The set of 3D polyhedra is available sorted by complexity
or by point-group symmetry
or by similarity (below).

### The Schwarz triangles oPoQoR*a and the general incidence matrices of the corresponding polyhedra

The set of possible Schwarz triangles can be achieved as is shown here. Based on those the kaleidoscopic construction from Wythoff produces (simple) polyhedra or more generally Grünbaumian polyhedra. Accordingly we have the following general types of possible Dynkin diagrams:

```xPoQoR*a
xPxQoR*a
xPxQxR*a
```

But for the context of incidence matrices we have to be more careful with numbers being 2, as trigons etc. are full valid faces, whereas a digon (x2o) degenerates to a (doubled) edge, and thus degenerates into an element of smaller dimension. On the other hand an alternate bi-digon (x2x), i.e. a rectangle, is still full dimensional. And finally, additional symmetry of the diagram like Q = R can be reflected in the matrix, as those rows resp. columns might be combined. Therefore we are left to consider the following 11 cases:

o,o,x
o,x,x
x,x,x

Next to the dihedral angles of the fundamental spherical triangle (the Schwarz triangle) the group order 'g' and the multiplicity 'µ' prove usefull. In this dimension they can be calculated from those angle numbers. If 'A' denotes the spherical area of the Schwarz triangle and 'ε' the spherical excess:

```A = ε
4πµ = gA
ε = π/p + π/q + π/r - π = πd/n + πb/m + πc/k - π
-> g = 4µ/(1/p + 1/q + 1/r -1) = 4nmkµ/(dmk + bnk + cnm - nmk)
```

The euclidean cases can be derived as the limiting case of vanishing curvature. So 'g' becomes generally infinite, whereas the denominators of the above resulting equation become zero.

Note, that any disconnected part of a Dynkin diagram, i.e. by having p=q=2 or the like, needs to have at least one knot ringed; else in the Wythoff construction the span of the resulting vertex set would become sub-dimensional.

It could be mentioned that the seed point of Wythoff's kaleidoscopic construction for xPxQxR*a happens to be the incenter of this fundamental region, i.e. the spherical triangle. Let ξi denote the angular distance from the according vertex of that spherical triangle (measured along the great circle), then those coordinates can be obtained from

```sin(ξ3) sin(π/2p) = sin(ξ1) sin(π/2q) = sin(ξ2) sin(π/2r)
```

### Snubs and holosnubs on oPoQoR*a groups

Beside of the usual node marks "x" and "o" for ringed nodes and unringed ones there are further node marks in use, for instance with respect to snubs. There in the full generality ringed nodes (denoted "x"), unringed nodes (denoted "o"), and rings without nodes (denoted "s") might co-exist. For holosnubs the nodemark "s" is replaced by "β". Recall that "s"-nodes are usable only iff all snub-nodes have adjoining link marks which are even or at least have even numerators. Else "β"-nodes are to be used throughout. In the former case they are called snubs, while in the latter they are holosnubs. The holosnub derivation is exactly the same as the snub derivation, only that the alternated faceting is rather applied to the Grünbaumian double cover of the starting figure than to the polyhedron itself.

Just as for Wythoff polyhedra the subdiagram x2o has to be considered separately, for snubs in addition the subdiagrams s2o, s4o, s4/3o, s2x, and s2s (respectively the corresponding ones for holosnubs) have to be considered separately too. Therefore for snubs we are left to consider the following 53 cases (the distinction between "s"-nodes and "β"-nodes will be done subsequently in the corresponding chapters; alike the distinction between s4o and s4/3o, the only group using both is o4/3o3o4*a):

o,o,s
o,s,x
o,s,s
s,x,x
s,s,x
s,s,s

The seed point of the kaleidoscopic construction for sPsQsR*a could be derived quite similarily, although the reflections themselves are missing here, only the rotational subgroup is to be considered. This is due to the fact that the additional snub face (triangle), occuring as the sefa (sectioning facet underneath) of the omitted vertex in the vertex alternating construction of snubbing has to be equilatteral too. Let ξi again denote the angular distance from the according vertex of the fundamental region (measured along the great circle), then those coordinates can be obtained from

```sin(ξ3) sin(π/p) = sin(ξ1) sin(π/q) = sin(ξ2) sin(π/r)
```

