Rank 3-4 Coxeter Groups, Quaternions
and Quasicrystals
Mehmet Koca
Department of Physics
College of Science
Sultan Qaboos University
Muscat-OMAN
kocam@squ.edu.om
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References
Polyhedra obtained from Coxeter groups and quaternions.
Koca M., Al-Ajmi M., Koc R. 11, November 2007, Journal
of Mathematical Physics, Vol. 48.
Catalan solids derived from 3D-root systems and quaternions.
Koca M., Koca N.O, Koc R. 4, s.l. : Journal of
Mathematical Physics, 2010, Vol. 51.
Chiral Polyhedra Derived from Coxeter Diagrams and
Quaternions, Mehmet Koca, Nazife Ozdes Koca and
Muna Al-Shueili, ", arXiv:1006.3149 [pdf], SQU Journal
for Science, 16 (2011) 63-82, 2011.
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Outline
1. Rank-3 Coxeter Groups with Quaternions and Polyhedra
1.1. Rank-3 Coxeter Groups with Quaternions
1.2. Quaternionic construction of vertices of Platonic and
Archimedean polyhedra with tetrahedral, octahedral and
icosahedral symmetries
1.3. Catalan solids as duals of the Archimedean solids
1.4. Novel construction of chiral polyhedra and their duals.
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Outline
2. Rank-4 Coxeter Groups with Quaternions and 4D polytopes
2.1. Representations of the group elements of rank-4
Coxeter groups with quaternions
2.2. 4D polytopes with W ( A 4 ) symmetry
2.3. 4D polytopes with W ( B 4 ) symmetry
2.4. 4D polytopes with W ( F 4 ) symmetry
2.5. 4D polytopes with W ( H 4 ) symmetry
2.6. Maximal Subgroups of W ( H 4 ) and associated
4D polytopes, snub 24-cell and Grand antiprism
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Outline
3. Quasicrystallography from higher dimensional lattices
3.1. Quasicrystals and aperiodic tiling of the plane
3.2. Maximal dihedral subgroups of the Coxeter groups
3.3. Projection of the lattices generated by the affine
Coxeter groups onto the Coxeter plane
3.4. Affine A4 and decagonal quasicrystals
3.5. Affine D6 and Icosahedral quasicrystals
3.6. Conclusion
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1.1. Rank-3 Coxeter Groups with Quaternions
O(4) transformations with quaternions
Let q  q0  qiei , i  1,2,3 be unit quaternion qq  qq  1
e1  i , e 2  j , e 3  k
e i e j   ij   ijk e k ,
The transformations
r  p rq : [ p , q ]
r  prq : [ p , q ] ;

with qq  pp  1 define an orthogonal group O(4)
leaving rr invariant.
If q  p then above transformations lead to O(3)
transformations.
Scalar product : ( p , q ) 
1
( pq  q p)
2
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Reflections

Quaternions can be used to represent reflections and rotations in Coxeter
Groups.
Let  be an arbitrary quaternionic simple root. Then the reflection of an arbitrary
vector  with respect to the plane orthogonal to the simple root  is given by
    2
       ,  

The reflection can be represented as:
r :      
1
2
 
     

2


  ,  


2
ʌ

   
 
,
 
2
2 
 2
N o tatio n s :ro tary reflectio n [ p , q ]
 
2
-α
α
*
p ro p er ro tatio n [ p , q ]
w h ere p an d q are arb itrary u n it q u atern io n s
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Finite Subgroups of Quaternions

(a) Cyclic group of order 2n generated by  p  exp(e1 n ) ,
(b) Dicyclic group of order 4n generated by

 p  exp(e1 ), e2  ,
n
(c) Binary Tetrahedral Group:
1
T  {1, e1, e2 , e3 , (1  e1  e2  e3 )}
2
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Finite Subgroups of Quaternions
(d) Binary Octahedral Group
Denote the set T   (V1  V2  V3 ) with
V1  { 12 (1  e1 ),
1
2
(e2  e3 )} , V2  { 12 (1  e2 ),
1 (  e  e )}
3
1 ,
2
V3  { 12 (1  e3 ), 12 (e1  e2 )}
Then O  T  T  represents the elements of the binary
octahedral group
(e) Binary icosahedral group I  b, c
1
c  (   e1  e2 ) , b  1 (   e1  e2 )
2
2

