Mehmet Koca Department of Physics College of Science Sultan Qaboos University Muscat-OMAN [email protected] Bangalore conference, 16-22 December, 2012 1.

Mehmet Koca Department of Physics

College of Science

Sultan Qaboos University

Muscat-OMAN

[email protected]

Bangalore conference, 16-22 December, 2012 1

Rank-4 Coxeter Groups with Quaternions

and 4D polytopes

Outline


2.1. Representations of the group elements of rank-4 Coxeter groups with quaternions 2.2. 4D polytopes with the 4( )W A symmetry 2.3. 4D polytopes with the 4( )W B symmetry 2.4. 4D polytopes with the 4( )W F symmetry 2.5. 4D polytopes with the 4( )W H symmetry 2.6. Maximal Subgroups of 4( )W H and associated 4D polytopes, snub 24-cell and Grandantiprism

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

are described with the quaternionic roots and the quaternionic group elements


4 4 4 4 4( ), ( ), ( ), ( ), ( ) ,W D W A W B W F W H 2 2( ) ( )I n I n

The group 4( )W D will be discussed in relation with the snub 24-cell


Let 2 2( ) ( )I n I n denotes the Coxeter diagram

Let 1exp( )p en

and 1 2exp( )q e e

n

be two orthogonal unit

quaternions generating the dicyclic group of order 4n. Then the following set of quaternions describes the root system

2 2( ) ( ) { , }; 1, 2, ..., 2k kI n I n p q k n

If , ( ( ) ( ))2 2

s t I n I n are arbitrary unit quaternions the group

2 2( ( ) ( )) {[ , ] [ , ] }Aut I n I n s t s t of order 4n 4n is

represented by the elements of the dicyclic group

nn

The Coxeter Dynkin diagram of A4


r1 r2 r3 r4

simple roots scaled by 2 can be taken as

)(2

1 , ),1(

2

1 ,1 32141332121 eeeeeee Coxeter-Weyl group

symbolically

4( ) {[ , ] [ , ] }W A I cIc I cIc

Here I represents the binary icosahedral group with Teec )(2

123 and )(~ pp is an

element of the binary icosahedral group I~ obtained from I by interchanging and .

The Dynkin diagram symmetry leads to the group extension, namely,

:)()( 44 AWAAut }][],{[ cp~cp,cp~cp

The F4 diagram with quaternionic reflection generators


4

1 1 2 3 1 2 3

1 1[ (1 ), (1 )]2 2

r e e e e e e , 2 3 3[ , ]r e e ,

3 2 3 2 3

1[( ), ( )]

2r e e e e , 4 1 2 1 2

1[( ), ( )]

2r e e e e

leads to the Coxeter-Weyl group

4( )W F A A

, ,

[ , ] [ , } { [ , ] [ , ]}a b i ja b i j

A T T T T V V V V ,

, ,

[ , ] [ , ] { [ , ] [ , ] }a b i ja b i j

A T T T T V V V V ,

, 0,a b ; , 1, 2,3i j

The F4 diagram with quaternionic reflection generators


The automorphism group

4 4( ) ( ) : {[ , ] [ , ] }Aut F W F O O O O

With

O T T

Dynkin diagram symmetry 2 3 2

1[ ( ), ]

2e e e leading to

1 1 1 2 2 3 3 2 3

1 1 1 11 (1 ), (1 ), ( ), ( )

2 2 2 2e e e e e e e e e


Bangalore conference, 16-22 December, 2012

9

The Coxeter-Weyl group 4( )W B is a subgroup of 4( )W F represented by quaternions as

4( )W B B C B C

0 0{[ , ] [ , ] [ , ]}B V V V V V V ,

1 1 2 2 3 3{[ , ] [ , ] [ , ]}C V V V V V V

4( ) {[ , ] [ , ] }, : Binary icosahedral group of order 120W H I I I I I

The Coxeter Group 4 )(W H 5

quaternionic simple roots

1 1 2 1 2 3 3 2 4 2 3

2 22 , ( ), 2 , ( )

2 2e e e e e e e

2.2. 4D polytopes with the W(A4) symmetry


443322114321 )( aaaaaaaa

)4321 ( , ,,,ii vectors of the dual basis satisfying ( , )i j ij For the "highest" weight vector the Dynkin indices are non-negative

integers 0ia

The Cartan matrix defined by ( , ) i j ijC and its inverse

)(),( 1ijji C given by

2100

1210

0121

0012

C ,

4321

3642

2463

1234

5

11C .

