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Schwarz triangles

Schwarz triangles describe the fundamental regions (not necessarily the elemental ones) of a finite reflectional symmetry group with 3 mirrors (generators). In order to be finite, the angles of those spherical triangles have to be all π/q, with q=n/d being some rational. Dually those are represented graphically by their Dynkin diagrams, here again triangles, the tips of which represent the 3 mirrors, the sides representing the pairwise intersection, and thus are marked by the dihedral angle, or rather by its submultiplicative rational. Thus we have

   spherical          Dynkin diagram / 
   triangle           Schwarz triangle
     o
     |\ pi/a            o---a---o
     | \                 \     /
     |  \                 c   b
     |   \ pi/b            \ /
pi/c o----o                 o

Adjoining such smaller spherical triangles to form larger ones clearly results in an addition for Schwarz triangles. Esp. if the summands are built from µ1 resp. µ2 elementary regions, the result will have an according multiplicity µ, which clearly is given as their sum:

    o           o           o
   / \         / \         / \
  p   q   =   p   x   +   x'  q
 /     \     /     \     /     \
o---r---o   o---r1--o   o---r2--o

where

1/r = 1/r1 + 1/r2
1 = 1/x + 1/x'
µ = µ1 + µ2

and moreover both added Schwarz triangles have to belong to the same symmetry group. – Note that this addition of Schwarz triangles not only provides an easy way to reproduce all possible fundamental domains by adjoining more and more elemental ones. In fact, the existance of an Schwarz triangle addition serves also as a condition for Goursat tetrahedra additions.

Note that the resulting triangle of such an addition needs not be a generator for the full symmetry group, it might also produce a subgroup only. Even so such cases are not valid fundamental domains, they are allowed to contribute for further additions (as the sum of a non-valid and a valid simplex might result in a valid one again). Such non-valid Schwarz triangles will be called semi-Schwarz triangles in what follows.

There is a further, more severe exclusion too. In order to adjoin fundamental triangles, the Schwarz triangle vertices next to the "+"-sign (below marked X and Y) represent the adjoined domain edges. Those have to match in their length clearly. For instance in tetrahedral symmetry the following would not work:

    o           o           o    
   / \         / \         / \     D'=2B does NOT exist,
  3  3/2  =   3  3/2  +   3  3/2   addition does NOT match,
 /     \     /     \     /     \   because of |X| ≠ |Y|
o--3/2--o   o---3---X   Y---3---o

Finally it should be noted, that we provide here additions only for convex spherical triangles, i.e. ones with vertex angles α < π. I.e., in the notation from the general addition rule above: r > 1.



Prismatic symmetries   (up)

This case can be provided by a single general addition picture:

    o           o           o
   / \         / \         / \
  2   2   =   2   2   +   2   2
 /     \     /     \     /     \
o---r3--o   o---r1--o   o---r2--o

where

r1 = n/d1
r2 = n/d2
r3 = n/d3
1/r3 = 1/r1 + 1/r2

and thus

d3/n = d1/n + d2/n    or   d3 = d1 + d2

– in other words:

r1 = n/d1
r2 = n/d2
r3 = n/(d1+d2)

Accordingly, here the general Schwarz triangle (2,2,n/d) obviously will have multiplicity µ=d.
The semi cases are given whenever gcd(n,d)>1.



Tetrahedral symmetries   (up)

    o     
   / \                             elementary
  3   3                            A
 /     \                           µ=1
o---2---o 
    o           o           o    
   / \         / \         / \     B=2A
  3   3   =   3   2   +   2   3    µ=2
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     C=B+A
  3   2   =   3  3/2  +   3   2    µ=3
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     D=B+C
 3/2  2   =  3/2  3   +  3/2  2    µ=5
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     E=2C
 3/2 3/2  =  3/2  2   +   2  3/2   µ=6
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
 


Octahedral symmetries   (up)