#### xPoQoR*a with p=n/d≠2, q=m/b≠2, r=k/c≠2, p<r   (up)

(alike p=r for colored faces)

```. . .    | g/2m |  2m |    m    m
---------+------+-----+----------
x . .    |    2 | g/2 |    1    1
---------+------+-----+----------
xPo .    |    n |   n | g/2n    *
x . oR*a |    k |   k |    * g/2k

vf is Archimedean 2q-gon: . x(p)-Q-x(r) :

special cases (p<r):
p=3/2 q=6   r=6   -> n=3 m=6 k=6 g=∞ µ=2
-> vf=[(3/2,6)6] -> "trat+3hexat"
(vertices identified, edges coincide by pairs)

p=5/2 q=5/4 r=3   -> n=5 m=5 k=3 g=120 µ=16
p=3/2 q=5   r=5/2 -> n=3 m=5 k=5 g=120 µ=8
p=5/3 q=5   r=3   -> n=5 m=5 k=3 g=120 µ=4
p=3/2 q=5/4 r=5/3 -> n=3 m=5 k=5 g=120 µ=32
-> vf=[(5/3,3)5] -> gacid ("sissid + gike")
(vertices identified, edges coincide by pairs)

p=5/2 q=3/2 r=5   -> n=5 m=3 k=5 g=120 µ=8
p=5/4 q=3   r=5/2 -> n=5 m=3 k=5 g=120 µ=16
p=5/3 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=4
p=5/4 q=3/2 r=5/3 -> n=5 m=3 k=5 g=120 µ=32
-> vf=[(5/3,5)3] -> ditdid

p=5/2 q=5/3 r=3   -> n=5 m=5 k=3 g=120 µ=10
p=3/2 q=5/2 r=5/2 -> n=3 m=5 k=5 g=120 µ=14
p=5/3 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=10
p=3/2 q=5/3 r=5/3 -> n=3 m=5 k=5 g=120 µ=26
-> vf=[(5/2,3)5]/3 -> gacid ("sissid + gike")
(vertices identified, edges coincide by pairs)

p=5/2 q=3   r=3   -> n=5 m=3 k=3 g=120 µ=2
p=3/2 q=3/2 r=5/2 -> n=3 m=3 k=5 g=120 µ=22
p=5/3 q=3/2 r=3   -> n=5 m=3 k=3 g=120 µ=18
p=3/2 q=3   r=5/3 -> n=3 m=3 k=5 g=120 µ=18
-> vf=[(5/2,3)3] -> sidtid

p=3   q=3/2 r=5   -> n=3 m=3 k=5 g=120 µ=6
p=5/4 q=3   r=3   -> n=5 m=3 k=3 g=120 µ=14
p=3/2 q=3   r=5   -> n=3 m=3 k=5 g=120 µ=6
p=5/4 q=3/2 r=3/2 -> n=5 m=3 k=3 g=120 µ=34
-> vf=[(3,5)3]/2 -> gidtid

p=3   q=5/4 r=5   -> n=3 m=5 k=5 g=120 µ=10
p=5/4 q=5   r=3   -> n=5 m=5 k=3 g=120 µ=10
p=3/2 q=5   r=5   -> n=3 m=5 k=5 g=120 µ=2
p=5/4 q=5/4 r=3/2 -> n=5 m=5 k=3 g=120 µ=38
-> vf=[(3/2,5)5] -> cid ("ike + gad")
(vertices identified, edges coincide by pairs)

p=3   q=4/3 r=4   -> n=3 m=4 k=4 g=48 µ=4
p=4/3 q=4   r=3   -> n=4 m=4 k=3 g=48 µ=4
p=3/2 q=4   r=4   -> n=3 m=4 k=4 g=48 µ=2
p=4/3 q=4/3 r=3/2 -> n=4 m=4 k=3 g=48 µ=14
-> vf=[(3/2,4)4]=[(3,4)4]/3 -> "oct+6{4}"
(vertices identified, edges coincide by pairs)

p=3   q=5/3 r=5   -> n=3 m=5 k=5 g=120 µ=4
p=5/4 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=16
p=3/2 q=5/2 r=5   -> n=3 m=5 k=5 g=120 µ=8
p=5/4 q=5/3 r=3/2 -> n=5 m=5 k=3 g=120 µ=32
-> vf=[(3,5)5]/3 -> cid ("ike + gad")
(vertices identified, edges coincide by pairs)

p=5/4 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=10
-> vf=[56]/2 -> "2doe"
(vertices identified, edges and faces both coincide by pairs)

p=5/4 q=5   r=5   -> n=5 m=5 k=5 g=120 µ=6
-> vf=[510]/4 -> "2gad"
(vertices identified, edges and faces both coincide by pairs)

p=4/3 q=3   r=4   -> n=4 m=3 k=4 g=48 µ=4
-> vf=[46]/2 -> "2cube"
(vertices identified, edges and faces both coincide by pairs)

p=3/2 q=5/3 r=3   -> n=3 m=5 n=3 g=120 µ=18
-> vf=[310]/2 -> "2ike"
(vertices identified, edges and faces both coincide by pairs)

p=3/2 q=3   r=3   -> n=3 m=3 k=3 g=24 µ=2
-> vf=[36]/2 -> "2tet"
(vertices identified, edges and faces both coincide by pairs)

p=3/2 q=5   r=3   -> n=3 m=5 k=3 g=120 µ=6
-> vf=[310]/4 -> "2gike"
(vertices identified, edges and faces both coincide by pairs)

p=5/3 q=5/3 r=5/2 -> n=5 m=5 k=5 g=120 µ=18
-> vf=[(5/2)10]/2 -> "2sissid"
(vertices identified, edges and faces both coincide by pairs)

p=5/3 q=3   r=5/2 -> n=5 m=3 k=5 g=120 µ=10
-> vf=[(5/2)6]/2 -> "2gissid"
(vertices identified, edges and faces both coincide by pairs)
```