The dual vectors i in terms of quaternions

c5

2e2e2

10

1 eee5

10

1

e2e210

1 ee5

10

1

32432

22

13

322321

.)(),(

),(),(


Branching of 4( )W A 3under ( )W A

})()()()()({)( 43

33

23334 dAWdAWdAWdAWAWAW

1 2 3 4d r r r r

4 3 3

3 3

1 2 3 4 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2 3 4

( ) ( ) ( ( 2 3 4 )) ( ( )) ( ( 2 3 ))

( ( ) ) ( ( 2 2 )) (( ) ) ( ( 3 2 ))

( )

A A A

A A

A

a a a a a a a a a a a a a a a a a a a

a a a a a a a a a a a a a a a a

a a a

3 1 2 3 4(4 3 2 ).a a a a


Examples

5-Cell:

4 3 3(1000) (100) ( 1) (0) (4)A A AO

Voronoi cell of the A*4 lattice

4 3 3 3 3 3(1111) (111) ( 10) (112) ( 5) (121) (0) (211) (5) (111) (10)A A A A A AO O O O

Runcinated 5-cell (Root system of A4 )

4 3 3 3(1001) (100) ( 5) (101) (0) (001) (5)A A A AO

These two polytopes have a larger symmetry :

:)()( 44 AWAAut

Duals of the A4 polytopes


Examples: Dual of the 5-cell

4(1000)A

Relevant maximal subgroup and cell: 3 1 2 3( ) , ,W A r r r , 3(100)A

The center of the tetrahedron 3(100)A can be represented by the

vector (0001) since it is left invariant by the group 3 1 2 3( ) , ,W A r r r .

Number of cells: 4

3

( ) 1205

( ) 24

W A

W A

The vertices of the dual cell : 4(0001)A which is another 5-cell.


Dual of the polytope 4 41 4(1001) ( )A A (runcinated 5-cell)

The generators 32 and rr fixing the vector 41 form a dihedral group of order 6 indicating that the polytope has 20 vertices. Numbers of edges, faces and cells are determined similarly:

0 1 2 320, 60 , 70 , 30N N N N 30 cells=10 tetrahedra+20 triangular prisms. The centers of two tetrahedra :

1 4 and Centers of six triangular prisms :

2 22 2 3 2 2 3 2 3 2 3 3 2 3 3, , ( ) ; , , ( )r r r r r r r r

These vectors constitute the hyperplane orthogonal to the vector

41


In a suitable quaternionic basis these 8 vectors form a rhombohedron with a dihedral symmetry D3

1 4

3 2 3 3

3 2 3 2

2 3 2 3 2 2

1,1, 1 , ω 1, 1,1

ω τ,σ,2 , r r ω σ, 2, τ ,

r r ω 2, τ, σ ), ω 2, τ,σ ,

r r ω σ,2, τ , r r ω τ, σ, 2 .

( ) ( )

( ) ( )

( ( )

( ) ( )


2.3. 4D polytopes with the 4( )W B symmetry Examples

16-Cell(tetrahedron):hyperoctahedron with 8

vertices

4 3 3 3(1000) (100) (0) (0) (1) (0) ( 1)B B B B

0 1 2 38, 24 , 32 , 16N N N N

Quaternionic vertices

0 1 2 3{ 1, , , }V e e e 8-Cell(cube):hypercube with 16 vertices

4 3 3

1 1(0001) (001) ( ) (001) ( )

2 2B B B

Quaternionic vertices

1 2 3

1{ 1 }

2V V e e e

It is dual to the hyperoctahedron


2.4. 4D polytopes with the 4( )W F symmetry: all roots of norm 2

4

2 1 0 0

1 2 2 0

0 2 2 1

0 0 1 2

FC

, 4

1

2 3 2 2 2

3 6 4 2 2 2( )

2 2 4 2 6 3

2 2 2 3 2

FC

The maximal Coxeter subgroups: 4( )W B , 3 1( ) ( )W B W A

Two subgroups 3 1 2 3 3 2 3 4( ) , , , ( ) , ,L RW B r r r W B r r r not conjugate in the group 4( )W F but conjugate in 4( )Aut F The octahedral group 3( )W B plays an important role when an arbitrary polytope of the group 4( )W F is projected into 3D.