For this symmetry should be noted additionally, that as rationals 4/2 and 2 surely are the same. But a spot on the sphere having 4fold symmetry and one having only 2fold symmetry does never interchange within the same symmetry setup. Therefore submultiples 4/2 and 2 have to be distinguished. Esp. no 4/d can be added to 2. – On the other hand, triangular mirror configurations with such an angle of 2π/4 look like those having π/2 instead. Accordingly octahedral symmetry gets reduced to some lesser subsymmetry. I.e. any such potential Schwarz triangle will get the attribute semi. – But also any other in here obtained Schwarz triangle, already being counted within above symmetries surely is semi.

    o     
   / \                             elementary
  3   4                            A
 /     \                           µ=1
o---2---o 
    o           o           o    
   / \         / \         / \     B=2A
  4   4   =   4   2   +   2   4    µ=2
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     C=2A
  3   3   =   3   2   +   2   3    µ=2
 /     \     /     \     /     \   semi
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     D=B+A
  4   2   =   4  3/2  +   3   2    µ=3
 /     \     /     \     /     \   semi
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     E=2C
  3   3   =   3  4/2  +  4/2  3    µ=4
 /     \     /     \     /     \   semi
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     F=C+B
  3   4   =   3   3   +  3/2  4    µ=4
 /     \     /     \     /     \   
o--4/3--o   o--4/2--o   o---4---o
                o           o    
               / \         / \     F=A+D
          =   3   2   +   2   4    
             /     \     /     \   
            o---4---o   o--4/2--o
    o           o           o    
   / \         / \         / \     G=F+A
  4   2   =   4  4/3  +   4   2    µ=5
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     H=2D
 4/2 4/2  =  4/2  2   +   2  4/2   µ=6
 /     \     /     \     /     \   semi
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     I=2D
  2   2   =   2  4/2  +  4/2  2    µ=6
 /     \     /     \     /     \   semi
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     J=E+C
  3  4/2  =   3  3/2  +   3  4/2   µ=6
 /     \     /     \     /     \   semi
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     J=F+B
  3  3/2  =   3  4/3  +   4  3/2   
 /     \     /     \     /     \   
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     K=F+D
  3   2   =   3  4/3  +   4   2    µ=7
 /     \     /     \     /     \   
o--4/3--o   o---4---o   o--4/2--o
                o           o    
               / \         / \     K=J+A
          =   3  3/2  +   3   2    
             /     \     /     \   
            o--4/2--o   o---4---o
    o           o           o    
   / \         / \         / \     L=H+D
 4/2  2   =  4/2 4/2  +  4/2  2    µ=9
 /     \     /     \     /     \   semi
o--4/3--o   o--4/2--o   o---4---o
                o           o    
               / \         / \     L=D+I
          =  4/2  2   +   2   2    
             /     \     /     \   
            o---4---o   o--4/2--o
    o           o           o    
   / \         / \         / \     L=G+F
  2  4/3  =   2  3/2  +   3  4/3   
 /     \     /     \     /     \   
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     M=J+E
 4/2 3/2  =  4/2 3/2  +   3  3/2   µ=10
 /     \     /     \     /     \   semi
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     M=2G
 3/2 3/2  =  3/2  2   +   2  3/2   
 /     \     /     \     /     \   
o--4/2--o   o---4---o   o---4---o
    o           o           o    
   / \         / \         / \     N=L+B
  2  3/2  =   2  4/3  +   4  3/2   µ=11
 /     \     /     \     /     \   
o--4/3--o   o--4/2--o   o---4---o
                o           o    
               / \         / \     N=G+J
          =   2  3/2  +   3  3/2   
             /     \     /     \   
            o---4---o   o--4/2--o
    o           o           o    
   / \         / \         / \     N=K+F
  2  4/3  =   2  4/3  +   4  4/3   
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     O=2J
 3/2 3/2  =  3/2 4/2  +  4/2 3/2   µ=12
 /     \     /     \     /     \   semi
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     P=2K
 4/3 4/3  =  4/3  2   +   2  4/3   µ=14
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     P=L+G
 4/3 3/2  =  4/3  2   +   2  3/2   
 /     \     /     \     /     \   
o--4/3--o   o--4/2--o   o---4---o
                o           o    
               / \         / \     P=F+M
          =  4/3  3   +  3/2 3/2   
             /     \     /     \   
            o---4---o   o--4/2--o