#### xPoQoP*a with p=n/d≠2, q=m/b≠2   (up)

According to the symmetrical Dynkin symbol the total symmetry order is increased: g+ = 2g

```. . .              | g/2m |  2m |  2m
-------------------+------+-----+----
x . .              |    2 | g/2 |   2
-------------------+------+-----+----
xPo .  &  x . oP*a |    n |   n | g/n

vf is Archimedean 2q-gon: . x(p)-Q-x(p) :

special cases:
p=5/2 q=3/2 -> n=5 m=3 g=120 µ=14
p=5/3 q=3/2 -> n=5 m=3 g=120 µ=26
-> vf=[(5/2)6]/2 -> "2gissid"
(vertices identified, edges and faces both coincide by pairs)

p=5/2 q=5/2 -> n=5 m=5 g=120 µ=6
p=5/3 q=5/2 -> n=5 m=5 g=120 µ=18
-> vf=[(5/2)10]/2 -> "2sissid"
(vertices identified, edges and faces both coincide by pairs)

p=3   q=5/4 -> n=3 m=5 g=120 µ=14
p=3/2 q=5/4 -> n=3 m=5 g=120 µ=34
-> vf=[310]/4 -> "2gike"
(vertices identified, edges and faces both coincide by pairs)

p=3   q=3/2 -> n=3 m=3 g=24 µ=2
p=3/2 q=3/2 -> n=3 m=3 g=24 µ=6
-> vf=[36]/2 -> "2tet"
(vertices identified, edges and faces both coincide by pairs)

p=3   q=5/2 -> n=3 m=5 g=120 µ=2
p=3/2 q=5/2 -> n=3 m=5 g=120 µ=22
-> vf=[310]/2 -> "2ike"
(vertices identified, edges and faces both coincide by pairs)

p=3   q=3   -> n=3 m=3 g=∞ µ=1
-> vf=[36] -> trat

p=4   q=3/2 -> n=4 m=3 g=48 µ=2
p=4/3 q=3/2 -> n=4 m=3 g=48 µ=14
-> vf=[46]/2 -> "2cube"
(vertices identified, edges and faces both coincide by pairs)

p=5   q=5/4 -> n=5 m=5 g=120 µ=6
p=5/4 q=5/4 -> n=5 m=5 g=120 µ=42
-> vf=[510]/4 -> "2gad"
(vertices identified, edges and faces both coincide by pairs)

p=5   q=3/2 -> n=5 m=3 g=120 µ=2
p=5/4 q=3/2 -> n=5 m=3 g=120 µ=38
-> vf=[56]/2 -> "2doe"
(vertices identified, edges and faces both coincide by pairs)

p=6   q=3/2 -> n=6 m=3 g=∞ µ=2
-> vf=[66]/2 -> "2hexat"
(vertices identified, edges and faces both coincide by pairs)
```