1 5
2
,

1 5
2
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Coxeter Diagrams A3, B3 and H3 with
quaternionic roots
A3 :
W ( A3 )  {[T , T ]  [T , T ]}:Symmetry of Tetrahedron
B 3:
W (B 3 )  A ut (A 3 )  {[T ,T ]  [T ,T ]  [T ,T ]  [T ,T ]}:Symmetry of Cube
H3 :
W ( H 3 )  {[ I , I ]  [ I , I ]}: Symmetry of Icosahedron
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Platonic solids (regular polyhedra)





Platonic solids are the five convex regular polyhedra. They consist of regular
polygons (triangle, square or pentagon) meeting in identical vertices. They have
identical faces of regular polygons and the same number of faces meeting at each
corner
In geometry, polyhedra are formed in pairs called duals, where the vertices of one
correspond to the faces of the other. The dual of each platonic solid is another
platonic solid:
Tetrahedron is self dual
Cube and Octahedron form a dual pair
Dodecahedron and Icosahedron form
a dual pair
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Archimedean Solids (semi-regular polyhedra)





Two or more types of regular polygons meet in identical vertices.
There are 13 Archimedean solids.
7 of the Archimedean solids can be obtained by truncation of the platonic solids.
4 of the Archimedean solids are obtained by expansion of platonic solids and
previous Archimedean solids.
The remaining 2 chiral solids are snub cube and snub dodecahedron
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,
Construction of polyhedra with Tetrahedral Symmetry
,
For any Coxeter
diagram,
the simple roots α i and their dual vectors
(  i ,  j )  C ij , (  i ,  j )  ( C
r1  [
1
2
( e1  e 2 ), 
1
2
1
) ij , (  i ,  j )   ij ; i, j  1,2,3.
( e1  e 2 )]

r2  [
1
2
( e3  e 2 ), 
ω i satisfy th e scalar product
 i  C ij  j ,  i  ( C
1
2
( e3  e 2 )]

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r3  [
1
2
1
) ij  j .
( e 2  e1 ), 
1
2
( e 2  e1 )]

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A method used to construct the polyhedra

We construct polyhedra using a method based on applying the group
elements of Coxeter-Weyl groups W(A3), W(B3) and W(H3) on a vector
representing one vertex of the polyhedron in the dual space denoted as
  a1 1  a 2 2  a 3 3  ( a1 a 2 a 3 )




This vector is called “highest weight” . It can be expressed as a linear
combination of imaginary quaternionic units.
Certain choices of the parameters of the highest weight vector lead to the
Platonic, Archimedean solids as well as the semi-regular polyhedra.
The set of vertices obtained by the action of Coxeter-Weyl group elements on
the highest weight defines a polyhedron and is called the “orbit”.
Denote by W(G)(a1a2a3)=(a1a2a3)G , the orbit of W(G)
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Construction of polyhedra with Tetrahedral Symmetry
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Construction of polyhedra with Octahedral Symmetry
T he C artan m atrix o f the C o xeter d iag ra m B 3 and its in verse m atrix are g iv e n b y
 2

C  1

 0
1
2



 2 ,

2 
0
2
C
1

 1

  1

 1
 2

1 

2

2.

3 
2 

1
2
2
T he g en erato rs,
r1  [
1
2
( e1  e 2 ), 
1
2

( e1  e 2 )] , r2  [
1
2
( e 2  e3 ), 
1
2

( e 2  e 3 )] , r3  [ e 3 ,  e 3 ]

)
g enerate the o ctahed ra l g ro u p w h ic h ca n be w r itten as


W ( B 3 )  A ut ( A3 )  S 4  C 2  {[ p , p ]  [ p , p ]  [ t , t ]  [ t , t ] } , p  T , t  T  .
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Construction of polyhedra with Octahedral Symmetry
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Construction of polyhedra with Icosahedral symmetry
𝑟1 =
 2