Icosahedral symmetries   (up)

    o     
   / \                             elementary
  3   5                            A
 /     \                           µ=1
o---2---o 
    o           o           o    
   / \         / \         / \     B=2A
  5   5   =   5   2   +   2   5    µ=2
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     C=2A
  3   3   =   3   2   +   2   3    µ=2
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     D=B+A
  5   2   =   5  3/2  +   3   2    µ=3
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     E=C+B
  3   5   =   3   3   +  3/2  5    µ=4
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     E=A+D
          =   3   2   +   2   5    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     F=2D
 5/2 5/2  =  5/2  2   +   2  5/2   µ=6
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     G=2D
  5   5   =   5   2   +   2   5    µ=6
 /     \     /     \     /     \   
o--5/4--o   o--5/2--o   o--5/2--o
                o           o    
               / \         / \     G=E+B
          =   5   3   +  3/2  5    
             /     \     /     \   
            o--5/3--o   o---5---o
    o           o           o    
   / \         / \         / \     H=E+C
  5   3   =   5  5/3  +  5/2  3    µ=6
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     I=D+E
  2   3   =   2  5/2  +  5/3  3    µ=7
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     J=H+C
  5  5/2  =   5  3/2  +   3  5/2   µ=8
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     J=G+B
  5  3/2  =   5  5/4  +   5  3/2   
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     K=J+A
  5   2   =   5  3/2  +   3   2    µ=9
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     K=G+D
          =   5  5/4  +   5   2    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     L=K+A
  5   3   =   5   2   +   2   3    µ=10
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     L=J+C
          =   5  3/2  +   3   3    
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     L=G+E
          =   5  5/4  +   5   3    
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     M=I+D
  3  5/2  =   3   2   +   2  5/2   µ=10
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     M=E+F
          =   3  5/3  +  5/2 5/2   
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     M=H+E
  3  5/3  =   3  3/2  +   3  5/3   
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     N=L+A
  5   2   =   5  5/4  +   5   2    µ=11
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     O=M+D
  3   2   =   3  5/3  +  5/2  2    µ=13
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     O=H+I
          =   3  3/2  +   3   2    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     P=J+H
 5/2 3/2  =  5/2 3/2  +   3  3/2   µ=14
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     P=2I
 5/2 5/2  =  5/2  2   +   2  5/2   
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     Q=2I
  3   3   =   3   2   +   2   3    µ=14
 /     \     /     \     /     \   
o--5/4--o   o--5/2--o   o--5/2--o
                o           o    
               / \         / \     Q=M+E
          =   3  5/2  +  5/3  3    
             /     \     /     \   
            o--5/3--o   o---5---o
    o           o           o    
   / \         / \         / \     R=H+M
  3  5/2  =   3  3/2  +   3  5/2   µ=16
 /     \     /     \     /     \   
o--5/4--o   o---5---o   o--5/3--o
                o           o    
               / \         / \     R=M+F
          =   3  5/3   +  5/2 5/2   
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     R=O+D
          =   3   2   +   2  5/2   
             /     \     /     \   
            o--5/3--o   o---5---o
    o           o           o    
   / \         / \         / \     R=L+G
  3  5/4  =   3  5/4  +   5  5/4   
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     S=J+K
 3/2  2   =  3/2 5/2  +  5/3  2    µ=17
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     S=M+I
 5/2  2   =  5/2 5/3  +  5/2  2    
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     T=J+L
 3/2  3   =  3/2  5   +  5/4  3    µ=18
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     T=B+R
          =  3/2  5   +  5/4  3    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     T=E+Q
 5/3  3   =  5/3  5   +  5/4  3    
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     U=P+E
 5/2 5/3  =  5/2 3/2  +   3  5/3   µ=18
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     U=J+M
          =  5/2 3/2  +   3  5/3   
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     U=2K
 5/3 5/3  =  5/3  2   +   2  5/3   
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     V=T+A