#### xPoQo with p=n/d≠2, q=m/b≠2   (up)

```. . . | g/2m |   m |    m
------+------+-----+-----
x . . |    2 | g/4 |    2
------+------+-----+-----
xPo . |    n |   n | g/2n

vf. is regular q-gon: . x(p)-Q-o

special cases:
p=5/4 q=3/2 -> n=5 m=3 g=120 µ=29
p=5/4 q=3   -> n=5 m=3 g=120 µ=19
p=5   q=3/2 -> n=5 m=3 g=120 µ=11
p=5   q=3   -> n=5 m=3 g=120 µ=1
-> vf=[53] -> doe

p=5/4 q=5/3 -> n=5 m=5 g=120 µ=27
p=5/4 q=5/2 -> n=5 m=5 g=120 µ=21
p=5   q=5/3 -> n=5 m=5 g=120 µ=9
p=5   q=5/2 -> n=5 m=5 g=120 µ=3
-> vf=[55]/2 -> gad

p=4/3 q=3/2 -> n=4 m=3 g=48 µ=11
p=4/3 q=3   -> n=4 m=3 g=48 µ=7
p=4   q=3/2 -> n=4 m=3 g=48 µ=5
p=4   q=3   -> n=4 m=3 g=48 µ=1
-> vf=[43] -> cube

p=3/2 q=5/4 -> n=3 m=5 g=120 µ=29
p=3/2 q=5   -> n=3 m=5 g=120 µ=11
p=3   q=5/4 -> n=3 m=5 g=120 µ=19
p=3   q=5   -> n=3 m=5 g=120 µ=1
-> vf=[35] -> ike

p=3/2 q=4/3 -> n=3 m=4 g=48 µ=11
p=3/2 q=4   -> n=3 m=4 g=48 µ=5
p=3   q=4/3 -> n=3 m=4 g=48 µ=7
p=3   q=4   -> n=3 m=4 g=48 µ=1
-> vf=[34] -> oct

p=3/2 q=3/2 -> n=3 m=3 g=24 µ=5
p=3/2 q=3   -> n=3 m=3 g=24 µ=3
p=3   q=3/2 -> n=3 m=3 g=24 µ=3
p=3   q=3   -> n=3 m=3 g=24 µ=1
-> vf=[33] -> tet

p=3/2 q=5/3 -> n=3 m=5 g=120 µ=23
p=3/2 q=5/2 -> n=3 m=5 g=120 µ=17
p=3   q=5/3 -> n=3 m=5 g=120 µ=13
p=3   q=5/2 -> n=3 m=5 g=120 µ=7
-> vf=[35]/2 -> gike

p=5/3 q=5/4 -> n=5 m=5 g=120 µ=27
p=5/3 q=5   -> n=5 m=5 g=120 µ=9
p=5/2 q=5/4 -> n=5 m=5 g=120 µ=21
p=5/2 q=5   -> n=5 m=5 g=120 µ=3
-> vf=[(5/2)5] -> sissid

p=5/3 q=3/2 -> n=5 m=3 g=120 µ=23
p=5/3 q=3   -> n=5 m=3 g=120 µ=13
p=5/2 q=3/2 -> n=5 m=3 g=120 µ=17
p=5/2 q=3   -> n=5 m=3 g=120 µ=7
-> vf=[(5/2)3] -> gissid

p=3   q=6   -> n=3 m=6 g=∞ µ=1
-> vf=[36] -> trat

p=4   q=4   -> n=4 m=4 g=∞ µ=1
-> vf=[44] -> squat

p=6   n=3   -> n=6 m=3 g=∞ µ=1
-> vf=[63] -> hexat
```

#### oPxQo with p=n/d≠2, q=m/b≠2, p<q   (up)

(alike p=q for colored faces)

```. . . | g/4 |   4 |    2    2
------+-----+-----+----------
. x . |   2 | g/2 |    1    1
------+-----+-----+----------
oPx . |   n |   n | g/2n    *
. xQo |   m |   m |    * g/2m

vf. is rectagle: x(p) . x(q)

special cases:
p=5/4 q=3/2 -> n=5 m=3 g=120 µ=29
p=5/4 q=3   -> n=5 m=3 g=120 µ=19
p=3/2 q=5   -> n=3 m=5 g=120 µ=11
p=3   q=5   -> n=3 m=5 g=120 µ=1
-> vf=[(3,5)2] -> id

p=5/4 q=5/3 -> n=5 m=5 g=120 µ=27
p=5/4 q=5/2 -> n=5 m=5 g=120 µ=21
p=5/3 q=5   -> n=5 m=5 g=120 µ=9
p=5/2 q=5   -> n=5 m=5 g=120 µ=3
-> vf=[(5/2,5)2] -> did

p=4/3 q=3/2 -> n=4 m=3 g=48 µ=11
p=4/3 q=3   -> n=4 m=3 g=48 µ=7
p=3/2 q=4   -> n=3 m=4 g=48 µ=5
p=3   q=4   -> n=3 m=4 g=48 µ=1
-> vf=[(3,4)2] -> co

p=3/2 q=5/3 -> n=3 m=5 g=120 µ=23
p=3/2 q=5/2 -> n=3 m=5 g=120 µ=17
p=5/3 q=3   -> n=5 m=3 g=120 µ=13
p=5/2 q=3   -> n=5 m=3 g=120 µ=7
-> vf=[(5/2,3)2] -> gid

p=3/2 q=3   -> n=3 m=3 g=24 µ=3
-> vf=[34] -> oct

p=3   q=6   -> n=3 m=6 g=∞ µ=1
-> vf=[(3,6)2] -> that
```

#### oPxPo with p=n/d≠2   (up)

According to the symmetrical Dynkin symbol the total symmetry order is increased: g+ = 2g

```. . .           | g/4 |   4 |   4
----------------+-----+-----+----
. x .           |   2 | g/2 |   2
----------------+-----+-----+----
oPx .  &  . xPo |   n |   n | g/n

vf. is square: x(p) . x(p)

special cases:
p=3/2 -> n=3 g=24 µ=5
p=3   -> n=3 g=24 µ=1
-> vf=[34] -> oct

p=4   -> n=4 g=∞ µ=1
-> vf=[44] -> squat
```