The quaternionic representations of the groups 3 2( )W B C :

3 2( ) {[ , ] [ , ] [ , ] [ , ] }RW B C T T T T T T T T

13 2

1( ) {[ , ] [ , ] [ , ] [ , ] },

2L

eW B C T qTq T qT q T qTq T qT q q

The group 3 2( )W B C of order 96 can be embedded in the group 4( )W F in 12 different ways,

In each embedding, a quaternion with sign left invariant 3 2( )RW B C leaves 1 invariant

3 2( )LW B C leaves 11( )

2

e invariant

the conjugates of 3 2( )RW B C leave the vectors of the set T invariant the conjugates of 3 2( )LW B C leave the vectors from T invariant.


Branching 4( )W F polytopes under the group 4( )W B

32

4 4 1 2 0 2 0 0 1 2 31

1( ) ( ) ; [1,1], [ ,1], [ ,1], (1 ).

2ii

W F W B g g g g e e e

4 4 4

4

1 2 3 4 1 2 3 4 3 2 3 4 2 3 1

2 3 4 3 1 2

( , , , ) {( 2 2 ), , , } { , , ( 2 ), }

{( 2 ), , , ( )} .

F B B

B

a a a a a a a a a a a a a a a

a a a a a a

24-cell:

4 4 4

0

(1,0,0,0) ( 2,0,0,0) (0,0,0,1)

, (1,0,0,0) 1(unit of quaternion)

F B B

T V V

24-cell consists of 24 octahedral cells every vertex is shared by 6 octahedra


Let the vertex represented by the quaternion 1. The following quaternions represent vertices of an octahedron

1 1 2 3

11, , (1 )

2e e e e

The center of this octahedron is represented, up to a scale factor,

by the quaternion 11

2

eT

Introduce the orthogonal vectors 1

0 0

1, , 1,2,3

2i i

ep p e p i

Vertices of above octahedron 0 , 1, 2,3ip p i

shifted in the 4th dimension 0p


The dual polytope 24-cell 4

(0,0,0,1)F decomposes under

the subgroup 4( )W B as

4 4

(0,0,0,1) (0,1,0,0)F B

vertices, edges, faces and cells respectively decompose as 24=24, 96=96, 96=32+64, 24=16+8


Branching 4( )W F polytopes and under 3( )RW B The right coset decomposition of 4( )W F under the

octahedral group 3( )RW B 24

4 31

( ) ( ) , [ ,1]R i ii

W F W B g g T

[ ,1] , [1, ]T T invariant subgroups of 4( )W F

4 3 3

3

3 3

3 421 2 3 4 2 3 4 1 2 3 1 2 3 4

3 4 32 22 3 4 1 2 3

2 12 3 4 1 2 3 1 2 3 4 3

23( , , , ) {( , , ) ( )} {( , , ( 2 2 ))

2 2

( )} {( , ( ), ( 2 2 )) ( )}2 22 2

{(( 2 ), , ( 2 2 )) ( } +{(( 2 2 ), , ) ( )}+2 2

F B B

B

B B

a aaa a a a a a a a a a a a a a

a a aa aa a a a a a

a aa a a a a a a a a a a

3 3

3

3 3

1 2 3 4 1 2 3 4 1 2 3 2 3 4

1 2 3 4 1 2 3 4 2 3 1 2 3

1 2 3 4 2 3 1 2 1 2 3 4

1 {(( ), , ) ( ( 3 2 2 2 )} {(( ), , ( 2 ))

21 1

( ( 2 2 )} {(( ), ( ), ( 2 )) ( 2 )}2 21 1

{( ( 2 ), , ( 2 )) ( )} {( , ( 2 ), )2 2

B B

B

B B

a a a a a a a a a a a a a a

a a a a a a a a a a a a a

a a a a a a a a a a a a

3

3

3 4 31 12 1 2 3 4 3 2

1 2 3 3 4 1 2 3 4

2( )} {( , ( 2 ), ) ( )}

2 22 21

{(( , ( 2 ), ( )) ( 2 2 2 )}.2

B

B

a a aa aa a a a a a a

a a a a a a a a a


Branching of 24-cell under 3( )RW B

4 3 3 3

1 1 1 1(1,0,0,0) {(0,0,0) } {(0,0,1) (0)} { (1,0,0) ( )}.

