [ 2 , 0,1], 2  [1,  2 , 2 ], r2 r32  [ ,   2,  ], (r2 r3 ) 2 2  [ 2 , 1,  2 ],
3  [1,1,  3 ],  (r3r2 )3  [ 2 , (  2), 0],  r2 r33  [2 ,  2 ,  ], 4  [ 2 , 0, 1]
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The solid defined by these 8 vertices has a dihedral symmetry D 3
Six faces consist of two types of kites, big and small
Three small kites meet at the top vertex and three big kites join to
the bottom vertex
At the other six vertices either two small kites and one big kite or two
big kites and one small kite get together.
A typical cell of the dual polytope of the polytope (1, 0, 0,1) H
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4
43
2.6. Maximal Subgroups of W (H 4 ) and associated 4D
polytopes: snub 24-cell and grand antiprism
Maximal subgroup; order
Aut ( A  A ) ; 144
Aut ( H  H ) ; 400
W ( H  A ) ; 240
Aut (SU (5))
; 240
W ( SO(8)) : C ; 576
2
2
2
3
Polytope; # of vertices
2
Grand antiprism; 100
1
3
Snub 24-cell;
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96
44
Snub 24-cell
Coxeter-Dynkin diagram of D 4
Quaternionic simple roots:
1  e 2  e 3 ,  2  e1  e 3 , 3  e 2  e 3 ,  4  1  e1
 2 1 0 0 
2



1 2 1 1
1 2
1

C 
, C 
 0 1 2 1
2 1



 0 0 1 2 
1
2 1 1

4 2 2
2 2 1

2 1 2
Quaternionic weight vectors:
1
2
1
2
1  (1  e1  e 2  e 3 ), 2  1  e1 , 3  (1  e1  e 2  e 3 ), 4  1
W (D 4 )  {[V 0 ,V 0 ]  [V  ,V  ]  [V  ,V  ]  [V 1 ,V 1 ]  [V 2 ,V 2 ]  [V 3 ,V 3 ]}
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Proper rotation subgroup of order 96
W (D 4 ) C 2  {[V 0 ,V 0 ]  [V  ,V  ]  [V  ,V  ]}  r1r2 , r3r2 , r4r2 
With the permutation symmetry S3 of D4.
proper rotation subgroup can be extended to the group
(W (D 4 ) C 2 ) : S 3  {[T ,T ]  [T ,T ]} of order 576
Extension of the group W (D 4 ) by the symmetric group S3
W (D 4 ) : S 3 W (F4 )  {[T ,T ]  [T ,T ]  [T ,T ]  [T ,T ]}
The group (W (D 4 ) C 2 ) : S 3 is also maximal subgroup of the Coxeter

W
(
H
)

{[
I
,
I
]

[
I
,
I
]
}
group
4
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46
Construction of the Snub 24-cell
Snub 24 –cell has 96 vertices and 144 cells (24 icosahedra+120 tetrahedra).
At each vertex there exist 3 icosahedra and 5 tedrahedra.
Recall that icosahedron can be generated as the orbit of the group
W (A 3 ) C 2  [T ,T ] acting on the either vector
 I  1  2  3  ( ,1, ) or  II  1  2  3  ( ,1,  )
Construction of three icosahedra by three proper tetrahedral group
 r1r2 , r2 r3 , r3r1   [T , 4T 4 ],  r3r2 , r2 r4 , r4 r3   [T , 1T 1 ],
 r4 r2 , r2 r1 , r1r4   [T , 3T 3 ]
by acting on the either vector
 I   (1  3  4 )  2 and  II   (1  3  4 )  2
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In terms of quaternions
1
1
1
1
I   2  I  (  e1   e 3 ), II   2  I I  (  e1   e 3 )
2
2
2
2
I  S ,
II  S  S (   )
1
1
1
S  { (  e1   e 3 ), (  e 2   e1 ), (  e 3   e 2 ),
2
2
2
1
1
1
(  e1   e 2 ), (  e 2   e 3 ), (  e 3   e1 ),
2
2
2
1
1
1
(1   e1   e 2 ), (1   e 2   e 3 ), (1   e 3   e1 ),
2
2
2
1
1
1
( e1   e 2  e 3 ), ( e 2   e 3  e1 ), ( e 3   e1  e 2 )}
2
2
2
The group
[T , qTq ], q T
can be embedded in the group [T ,T ] in 12 different
ways
 24  8  8  8 icosahedral cells of the snub 24-cell
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Centers of icsahedra are represented by the vectors q T
The sets of vectors form five tetrahedra joined to the vertex  I
P (1)  (I , r1r3I , r3r4 I , r4 r1I ) P (2)  (I , r2 r1I , r2 r3I , r2r4 I )
P (3)  (I , r3r2 I , r3r1I , r3r4 I )
P (4)  (I , r4 r2 I , r4 r3I , r4 r1I ) P (5)  (I , r1r2 I , r1r3I , r1r4 I )
two tetrahedra P (1) and P (2) are left invariant by the permutation symmetry S3
but three tetrahedra P (3), P (4) and P (5) are permuted by the symmetric group S3
The Klein four-group r1r3 , r1r4   C 2 C 2 can be extended by the permutation group S3 to the
tetrahedral groupT d  (C 2 C 2 ) : S 3  {[T , 2T 2 ]  [T , 2T 2 ] leaving the vector 2 invariant which
represents the center, up to a scale factor, of the tetrahedron P (1) symmetry  24 tetrahedra ;
centers can be represented by q T 
Centers of the five tetrahedra
c (1) 
4
1
1
4
P (1) 
(1  e1 ) , c (2)   P (2)  1 [(   )   e1   e3 ) c (3)   P (3)  1 [(   )   e1   e 2 ]
2
2
2 2
2 2
2 2
2 2
c (4) 
4
2 2
P (4) 
1
2 2
4
[  (   )e1   e 3 ] , c (5)   P (5)  1 [(   )   e1   e 2 ]
2 2
2 2
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Dual of the snub 24-cell
Centers of 3 icosahedra and 5 tetrahedra