#### xPxQoR*a with p=n/d, q=m/b≠2, r=k/c≠2, q<r   (up)

(alike q=r for colored faces or if node x1≠x2)

```. . .    | g/2 |   2   2 |    2    1    1
---------+-----+---------+---------------
x . .    |   2 | g/2   * |    1    1    0
. x .    |   2 |   * g/2 |    1    0    1
---------+-----+---------+---------------
xPx .    |  2n |   n   n | g/2n    *    *
x . oR*a |   k |   k   0 |    * g/2k    *
. xQo    |   m |   0   m |    *    * g/2m

vf. is trapezium: x(q) || x(r) #(2p)

special cases with x1=x2:
cases 1/q + 1/r ≠ 1:
p=5/4 q=5/4 r=3/2 -> n=5 m=5 k=3 g=120 µ=38
p=5/4 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=10
-> vf=3[3,10/4,5/4(!),10/4] -> "sidtid+ditdid"
(vertices coincide by three, edges coincide by pairs)

p=5/4 q=3/2 r=5/3 -> n=5 m=3 k=5 g=120 µ=32
p=5/4 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=16
-> vf=5[3,10/4,5/2,10/4] -> "3sissid+gike"
(vertices coincide by five, edges coincide by four, {5/2} and {10/2} coincide)

p=4/3 q=4/3 r=3/2 -> n=4 m=4 k=3 g=48 µ=14
p=4/3 q=3   r=4   -> n=4 m=3 k=4 g=48 µ=4
-> vf=[3,8/3,4,8/3] -> gocco

p=3/2 q=5/4 r=3/2 -> n=3 m=5 k=3 g=120 µ=34
p=3/2 q=3   r=5   -> n=3 m=3 k=5 g=120 µ=6
-> vf=5[3,6/2,5,6/2] -> "3ike+gad"
(vertices coincide by five, edges coincide by four, {3} and {6/2} coincide)

p=3/2 q=5/4 r=5/3 -> n=3 m=5 k=5 g=120 µ=32
p=3/2 q=5/2 r=5   -> n=3 m=5 k=5 g=120 µ=8
-> vf=3[5/2,6/2,5,6/2] -> "sidtid+gidtid"
(vertices coincide by three, edges coincide by pairs)

p=3/2 q=3/2 r=5/2 -> n=3 m=3 k=5 g=120 µ=22
p=3/2 q=5/3 r=3   -> n=3 m=5 k=3 g=120 µ=18
-> vf=5[5/3,6/2,3,6/2] -> "sissid+3gike"
(vertices coincide by five, edges coincide by four, {3} and {6/2} coincide)

p=5/3 q=5/4 r=3/2 -> n=5 m=5 k=3 g=120 µ=32
p=5/3 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=4
-> vf=[3,10/3,5,10/3] -> gidditdid

p=5/3 q=3/2 r=5/3 -> n=5 m=3 k=5 g=120 µ=26
p=5/3 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=10
-> vf=[5/2,10/3,3,10/3] -> gaddid

p=2   q=5/4 r=3/2 -> n=2 m=5 k=3 g=120 µ=29
p=2   q=3   r=5   -> n=2 m=3 k=5 g=120 µ=1
-> vf=[3,4,5,4] -> srid

p=2   q=5/4 r=5/3 -> n=2 m=5 k=5 g=120 µ=27
p=2   q=5/2 r=5   -> n=2 m=5 k=5 g=120 µ=3
-> vf=[5/2,4,5,4] -> raded

p=2   q=5/4 r=5/2 -> n=2 m=5 k=5 g=120 µ=21
p=2   q=5/3 r=5   -> n=2 m=5 k=5 g=120 µ=9
-> vf=3[5/3,4,5,4] -> cadditradid ("ditdid + rhom")
(vertices coincide by three, edges coincide by pairs)

p=2   q=5/4 r=3   -> n=2 m=5 k=3 g=120 µ=19
p=2   q=3/2 r=5   -> n=2 m=3 k=5 g=120 µ=11
-> vf=3[3/2,4,5/4(!),4] -> gicdatrid ("gidtid + rhom")
(vertices coincide by three, edges coincide by pairs)

p=2   q=4/3 r=3/2 -> n=2 m=4 k=3 g=48 µ=11
p=2   q=3   r=4   -> n=2 m=3 k=4 g=48 µ=1
-> vf=[3,4,4,4] -> sirco

p=2   q=4/3 r=3   -> n=2 m=4 k=3 g=48 µ=7
p=2   q=3/2 r=4   -> n=2 m=3 k=4 g=48 µ=5