2 2 2 2 2F B B B

1 2 3 1 2 3 1 2 3

1 1{ 1} { , , } (1 ) ( 1 )

2 2T e e e e e e e e e

The dual 24-cell 4

1(0,0,0,1)

2F T decomposes as

4 3 3

1 1 1 1(0,0,0,1) { (0,0,1) } { (0,1,0) (0)}

22 2 2F B B

with quaternions

1 2 3 1 2 2 3 3 1

1 1 1 1 1 1{ ( 1 ), ( 1 ), ( 1 )} { ( ), ( ), ( )}.

2 2 2 2 2 2T e e e e e e e e e

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


Dual polytope of the polytope 4 4 1 4(1,0,0,1) ( )( )F W F The polytope has

0 1 2 3144 vertices, 576 edges, 672 faces, and 240 cells.N N N N 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( )

2 3 2 3

1 4 2 2 3 2 2 3 2 2 3 2 3 2 3 3 2 3 3 2 3 3, , , , ( ) , ( ) , , , ( ) , ( )r r r r r r r r r r r r

2 2

2

This cell of the dual polytope is tetragonal trapezohedron with 10 vertices with the symmetry 4 2:D C of order 16


The dihedral group 4D generated by 4 2 3 2, and D r r C is the group generated by the diagram symmetry

Define 1 40

1 4

p

0 and , 1, 2,3i ip e p i

1 1 4 1

2 1 2 3

2 3 2 1 2 3

22 3 2 1 2 3

32 3 2 1 2 3

3 1 2 3

2 3 3 1 2 3

22 3 3 1 2 3

,

(2 2 3) ( 2 1)

(2 2 3) ( 2 1)

( ) (2 2 3) ( 2 1)

( ) (2 2 3) ( 2 1)

(3 2 2) ( 2 1)

(3 2 2) ( 2 1)

( ) (3 2 2) ( 2 1)

(

p p

p p p

r r p p p

r r p p p

r r p p p

p p p

r r p p p

r r p p p

r

3

2 3 3 1 2 3) (3 2 2) ( 2 1)r p p p


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

4 4 4 4(1,0,0,0) (0,0,0,1) (0,1,0,0) (0,0,1,0)F F F F

The vertices of the dual polytope is on two concentric radii of the spheres

3S with radii 1 4 2 32.414 and 2.449.r r r r

Bangalore conference, 16-22 December, 2012

27

Dual polytope of the polytope 4 4 2 3(0,1,1,0) ( )( )F W F

The polytope has

0 1 2 3288 vertices, 576 edges, 336 faces, and 48 cellsN N N N

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 1 1 4 4 4, , , r r It is a solid with four vertices which has a symmetry 2 2:D C

2 1 4 2, and is generated by the diagram symmetry D r r C

2 3 2

1[ ( ), ]

2D e e e .


The cell of the dual polytope consist of four faces of isosceles triangles

of two sides 4 2 2 and one side 2 .

Define

2 30

2 3

p

0 and , 1, 2,3i ip e p i

1 1 2 3 1 1 1 2 3

4 1 2 3 4 4 1 2 3

(1 2) (1 2) , (1 2) (1 2)

(1 2) (1 2) , (1 2) (1 2)

p p p r p p p

p p p r p p p

The dual polytope is the union of the vertices determined by the polytopes 4 4

(1,0,0,0) (0,0,0,1) .F F

The 48 vertices of the dual polytope are on the same 3S sphere

with radius 2 It is cell and vertex transitive under the group 4( )Aut F

2.5. 4D polytopes with the H4 symmetry


5

1 1 2 1 2 3 3 2 4 2 3

2 22 , ( ), 2 , ( )

2 2e e e e e e e

4 4

1 4

2

2 0 0 4 3 2

2 1 0 3 6 4 2, ( )

0 1 2 1 2 4 2(2 ) 2

0 0 1 2 2 2 2

H HC C

.

Quaternionic representations of the dual vectors

4 2 31 1 3 2 3

3 2 3 4

1( ), 2( ),

21

( ( 2) ), 2 2

e e e

e e

4( ) [ , ] [ , ]W H I I I I , 3 5 2( ) [ , ] [ , ] C W H I I I I A


Maximal subgroups

3 1 5 2 2( ) ( ) [ , ] [ , ] A C C W H W A I I I I

All conjugates

3 1{ ( ) ( )} [ , ] [ , ] qW H W A I qIq I qIq , q I

r1 r2 r3 r4

Coxeter diagram of 4A

One possible choice of the simple roots the set I

1 2 1 2 3 3 1 4 1 2 3

2 22, (1 ), 2 , ( )