  3   2   =   3  3/2  +   3   2    µ=19
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     V=R+D
          =   3  5/4  +   5   2    
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     V=L+K
          =   3  5/4  +   5   2    
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     W=U+D
 5/2  2   =  5/2 5/3  +  5/2  2    µ=21
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     W=P+I
          =  5/2 3/2  +   3   2    
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     W=J+O
          =  5/2 3/2  +   3   2    
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     W=L+N
 5/4  2   =  5/4  3   +  3/2  2    
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     X=2N
 3/2 3/2  =  3/2  2   +   2  3/2   µ=22
 /     \     /     \     /     \   
o--5/2--o   o---5---o   o---5---o
    o           o           o    
   / \         / \         / \     X=R+H
 5/2 3/2  =  5/2 5/4  +   5  3/2   
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     Y=M+O
 5/3  2   =  5/3 5/2  +  5/3  2    µ=23
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     Y=P+K
 3/2  2   =  3/2 5/2  +  5/3  2    
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     Y=H+S
          =  3/2  3   +  3/2  2    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     Z=2O
 5/3 5/3  =  5/3  2   +   2  5/3   µ=26
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     Z=S+K
 3/2 5/3  =  3/2  2   +   2  5/3   
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     Z=J+U
          =  3/2 5/2  +  5/3 5/3   
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     AA=R+N
 5/4  2   =  5/4  3   +  3/2  2    µ=27
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     AA=G+W
          =  5/4  5   +  5/4  2    
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     AA=U+K
 5/3  2   =  5/3 5/2  +  5/3  2    
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AA=M+S
          =  5/3  3   +  3/2  2    
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     AA=E+O
          =  5/3  3   +  3/2  2    
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     AB=L+V
 5/4  2   =  5/4  5   +  5/4  2    µ=29
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     AB=T+N
 3/2  2   =  3/2  3   +  3/2  2    
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AB=J+W
          =  3/2  5   +  5/4  2    
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     AB=B+AA
          =  3/2  5   +  5/4  2    
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     AC=W+N
 5/4 3/2  =  5/4  2   +   2  3/2   µ=32
 /     \     /     \     /     \   
o--5/3--o   o--5/2--o   o---5---o
                o           o    
               / \         / \     AC=L+X
          =  5/4  3   +  3/2 3/2   
             /     \     /     \   
            o---5---o   o--5/2--o
    o           o           o    
   / \         / \         / \     AC=Q+T
 5/4 5/3  =  5/4  3   +  3/2 5/3   
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     AC=Y+K
 3/2 5/3  =  3/2  2   +   2  5/3   
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AC=P+U
          =  3/2 5/2  +  5/3 5/3   
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     AC=H+Z
          =  3/2  3   +  3/2 5/3   
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     AD=Z+J
 3/2 3/2  =  3/2 5/3  +  5/2 3/2   µ=34
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AD=2S
          =  3/2  2   +   2  3/2   
             /     \     /     \   
            o--5/2--o   o--5/2--o
    o           o           o    
   / \         / \         / \     AD=R+T
 5/4 3/2  =  5/4 5/2  +  5/3 3/2   
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     AE=2V
 5/4 5/4  =  5/4  2   +   2  5/4   µ=38
 /     \     /     \     /     \   
o--3/2--o   o---3---o   o---3---o
    o           o           o    
   / \         / \         / \     AE=AC+G
 3/2 5/4  =  3/2 5/4  +   5  5/4   
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AE=X+R
          =  3/2 3/2  +   3  5/4   
             /     \     /     \   
            o--5/2--o   o--5/2--o
                o           o    
               / \         / \     AE=N+AA
          =  3/2  2   +   2  5/4   
             /     \     /     \   
            o---5---o   o--5/3--o
    o           o           o    
   / \         / \         / \     AF=AC+L
 5/4 5/4  =  5/4 3/2  +   3  5/4   µ=42
 /     \     /     \     /     \   
o--5/4--o   o--5/3--o   o---5---o
                o           o    
               / \         / \     AF=2W
          =  5/4  2   +   2  5/4   
             /     \     /     \   
            o--5/2--o   o--5/2--o


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