-> vf=[3/2,4,4,4] -> querco

p=2   q=3/2 r=5/3 -> n=2 m=3 k=5 g=120 µ=23
p=2   q=5/2 r=3   -> n=2 m=5 k=3 g=120 µ=7
-> vf=3[5/2,4,3,4] -> sicdatrid ("sidtid + rhom")
(vertices coincide by three, edges coincide by pairs)

p=2   q=3/2 r=5/2 -> n=2 m=3 k=5 g=120 µ=17
p=2   q=5/3 r=3   -> n=2 m=5 k=3 g=120 µ=13
-> vf=[5/3,4,3,4] -> qrid

p=2   q=3   r=6   -> n=2 m=3 k=6 g=∞ µ=1
-> vf=[3,4,6,4] -> rothat

p=5/2 q=5/4 r=3   -> n=5 m=5 k=3 g=120 µ=16
p=5/2 q=3/2 r=5   -> n=5 m=3 k=5 g=120 µ=8
-> vf=5[3/2,10/2,5,10/2] -> "ike+3gad"
(vertices coincide by five, edges coincide by four, {5} and {10/2} coincide)

p=5/2 q=3/2 r=5/2 -> n=5 m=3 k=5 g=120 µ=14
p=5/2 q=5/3 r=3   -> n=5 m=5 k=3 g=120 µ=10
-> vf=3[5/3,10/2,3,10/2] -> "ditdid+gidtid"
(vertices coincide by three, edges coincide by pairs)

p=3   q=5/4 r=5/2 -> n=3 m=5 k=5 g=120 µ=16
p=3   q=5/3 r=5   -> n=3 m=5 k=5 g=120 µ=4
-> vf=[5/3,6,5,6] -> ided

p=3   q=5/4 r=3   -> n=3 m=5 k=3 g=120 µ=14
p=3   q=3/2 r=5   -> n=3 m=3 k=5 g=120 µ=6
-> vf=[3/2,6,5,6] -> giid

p=3   q=3/2 r=5/3 -> n=3 m=3 k=5 g=120 µ=18
p=3   q=5/2 r=3   -> n=3 m=5 k=3 g=120 µ=2
-> vf=[5/2,6,3,6] -> siid

p=4   q=4/3 r=3   -> n=4 m=4 k=3 g=48 µ=4
p=4   q=3/2 r=4   -> n=4 m=3 k=4 g=48 µ=2
-> vf=[3/2,8,4,8] -> socco

p=5   q=5/4 r=3   -> n=5 m=5 k=3 g=120 µ=10
p=5   q=3/2 r=5   -> n=5 m=3 k=5 g=120 µ=2
-> vf=[3/2,10,5,10] -> saddid

p=5   q=3/2 r=5/2 -> n=5 m=3 k=5 g=120 µ=8
p=5   q=5/3 r=3   -> n=5 m=5 k=3 g=120 µ=4
-> vf=[5/3,10,3,10] -> sidditdid

p=6   q=3/2 r=6   -> n=6 m=3 k=6 g=∞ µ=2
-> vf=[3/2,12,6,12] -> shothat (euclidean analogon of socco resp. saddid)

cases 1/q + 1/r = 1:
p=5/3 q=3/2 r=3   -> n=5 m=3 k=3 g=120 µ=18
-> vf=2[3/2,10/3,3,10/3] -> "2geihid"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=5/3 q=5/3 r=5/2 -> n=5 m=5 k=5 g=120 µ=18
-> vf=2[5/3,10/3,5/2,10/3] -> "2gidhid"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=2   q=3/2 r=3   -> n=2 m=3 k=3 g=24 µ=3
-> vf=2[3/2,4,3,4] -> "2thah"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=3   q=5/4 r=5   -> n=3 m=5 k=5 g=120 µ=10
-> vf=2[5/4,6,5,6] -> "2gidhei"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=3   q=4/3 r=4   -> n=3 m=4 k=4 g=48 µ=4
-> vf=2[4/3,6,4,6] -> "2cho"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=3   q=3/2 r=3   -> n=3 m=3 k=3 g=24 µ=2
-> vf=[3/2,6,3,6] -> oho

p=3   q=5/3 r=5/2 -> n=3 m=5 k=5 g=120 µ=10
-> vf=2[5/3,6,5/2,6] -> "2sidhei"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=5   q=5/4 r=5   -> n=5 m=5 k=5 g=120 µ=6
-> vf=2[5/4,10,5,10] -> "2sidhid"
(vertices coincide by pairs, edges and faces both coincide by pairs)

p=5   q=3/2 r=3   -> n=5 m=3 k=3 g=120 µ=6
-> vf=2[3/2,10,3,10] -> "2seihid"
(vertices coincide by pairs, edges and faces both coincide by pairs)
```