2 2e e e e e e e

Let Teec )(2

123 be a fixed quaternion


The Dynkin diagram symmetry 1 4 2 3: , can be represented by the quaternions

1 2 3 1 2 3

1 1[ , ] with ( ) and ( )

2 2a b a e e e b e e e

leading to the

4 4 2 5 2 2( ) ( ) : {[ , ] [ , ] } ( : ) :Aut A W A C I cIc I cIc A C C

4( ) [ , ] [ , ]W H I I I I


The 4( )W H polytopes and branching under the group 3( )W H

The orbit 44( ) HW H decomposes as

4

120

31

( ) iH

i

W H g

; [ ,1]ig I , 1, 2,...,120i

The 600-cell

4 3 3

3 3 3

(0,0,0,1) (0,0,0) ( 1) (0,0,1) ( )2

1(0,0, ) ( ) (1,0,0) ( ) (0,1,0) (0)

2 2

H H H

H H H

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


4 3 3

3 3 3

3

22

2(1,0,0 ,0) (1,0,0) ( ) ( ,0,0) ( )

2 2

( ,0,0) ( ) (0, ,0) ( ) (0,1, ) ( )2 2

H H H

H H H

3 3 3

21(0, ,1) ( ) ( ,0, ) ( ) (1,0, ) (0)

2 2H H H

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

4

, 0

i j

i j

J p T p


The 720 cell

4( )(0,0,1,0)W H has 720 vertices. It consists of 600 octahedra and 120 icosahedra.

4 3 3

3 3 3

22(0,0,1,0) (0,0,1) ( ) (0,0, ) ( )

2 2

(0,1,0) ( ) (0,0,2 ) (0) (0,1, ) ( )2

H H H

H H H

3 3 3 3 3

2 1(1,0,1) ( ) (1,0, ) ( ) (1,1,0) ( ) ( ,0,1) (0) (0, ,0) ( 1)

2 2 2H H H H H


Dual polytopes of the uniform polytopes of the Coxeter-Weyl group 4( )W H

Examples

Dual polytope of the 600-cell 4 4 4(0001) ( )2H W H I

4 40 1

3 4 1 2

4 42 3 0 1 2 3

3 4 1 1 2 3

( ) ( )120, 720,

( ) ,

( ) ( )1200, 600; 0

, , ,

W H W HN N

W H r r r

W H W HN N N N N N

r r r r r r

The subgroup 3 2 3 4( ) , ,W A r r r 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

3

2 3

( ) 12020

, ) 6

W H

r r

whose centers constitute the vertices of a dodecahedron 3 1( )W H


Dual polytope (120- cell): 44 1( ) (1,0,0,0)HW H

At each vertex there are 3

2 3

( ) 244 dodecahedra

, ) 6

W A

r r


Dual polytope of the polytope 720'-cell

4 4(0,0,1,0) ( )(0,0,1,0)H W H

4 4 4 40 1 2

1 2 4 3 1 3 4 1 2 3

4 43 0 1 2 3

1 2 3 2 3 4

( ) ( ) ( ) ( )720, 3600, 3600,

, , ,

( ) ( ) 720; 0

, , , ,

W H W H W H W HN N N

r r r r r r r r r r

W H W HN N N N N

r r r r r r

The number of cells sharing the vector 3 as a vertex consists of 2 icosahedra and 5 octahedra. The centers of the cells

Centers of octahedra2 3 4

1 1 2 1 1 2 1 1 2 1 1 2 1, , ( ) , ( ) , ( )r r r r r r r r

Centers of icosahedra 4 4 4 , r up to a scale factor

Scale factor is determined from the relation

4 1 3( ). 0 2

2


These seven vectors describe a dipyramid with pentagonal base

with edge length 2 1.41 and the other edges of length

4 1

82.05

5

Dual polytope consist of 720 vertices represented by the union of two orbits

4 4

2(1,0,0,0) (0,0,0,1)

2H H

These two orbits define two concentric spheres 3S with the ratio

of radii 4

11.023R

R

Define 30 1 1 0 2 2 0 3 3 0

3

, , , p p e p p e p p e p

after deleting the common components 3

03

p

vertices of the dipyramid

2 2

2 2

5 5 5(0,1, ), (0, 1, ), ( 1, , ), ( ,1, ), ( 2 ,0,0),

2 2 2

5 5( , 1, ), ( 1, , ).

2 2


(a) (b)

(a) A pentagonal dipyramid, a typical cell of the polytope 2

(1,0,0,0) (0,0,0,1)3

O O

(b) top view. .