C  

 0

𝑒1 , −𝑒1 ∗ , 𝑟2
1
2
2
1
=
0 

1

2 
𝐶𝐻3 −1
𝜏𝑒1 + 𝑒2 + 𝜎𝑒3 , −
2
1 3𝜏
=
2𝜏 3
2
𝜏3
1
(𝜏𝑒1
2
2𝜏 3
4𝜏 2
2𝜏 2
∗
𝜏3
2𝜏 2 .
𝜏+2
+ 𝑒2 + 𝜎𝑒3 ) , 𝑟3 = 𝑒2 , −𝑒2
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∗
.
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Construction of polyhedra with Icosahedral Symmetry
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Catalan Solids (Duals of Archimedean Solids)
Face transitive (faces are transformed to each other by the Coxeter-Weyl group) .
Faces are non regular polygons: scalene triangles, isosceles triangles, rhombuses,
kites or irregular pentagons. Two Catalan solids are Chiral.
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The method to generate the dual polyhedra



It is based on examining the CoxeterDynkin diagram of the polyhedron
which helps in determining the type
of its faces and the center of a
representative face which
corresponds to the dual’s vertex.
The center is a vector that is left
invariant under the action of the
dihedral subgroup that generated the
face
For example, the Coxeter-Dynkin
diagram of 𝑊(𝐵3 ) can generate
polyhedra with faces listed in Table
depending on the components
(Dynkin indices) of the highest weight
Λ and the dihedral subgroups of
𝑊(𝐵3 ).
Example:
𝜔1 is left unchanged by
𝑟2 , 𝑟3 because
𝑟2 𝜔1 = 𝜔1
𝑟3 𝜔1 = 𝜔1 .
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The method to generate the dual polyhedra
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Catalan Solid Possessing Tetrahedral Symmetry







There is only one Catalan solid that possesses tetrahedral symmetry. It is the triakis
tetrahedron (dual of truncated tetrahedron)which has 8 faces (4 triangles and 4 hexagons).
The vertices of the truncated tetrahedron were obtained as the orbit 𝑾 𝑨𝟑 (𝟏𝟏𝟎).
A triangle of the truncated tetrahedron is generated as the orbit 𝒓𝟐 , 𝒓𝟑 (𝟏𝟎), so its
representative center will be 𝝎𝟏 .
On the other side a hexagon is generated by 𝒓𝟏 , 𝒓𝟐 (𝟏𝟏)and
its center will be 𝝎𝟑 .
The line joining the two centers should be orthogonal to Λ = (110),
so one of the centers should be scaled by 𝜆.
𝜆𝜔1 − 𝜔3 . Λ = 0


𝝎 .𝜦
𝟏
𝟏+𝟐
𝟑
𝟏
𝟏
𝟐
The triakis tetrahedron vertices are the union of the following two orbits
𝜆𝜔1

𝝎 .(𝝎 +𝝎 )
We can determine 𝝀 = 𝝎𝟑.𝜦 = 𝝎𝟑.(𝝎𝟏+𝝎𝟐) = 𝟑+𝟐 = 𝟓 .
3
=
5
1
1
𝑒1 + 𝑒2 + 𝑒3 , 𝑒1 − 𝑒2 − 𝑒3 ,
2
2
1
1
−𝑒1 − 𝑒2 + 𝑒3 , −𝑒1 + 𝑒2 − 𝑒3
2
2
, 𝜔3 =
1
1
−𝑒1 + 𝑒2 + 𝑒3 , 𝑒1 − 𝑒2 + 𝑒3 ,
2
2
.
1
1
𝑒 + 𝑒2 − 𝑒3 , − 𝑒1 + 𝑒2 + 𝑒3
2 1
2
the two orbits comprising the triakis tetrahedron are the vertices of two mirror images tetrahedra.
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Catalan Solid Possessing Octahedral Symmetry






There are 5Catalan solids that possesses octahedral symmetry.
Example:Rhombic dodecahedron (dual of cuboctahedron)
The cuboctahedron was obtained as the orbit 𝑊 𝐵3 (010).
A triangle will be generated as 𝑟1 , 𝑟2 (01) with a center represented by 𝜔3
Asquare is obtained as 𝑟2 , 𝑟3 (10) with a center 𝜔1 .
For the orthogonality condition, the scale factor multiplying 𝜔3 can be calculated to be
𝜆=