#### xPxQoQ*a with p=n/d, q=m/b≠2, and x1=x2   (up)

According to the symmetrical Dynkin symbol the total symmetry order is increased: g+ = 2g

```. . .              | g/2 |  4 |    2   2
-------------------+-----+----+---------
x . .     &  . x . |   2 |  g |    1   1
-------------------+-----+----+---------
xPx .              |  2n | 2n | g/2n   *
x . oQ*a  &  . xQo |   m |  m |    * g/m

vf. is rectangle: x(q) || x(q) #(2p)

special cases:
p=5/4 q=5/4 -> n=5 m=5 g=120 µ=42
p=5/4 q=5   -> n=5 m=5 g=120 µ=6
-> vf=2[(10/4,5)2] -> "2did"
(vertices, edges, and {5} all coincide by pairs)

p=5/4 q=3/2 -> n=5 m=3 g=120 µ=34
p=5/4 q=3   -> n=5 m=3 g=120 µ=14
-> vf=2[(10/4,3)2] -> "2gid"
(vertices, edges, and {3} all coincide by pairs)

p=3/2 q=4/3 -> n=3 m=4 g=48 µ=14
p=3/2 q=4   -> n=3 m=4 g=48 µ=2
-> vf=2[(6/2,4)2] -> "2co"
(vertices, edges, and {4} all coincide by pairs)

p=3/2 q=3/2 -> n=3 m=3 g=24 µ=6
p=3/2 q=3   -> n=3 m=3 g=24 µ=2
-> vf=2[(6/2,3)2] -> "2oct"
(vertices, edges, and {3} all coincide by pairs)

p=3/2 q=5/3 -> n=3 m=5 g=120 µ=26
p=3/2 q=5/2 -> n=3 m=5 g=120 µ=14
-> vf=2[(5/2,6/2)2] -> "2gid"
(vertices, edges, and {5/2} all coincide by pairs)

p=3/2 q=5/4 -> n=3 m=5 g=120 µ=38
p=3/2 q=5   -> n=3 m=5 g=120 µ=2
-> vf=2[(6/2,5)2] -> "2id"
(vertices, edges, and {5} all coincide by pairs)

p=3/2 q=6   -> n=3 m=6 g=∞ µ=2
-> vf=2[(6/2,6)2] -> "2that"
(vertices, edges, and {6} all coincide by pairs)

p=2   q=3/2 -> n=2 m=3 g=24 µ=5
p=2   q=3   -> n=2 m=3 g=24 µ=1
-> vf=[(3,4)2] -> co

p=2   q=4   -> n=2 m=4 g=∞ µ=1
-> vf=[44] -> squat

p=5/2 q=3/2 -> n=5 m=3 g=120 µ=22
p=5/2 q=3   -> n=5 m=3 g=120 µ=2
-> vf=2[(3,10/2)2] -> "2id"
(vertices, edges, and {3} all coincide by pairs)

p=5/2 q=5/3 -> n=5 m=5 g=120 µ=18
p=5/2 q=5/2 -> n=5 m=5 g=120 µ=6
-> vf=2[(5/2,10/2)2] -> "2did"
(vertices, edges, and {5/2} all coincide by pairs)

p=3   q=3   -> n=3 m=3 g=∞ µ=1
-> vf=[(3,6)2] -> that
```