This polytope is the projection of the Voronoi cell of 8( )W E with 19,440 vertices into

4D space with the residual symmetry 4( )W H


Dual polytope of the polytope

4 4 1 4(1,0,0,1) ( )( )H W H

4 4 4

0 12 3 1 3 4 2

4 4 42

1 2 3 4 1 4

4 4 4 43

1 2 3 2 3 4 1 2 4 1 3 4

0 1 2 3

( ) ( ) ( )2400, 7200,

,

( ) ( ) ( )7440,

, , ,

( ) ( ) ( ) ( )2640

, , , , , , , ,

0

W H W H W HN N

r r r r r r

W H W H W HN

r r r r r r

W H W H W H W HN

r r r r r r r r r r r r

N N N N


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

2 2 3 2 2 3 2, ( ) , ( )r r r r

Pentagonal prisms : 3 centers represented by 2

3 2 3 3 2 3 3, ( ) , ( )r r r r

These vectors are defined up to some scale vectors

23 2 3 3 2 3 3, ( ) , ( ) ,r r r r 1 4 and

4 4 4

, and 3 4 3


The 2640 vertices of the dual polytope

4 4 4 4(1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1)H H H H

These vertices lie on the concentric four3S spheres with the ratio of radii

2 3 4 1: : : 2.45 : 2.47 : 2.52 : 2.97.R R R R 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 1 4

0 1 1 0 2 2 0 3 3 01 4

, , , p p e p p e p p e p

Excluding the common components

in the direction of 0p they can be determined in 3 dimensions as follows:

2 2 2 21 2 2 3 2 2 3 2[ ,0,1], [ 1, , 2 ], [ , 2, ], ( ) [ 2 , 1, ],r r r r

3 2 2 2

3 3 2 3 2 3 3 4[ 1,1, ], ( ) [ , ( 2),0], [ 2 , , ], [ ,0, 1]r r r r


The solid defined by these 8 vertices has a dihedral symmetry 3D 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

4(1,0,0,1)H


2.6. Maximal Subgroups of 4( )W H and associated 4D polytopes: snub 24-cell and grand antiprism

Polytope; # of vertices Maximal subgroup; order ; 144 )(

22AAAut

Grand antiprism; 100 ; 400 )(22

HHAut ; 240 )(

13AHW

; 240 ( (5))Aut SU Snub 24-cell; 96 ; 576

38 CSOW :))((


Snub 24-cell

Coxeter-Dynkin diagram of 4D

Quaternionic simple roots:

1 2 3 2 1 3 3 2 3 4 1, , , 1e e e e e e e

1

2 1 0 0 2 2 1 1

1 2 1 1 2 4 2 21,

0 1 2 1 1 2 2 12

0 0 1 2 1 2 1 2

C C

Quaternionic weight vectors:

1 1 2 3 2 1 3 1 2 3 4

1 1(1 ), 1 , (1 ), 1

2 2e e e e e e e

4 0 0 1 1 2 2 3 3( ) {[ , ] [ , ] [ , ] [ , ] [ , ] [ , ] }W D V V V V V V V V V V V V


Proper rotation subgroup of order 96

4 2 0 0 1 2 3 2 4 2( ) {[ , ] [ , ] [ , ]} , ,W D C V V V V V V r r r r r r With the permutation symmetry S3 of D4. proper rotation subgroup can be extended to the group

4 2 3( ( ) ) : {[ , ] [ , ] }W D C S T T T T of order 576 Extension of the group 4( )W D by the symmetric group S3

4 3 4( ) : ( ) {[ , ] [ , ] [ , ] [ , ] }W D S W F T T T T T T T T The group 4 2 3( ( ) ) :W D C S is also maximal subgroup of the Coxeter

group 4 ( ) {[ , ] [ , ] }W H I I I I


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

3 2( ) [ , ]W A C T T acting on the either vector

1 2 3 ( ,1, )I or 1 2 3 ( ,1, )II Construction of three icosahedra by three proper tetrahedral group

1 2 2 3 3 1 4 4 3 2 2 4 4 3 1 1

4 2 2 1 1 4 3 3

, , [ , ], , , [ , ],

, , [ , ]

r r r r r r T T r r r r r r T T

r r r r r r T T

by acting on the either vector

1 3 4 2( )I and 1 3 4 2( )II


In terms of quaternions 2 2

1 3 1 3

1 1 1 1( ), ( )