𝜔1 . 𝜔2
2
1
=
=
.
𝜔3 . 𝜔2 2 2
2
The vertices of the dual are given as the union of the following two orbits
𝑂 𝜔1 = ±𝑒1 , ±𝑒2 , ±𝑒3 , 𝑂 𝜆𝜔3 =
1
2
±𝑒1 ± 𝑒2 ± 𝑒3 .
The first orbit contains vertices of an octahedron and
the second orbit contains vertices of a cube. The two orbits lie on two concentric spheres.
Since 4 faces of the cuboctahedron meet at one vertex, then the dual’s face will be of 4 vertices
(that is a rhombus).
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Chiral Archimedean and Catalan solids
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Chirality



Objects or molecules which cannot be superimposed with their mirror image
are called chiral. Human hands are one of the example of chirality. Achiral
(not chiral) objects are objects that are identical to their mirror image.
In three dimensional Euclidean space the chirality is defined as follows:
The object which can not be transformed to its mirror image by proper rotations
and translations is called a chiral object.
Chirality is a very interesting topic in
i) molecular chemistry
A number of molecules display one type of chirality; they are either left-oriented
or right-oriented molecules.
ii) In fundamental physics chirality plays very important role. For example:
 A massless Dirac particle has to be either in the left handed state or in the
right handed state.
 The weak interactions which is described by the standard model of high
energy physics is invariant under one type of chiral transformations
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Snub Cube
T he proper rotational subgroup is W ( B 3 ) C 2  S 4 (S ym m etric group of order 4!= 24)
T hey can be generated by the generators a  r1 r2 and b  r2 r3 , a  b
3
4
 ( ab )  1.
2
Let   a1 1  a 2  2  a 3 3  ( a1a 2 a 3 ) be a vector in the d ual space.
T he follow ing sets of vertices form an equ ilateral triangle and a sq uare respecti vely
(  , r1 r2  , ( r1 r2 )  ), (  , r2 r3  , ( r2 r3 )  ), ( r2 r3 )  ) w ith the respective
2
2
3
r3 r2 
r1 r3 
square of edge len g t hs:
2( a1 + a1a 2  a 2 ) and 2( a 2 +
2
2
2
5
2 a 2 a 3  a 3 ).
2
W e have another vertex r1 r3   r3 r1  . T o have all edg e
r1 r2 
4
2
2
2
Factoring by a 2 a nd d efining x 
2
x
3
x
2
a1
a2
2
r2 r1 
and y 
2
a3
one obtains y 
a2
(r3r2 ) 
2
2
2 a 2 a 3  a 3 )  ( a1  a 3 )
2
1
3
length equal the fol low ing equations m ust be satisfied
( a 1 + a 1a 2  a 2 )  ( a 2 +

x
2
1
r2 r3 
and the cubic equation
2
 x  1  0 w hich has one real solu tio n x  1 .83 9 3.
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Snub Cube
The first vertex  and its mirror image   r  can be derived from
the vector   a ( x +   y ) and  II  r1  I  a 2 ( x (1 - 1 ) +  2  y 3 )
II
I
I
2
1
2
1
I
3
can be written in terms of quaternionic units as
a 2 ( x  1)
I 
2
a 2 ( x  1)
2
2
1
( xe1 +e 2  x e3 )  II 
2
deleting the overall scale factor the orbits
can easily be determined as
1
1
( e1  xe 2  x e 3 )
W ( B 3 ) / C 2 (  I ) and W ( B 3 ) / C 2 (  II )
1
1
O (  I )  {(  xe1  e 2  x e 3 ), (  xe 2  e 3  x e1 ), (  xe 3  e 1  x e 2 )}
1
1
1
O (  II )  {(  e1  x e 2  x e3 ), (  e 2  x e 3  x e1 ), (  e 3  x e 1  x e 2 )}
The snub cubes represented by these sets of vertices are shown
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Dual solid of the snub cube
We can determine the centers of the faces in figure below:
The faces 1 and 3 are represented by the vectors  and  up to some scale
factors.
 is invariant under the rotation represented by r1r2 . In other words the
triangle 3 is rotated to itself by a rotation around the vector  .
1
3
3
3
The vectors representing the centers of the faces 2, 4 and 5 can be
determined by averaging the vertices representing these faces and they lie in
the same orbit under the proper octahedral group.
r3 r2 
r1 r3 
The vector representing the center of the face 2 is
c 2  (2 x  1) e1 +e 2  x e3
2
r1 r2 
The scale factors multiplying the vectors
and  
can be determined as  
2
plane containing these five points .
4
2
2
when
1
3
r2 r1 
x