[
1 ,
3 ,
4 , c1 , c2 , c3 , c4 , c5 ]
2
2
2
Define new quaternionic units
( p 0  I , p1  e1 p 0 , p 2  e 2 p 0 , p3  e 3 p 0 )
The cell represented by 8 vertices
1
1
1
1
1
( , 0,1),
(0, 1,  ),
(1, , 0),
(  ,  ,  ),
( ,  ,  ),
2 2
2 2
2 2
2 2
2 2
1
1
1
( 2 , 0,1),
(1,  2 , 0),
(0, 1,  2 )
2 2
2 2
2 2
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Dual of the snub 24-cell
2
Three faces are made of kites of sides
of isosceles triangles of sides

2
,
2
,
1
2 and six triangular faces form
1
2
Cell has a symmetry S two mirror images of the snub 24 cell form the set of
3
vectors S  S the symmetry of the new polytope is represented by
W (D 4 ) : S 3 W (F4 )
The polytope has 192 vertices determined by the orbit (0, 0, ,1) F4
It consist of the cells of cubes and semiregular truncated octahedra
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Grand Antiprism
H2  H2  {bm , e3bm;m=0,1,...,9}
1
b  (   e1  e2 ) 12(1) ,
2
W ( H 2 )  {[b, b],[1, 1]};
b5  1
W ( H 2 )  {[b , b],[e3, e3 ]}
Aut ( H 2  H 2 )  {[ p, q]  [ p, q];
p,q  (H 2  H 2 )}
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Construction of the vertices of the Grandantiprism
Define new quaternionic units:
1  1,
e1 
 e1  e2
,
2 

e2  e3e1 
( e1)m
H 2  H 2  {exp 5
,
e1   e2
, e3  e3
2 
H 2 : Plane generated by
H 2: Plane