2 2 2 2

, ( )

I I II I I

I II

e e e e

S S S

1 3 2 1 3 2

1 2 2 3 3 1

1 2 2 3 3 1

1 2 3 2 3 1 3 1 2

1 1 1{ ( ), ( ), ( ),

2 2 21 1 1

( ), ( ), ( ),2 2 21 1 1

( 1 ), ( 1 ), ( 1 ),2 2 21 1 1

( ), ( ), ( )}2 2 2

S e e e e e e

e e e e e e

e e e e e e

e e e e e e e e e

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


Centers of icsahedra are represented by the vectors q T

The sets of vectors form five tetrahedra joined to the vertex I

1 3 3 4 4 1(1) ( , , , )I I I IP r r r r r r 2 1 2 3 2 4(2) ( , , , )I I I IP r r r r r r

3 2 3 1 3 4(3) ( , , , )I I I IP r r r r r r

4 2 4 3 4 1(4) ( , , , )I I I IP r r r r r r 1 2 1 3 1 4(5) ( , , , )I I I IP r r r r r r

two tetrahedra (1) and (2)P P are left invariant by the permutation symmetry S3 but three tetrahedra (3), (4) and (5)P P P are permuted by the symmetric group S3 The Klein four-group 1 3 1 4 2 2,r r r r C C can be extended by the permutation group S3 to the tetrahedral group 2 2 3 2 2 2 2( ) : {[ , ] [ , ]dT C C S T T T T leaving the vector 2 invariant which represents the center, up to a scale factor, of the tetrahedron (1)P symmetry 24 tetrahedra ;

centers can be represented byq T Centers of the five tetrahedra

1

1 1(1) (1) (1 )

2 2c P e ,

4

1 3

1(2) (2) [( ) )

2 2 2 2c P e e

4

1 2

1(3) (3) [( ) ]

2 2 2 2c P e e

4

1 3

1(4) (4) [ ( ) ]

2 2 2 2c P e e

,4

1 2

1(5) (5) [( ) ]

2 2 2 2c P e e


Dual of the snub 24-cell

Centers of 3 icosahedra and 5 tetrahedra

1 3 4 1 2 3 4 5[ , , , , , , , ]2 2 2

c c c c c

Define new quaternionic units 0 1 1 0 2 2 0 3 3 0( , , , )Ip p e p p e p p e p The cell represented by 8 vertices

2 2 2

1 1 1 1 1( ,0,1), (0, 1, ), (1, ,0), ( , , ), ( , , ),

2 2 2 2 2 2 2 2 2 21 1 1

( ,0,1), (1, ,0), (0, 1, )2 2 2 2 2 2


Dual of the snub 24-cell

Three faces are made of kites of sides 2 1

, 2 2

and six triangular faces form

of isosceles triangles of sides 1

, 2 2

Cell has a symmetry 3S two mirror images of the snub 24 cell form the set of

vectors S S the symmetry of the new polytope is represented by

4 3 4( ) : ( )W D S W F

The polytope has 192 vertices determined by the orbit 4(0,0, ,1)F

It consist of the cells of cubes and semiregular truncated octahedra


Grand Antiprism

2 2 3{ , e ;m=0,1,...,9}m mH H b b

51 2

1 ( ) 12(1) , b 12

b e e

2 2 3 3 ( ) {[ , ],[1, 1] }; ( ) {[ , ],[ , ] } W H b b W H b b e e

2 2 2 2( ) {[ , ] [ , ] ; p,q ( )}Aut H H p q p q H H


Construction of the vertices of the Grandantiprism

Define new quaternionic units:

1 2 1 21 2 3 1 3 31 1, e , e e , e

2 2e e e e

e e

1 12 2 3

( ) ( )5 5{exp , e exp }, m=0,1,...,9

m me eH H

2

2

2 33

1Plane

: Plane

: generated by 1,

generated by e , e

e

H

H

The quaternions 1 21 ( ) 12(1) 2

b e e and

1 21 ( ) 12(1)2

c e e generate the set I of 120 quaternions by

multiplication.