(r3r2 ) 
2
2
 1 ,  3 and  c 2
x
5
I
r2 r3 
represents the normal of the
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Dual solid of the snub cube
Then 38 vertices of the dual solid of the snub
cube, the pentagonal icositetrahedron, are given
in three orbits
x
O ( 1 ) 
2
O ( 3 ) 
1
O ( c 2 ) 
x
2
{  e1 ,  e 2 ,  e 3 }
(  e1  e 2  e 3 )
2
{[  (2 x  1) e1  e 2  x e 3 ],[  (2 x  1) e 2  e 3  x e1 ],[  (2 x  1) e 3  e1  x e 2 ]}
2
2
2
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2
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Snub Dodecahedron
The proper rotational subgroup W(H3)/C2 is the simple finite subgroup of
order 60.
They can be generated by the generators a  r1 r2 and b  r2 r3 , a 5  b 3  ( ab ) 2  1
Let   ( a1 a 2 a 3 ) be a general vector in the dual basis.
The following sets of vertices form
a pentagon and an equilateral triangle
r2 r1 
( r1 r2 ) 
r2 r3 
3
(  , r1 r2  , ( r1 r2 )  , ( r1 r2 )  , ( r1 r2 )  ),
2
3
(  , r2 r3  , ( r2 r3 )  ),
4
2
with the respective square of edge lengths:
2( a1   a1 a 2  a 2 ) and 2( a 2  a 2 a 3  a 3 )
2
2
2
2
2
1

3
4
( r1 r2 ) 
2
rr 
r3 r2 
5
r1 r3 
1 2
We have another vertex : r1 r3   r3 r1 
Let all edge lengths be the same.
The following equation is satisfied ( a1 2   a1 a 2  a 2 2 )  ( a 2 2  a 2 a 3  a 3 2 )  ( a1 2  a 3 2 )
The equation
x  x  x   0
3
2
has the real solution
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x  1 .9 4 3 1 5
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Snub Dodecahedron
The first orbit and its mirror image can be obtained from the vectors
expressed in terms of quaternionic units as
I 
a2
2
[ ( x  1) e1  x e 2  (1   x ) e3 ]
2
3
 II 
a2
2
[  ( x  1) e1  x e 2  (1   x ) e3 ]
2
3
The snub dodecahedrons represented by the orbits of these vectors
W ( H 3 ) / C 2 (  I ) and W ( H 3 ) / C 2 (  II )
are shown:
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Dual solid of the snub dodecahedron
The vertices of the dual solid of the snub dodecahedron represented byW ( H
can be given as the union of three orbits of the group W(H3)/C2.
The first orbit O ( ) consists of 20 vertices of a dodecahedron.
The second orbit consists of 12 vertices of an
icosahedron  O ( 3 ) where
3
) / C 2 ( I )
1
r2 r1 
 3 x   x  2  
2
 
( r1 r2 ) 
2
x  (2   ) x  1
2
3
The third orbit  O ( c 2 ) involves the vertices
including the centers of the faces 2, 4 and 5 where
the vector  c 2 is given by
 c2 
1
3 x   x    2
2
1

5
4
( r1 r2 ) 
2
r3 r2 
r1 r2 
r1 r3 
3
{[(2  1) x   x  1]e1  (  x  3 x  3) e 2   ( x   ) e 3 }
2
2 (21  20) x  (21  17 ) x  21  11
2
r2 r3 
3
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2
3
3
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Dual solid of the snub dodecahedron
Applying the group A  [ I , I ] on the vector  c 2 one generates an
orbit of size 60.
The 92 vertices consisting of these three orbits constitute dual
solid of the snub dodecahedron, pentagonal hexecontahedron.
It is one of the face transitive Catalan solid which has 92
vertices, 180 edges and 60 faces.
5
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Summary
In this work a systematic construction of all Platonic,
Archimedean and Catalan solids and chiral polyhedra, the snub
cube, snub dodecahedron and their duals have been presented.
The Coxeter diagrams A3, B3 and H3 were used to represent
the symmetries of the polyhedra.
A number of programs were developed to generate
Coxeter group elements in terms of quaternions, quaternionic
vertices of polyhedra and to plot the polyhedra.
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