( e1)m
e3 exp 5
}, m=0,1,...,9
1,
e1
generated by e2 , e3
The quaternions b  1 (   e1  e2 ) 12(1) and
2
1
c  (  e1  e2 ) 12(1) generate the set I of 120 quaternions by
2
multiplication.
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Vertices of the Grandantiprism
GA={bmcbn ,bme3cbn}, m,n=0,1,...,9
The set GA consists of the 100 elements of the set of
quaternions I from which the set of 20 quaternions removed
H2  H2  {bm , e3bm;m=0,1,...,9} are removed.
The set of quaternions 12 (1) in Table 1 represents an
icosahedron :
b
bcb
c
cb
bc
b2cb2
b2cb
b
b 2cb 2
-b2cb2
bcb
bcb2
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The quaternions b and b are removed from GA. Then 10
quaternions
c
cb
b 2cb2
bcb
bc
b2cb
b 2cb 2
-b 2cb2
bcb
bcb2
represent a pentagonal antiprism
These pentagonal antiprisms form two rings:
R1 {bmcbn },
R2  {bme3cbn },
m,n=0,1,...,9 ;
m,n=0,1,...,9
The ring R1 :
.
.
.
.
.
cb, bcb2 , -b2cb2 , b2cb, bc
c, bcb, b2cb2 , b2cb2 , bcb
bc, b2cb, -b2cb2 , bcb2 , cb
.
.
.
.
.
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Two rings can be transformed to each other:
e3R1  R1e3  R2 , e3R2  R2e3  R1
The centers of 20 pentagonal antiprisms can be represented
by the quaternions:
 bm ,
2
 e bm , m=0,1,...,9
3
2
The grand antiprism has 300 tetrahedral cells connecting the
vertices of two rings.
Define the icosahedral set 12 (c)  12 (1)c  c(12 (1)) by
cb
c2 , cbcb, cb 2cb 2 , b,
cbc,
cb 2cb,
cbcb
1, cbcb 2 ,
c2 b
cb
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If b and 1 are removed the rest belongs to two rings. Removing these vertices
from the above set reduces 20 faces of icosahedron to 12 faces. 10 vertices
above form 12 tetrahedra with the vertex c. Those 10 vertices also form the
vertex figure (dissected icosahedron) of the grandantiprism
The 12 tetrahedra can be classified as follows:
Four tetrahedra:
(c, q2; q6 , q9 ), (c, q3; q4 , q9 ), (c,q7 ; q1, q4 ), (c,q10 ; q1, q6 )
Six tetrahedra:
(c,q2 , q5; q9 ), (c, q2 , q10 ; q6 ), (c,q3 , q5; q9 ),
(c, q3 , q7 ; q4 ), (c, q7 , q8; q1), (c,q8 , q10 ; q1)
Two tetrahedra: (c; q1, q4 , q6 ), (c;q4 , q6 , q9 ) .
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When we consider the tetrahedral structures formed by the quaternion e3c and
the set of quaternions 12 (e3c) , the above structures will be reversed because
multiplication by the quaternion e3 reverses the sets of rings. Then the 24
tetrahedra can be written:
24=8+8+8
8: 2 vertices from R1 + 2 vertices from R2
8: 3 vertices from R1 + 1 vertex from R2
8:
1 vertex from R1 + 3 vertices from R2
Number of Tetrahedra: 12100  300
4
Number of Edges : 10100  500
2
Number of Triangular Faces: 21100  700
3
Number of Pentagonal Faces : 1100  20
5
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Centers of two icosahedra:
c13 
 1, c   b
14
2
2
Centers of the 12 tetrahedra
1
(  e1  2e2   e3 ),
1
c2 
(   e1  5e2 ) ,
1
(2   e1   e2  e3 )
2 2
2 2
2 2
1
1
1
c4 
(2   e1  e2   e3 ), c5 
( 5   e1   e2 ),
c6 
( 5  e1  e2  e3 )
2 2
2 2
2 2
1
1
1
c7 
(   e1   e2   2e3 ), c8 
(  e1  2e2   e3 ), c9 
(   e1   e2   2e3 )
2 2
2 2
2 2
1
1
1
c10 
(2   e1  e2   e3 ), c11 
( 5  e1  e2  e3 ) , c12 
(2   e1   e2  e3 )
2 2
2 2
2 2
c1 
c3 
14 vertices form one cell of the dual polytope; these vertices in the hyperplane
orthogonal to the unit quaternion c are :
(0,  , 1), (- , 1,0), ( ,1,0),
(1,0,   ), (-1,0, )
(1,0, ), ( ,-1,0)
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The dual cell consists of
4 Pentagons:
(c1, c2 , c8, c7 , c9 ), (c8, c12 , c6 , c4 , c7 ), (c7 , c4 , c5, c10, c9 ), (c9, c10, c11, c3, c1)
4 Kites:
(c14 , c2 , c1, c3 ), (c14 , c2 , c8 , c12 ), (c13, c5, c4 , c6 ), (c13, c11, c10 , c5 )
2 Isosceles trapezoids:
(c13, c14 , c12 , c6 ), (c13, c14 , c3, c11)
It has a C2  C2 symmetry.
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320 vertices of the dual polytope :
5
200 vertices : J1   (biV1b j )
i, j 0
5
100 vertices : J3   (bi 1 (1 e3 )b j )
2
i, j 0
20 vertices : {  bm ,  e3bm;m=0,1,...,9}
2
2
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Some polyhedra from the projection of the grand antiprism
into 3D space:
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