Vertices of the Grandantiprism

m m3 GA={b ,b }, m,n=0,1,...,9n ncb e cb

The set GA consists of the 100 elements of the set of quaternions I from which the set of 20 quaternions removed 2 2 3{ , e ;m=0,1,...,9}m mH H b b are removed. The set of quaternions 12 (1) in Table 1 represents an icosahedron :

2 2 2 2

2 2 2 2

b

c bcb b b bcb

cb bc b -b bcb

b

cb cb

cb cb


The quaternions and bb are removed from GA. Then 10 quaternions

2 2 2 2

2 2 2 2

c bcb b b bcb

cb bc b -b bcb

cb cb

cb cb

represent a pentagonal antiprism

These pentagonal antiprisms form two rings:

1 { }, m,n=0,1,...,9m nR b cb ;

32 { }, m,n=0,1,...,9m neR b cb

The ring 1R :

2 2 2 2

2 2 2 2

. . . . .

cb, bcb , -b cb , b cb, bc

c, bcb, b cb , b cb , bcb

bc, 2 2 2 2 b cb, -b cb , bcb , cb . . . . .


Two rings can be transformed to each other: 3 1 1 3 2 3 2 2 3 1 , e R R e R e R R e R

The centers of 20 pentagonal antiprisms can be represented by the quaternions: 3, , m=0,1,...,9

2 2m mb e b

The grand antiprism has 300 tetrahedral cells connecting the vertices of two rings.

Define the icosahedral set 12 ( ) 12 (1) (12 (1))c c c by

2 2 2

2 2 2

cb

c , cbcb, cb cb , b, cbcb

cbc, cb cb, 1, cbcb , c b

cb


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:

72 6 9 3 4 9 1 4 10 1 6( , ; , ), (c, q ; , ), (c,q ; , ), (c,q ; , ) c q q q q q q q q q

Six tetrahedra:

5 52 9 2 10 6 3 9

7 73 4 8 1 8 10 1

(c,q , ; ), ( , , ; ), (c,q , ; ),

(c, q , ; ), (c, q , ; ), (c,q , ; )

q q c q q q q q

q q q q q q

Two tetrahedra: 1 4 6 4 6 9( ; , , ), (c;q , , )c q q q q q .


When we consider the tetrahedral structures formed by the quaternion 3e c and the set of quaternions 312 ( )e c , the above structures will be reversed because multiplication by the quaternion 3e reverses the sets of rings. Then the 24 tetrahedra can be written: 24=8+8+8 8: 2 vertices from 1R + 2 vertices from 2R

8: 3 vertices from 1R + 1 vertex from 2R

8: 1 vertex from 1R + 3 vertices from 2R

Number of Tetrahedra: 12 100 3004

Number of Edges : 10 100 5002

Number of Triangular Faces: 21 100 7003

Number of Pentagonal Faces : 1 100 20

5


Centers of two icosahedra:

13 12

c , 14 2c b

Centers of the 12 tetrahedra

1 1 2 3 2 1 2 3 1 2 3

54 1 2 3 1 2 6 1 2 3

27 1 2 3 8 1 2 3 9 1

1 1 1( 2 ), ( 5 ) , (2 )2 2 2 2 2 2

1 1 1(2 ), ( 5 ), ( 5 )2 2 2 2 2 2

1 1 1( ), ( 2 ), (2 2 2 2 2 2

c e e e c e e c e e e

c e e e c e e c e e e

c e e e c e e e c e

22 3

10 1 2 3 11 1 2 3 12 1 2 3

)

1 1 1(2 ), ( 5 ) , (2 )2 2 2 2 2 2

e e

c e e e c e e e c e e e

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)


The dual cell consists of 4 Pentagons:

7 7 7 51 2 8 9 8 12 6 4 4 10 9 9 10 11 3 1( , , , , ), ( , , , , ), ( , , , , ), ( , , , , )c c c c c c c c c c c c c c c c c c c c

4 Kites: 5 514 2 1 3 14 2 8 12 13 4 6 13 11 10( , , , ), ( , , , ), ( , , , ), ( , , , )c c c c c c c c c c c c c c c c

2 Isosceles trapezoids: 13 14 12 6 13 14 3 11( , , , ), ( , , , )c c c c c c c c

It has a 2 2C C symmetry.


320 vertices of the dual polytope :

200 vertices : 5

1 1, 0

)( ji

i jJ b V b

100 vertices : 5

3 3, 0

)1( ( 1 )2

ji

i jJ b e b

20 vertices : 32 2{ , e ;m=0,1,...,9}m mb b

Some polyhedra from the projection of the grand antiprism into 3D space:


Mehmet Koca Department of Physics College of Science Sultan Qaboos University Muscat-OMAN [email protected] Bangalore conference, 16-22 December, 2012 1.

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