#### xPxQo with p=n/d, q=m/b≠2   (up)

```. . . | g/2 |   1   2 |    2    1
------+-----+---------+----------
x . . |   2 | g/4   * |    2    0
. x . |   2 |   * g/2 |    1    1
------+-----+---------+----------
xPx . |  2n |   n   n | g/2n    *
. xQo |   m |   0   m |    * g/2m

vf. is isoceles triangle: pt || x(q) #(2p)

special cases with x1=x2:
p=5/4 q=3/2 -> n=5 m=3 g=120 µ=29
p=5/4 q=3   -> n=5 m=3 g=120 µ=19
-> vf=5[3,10/4,10/4] -> "2sissid+gike"
(vertices coincide by five, edges coincide by three)

p=5/4 q=5/3 -> n=5 m=5 g=120 µ=27
p=5/4 q=5/2 -> n=5 m=5 g=120 µ=21
-> vf=3[5/2,10/4,10/4] -> "3gissid"
(vertices and edges both coincide by three, {5/2} and {10/4} coincide)

p=4/3 q=3/2 -> n=4 m=3 g=48 µ=11
p=4/3 q=3   -> n=4 m=3 g=48 µ=7
-> vf=[3,8/3,8/3] -> quith

p=3/2 q=5/4 -> n=3 m=5 g=120 µ=29
p=3/2 q=5   -> n=3 m=5 g=120 µ=11
-> vf=5[5,6/2,6/2] -> "2ike+gad"
(vertices coincide by five, edges coincide by three)

p=3/2 q=4/3 -> n=3 m=4 g=48 µ=11
p=3/2 q=4   -> n=3 m=4 g=48 µ=5
-> vf=4[4,6/2,6/2] -> "2oct+6{4}"
(vertices coincide by four, edges coincide by three)

p=3/2 q=3/2 -> n=3 m=3 g=24 µ=5
p=3/2 q=3   -> n=3 m=3 g=24 µ=3
-> vf=3[3,6/2,6/2] -> "3tet"
(vertices and edges both coincide by three, {3} and {6/2} coincide)

p=3/2 q=5/3 -> n=3 m=5 g=120 µ=23
p=3/2 q=5/2 -> n=3 m=5 g=120 µ=17
-> vf=5[5/2,6/2,6/2] -> "2gike+sissid"
(vertices coincide by five, edges coincide by three)

p=5/3 q=5/4 -> n=5 m=5 g=120 µ=27
p=5/3 q=5   -> n=5 m=5 g=120 µ=9
-> vf=[5,10/3,10/3] -> quit sissid

p=5/3 q=3/2 -> n=5 m=3 g=120 µ=23
p=5/3 q=3   -> n=5 m=3 g=120 µ=13
-> vf=[3,10/3,10/3] -> quit gissid

p=2   q=m/b -> n=2 m=m g=4m µ=b
p=2 q*=m/(m-b) -> n=2 m=m g=4m µ=m-b
-> vf=[q,4,4] -> q-prism
(degeneracy depends only on degeneracy of {q})

p=5/2 q=5/4 -> n=5 m=5 g=120 µ=21
p=5/2 q=5   -> n=5 m=5 g=120 µ=3
-> vf=3[5,10/2,10/2] -> "3doe"
(vertices and edges both coincide by three, {5} and {10/2} coincide)

p=5/2 q=3/2 -> n=5 m=3 g=120 µ=17
p=5/2 q=3   -> n=5 m=3 g=120 µ=13
-> vf=5[3,10/2,10/2] -> "2gad+ike"
(vertices coincide by five, edges coincide by three)

p=3   q=5/4 -> n=3 m=5 g=120 µ=19
p=3   q=5   -> n=3 m=5 g=120 µ=1
-> vf=[5,6,6] -> ti

p=3   q=4/3 -> n=3 m=4 g=48 µ=7
p=3   q=4   -> n=3 m=4 g=48 µ=1
-> vf=[4,6,6] -> toe

p=3   q=3/2 -> n=3 m=3 g=24 µ=3
p=3   q=3   -> n=3 m=3 g=24 µ=1
-> vf=[3,6,6] -> tut

p=3   q=5/3 -> n=3 m=5 g=120 µ=13
p=3   q=5/2 -> n=3 m=5 g=120 µ=7
-> vf=[5/2,6,6] -> tiggy

p=3   q=6   -> n=3 m=6 g=∞ µ=1
-> vf=[63] -> hexat

p=4   q=3/2 -> n=4 m=3 g=48 µ=5
p=4   q=3   -> n=4 m=3 g=48 µ=1
-> vf=[3,8,8] -> tic

p=4   q=4   -> n=4 m=4 g=∞ µ=1
-> vf=[4,8,8] -> tosquat

p=5   q=3/2 -> n=5 m=3 g=120 µ=11
p=5   q=3   -> n=5 m=3 g=120 µ=1
-> vf=[3,10,10] -> tid

p=5   q=5/3 -> n=5 m=5 g=120 µ=9
p=5   q=5/2 -> n=5 m=5 g=120 µ=3
-> vf=[5/2,10,10] -> tigid

p=6   n=3   -> n=6 m=3 g=∞ µ=1
-> vf=[3,12,12] -> toxat
```

#### xPxQxR*a with p=n/d, q=m/b, r=k/c, p<q<r   (up)

(alike p<q=r for colored faces or if node x2<x3)
(alike p=q<r for colored faces or if node x1<x2)
(alike p=q=r for colored faces or if node x1<x2<x3)

```. . .    |  g |   1   1   1 |    1    1    1
---------+----+-------------+---------------
x . .    |  2 | g/2   *   * |    1    1    0
. x .    |  2 |   * g/2   * |    1    0    1
. . x    |  2 |   *   * g/2 |    0    1    1
---------+----+-------------+---------------
xPx .    | 2n |   n   n   0 | g/2n    *    *
x . xR*a | 2k |   k   0   k |    * g/2k    *
. xQx    | 2m |   0   m   m |    *    * g/2m

vf. is triangle: pt || pt || pt : #(2p,2r,2q)

special cases with x1=x2=x3:
p=5/4 q=3/2 r=5/3 -> n=5 m=3 k=5 g=120 µ=32
-> vf=4[6/2,10/3,10/4] -> "gid+geihid+gidhid"
(vertices coincide by 4, edges coincide by three, {10/3} coincide by pairs)

p=5/4 q=3/2 r=2   -> n=5 m=3 k=2 g=120 µ=29
-> vf=6[4/3(!),6/2,10/4] -> "2sidtid+rhom"
(vertices coincide by 6, edges coincide by three)

p=5/4 q=5/3 r=2   -> n=5 m=5 k=2 g=120 µ=27
-> vf=2[4,10/3,10/4] -> "gird+12{10/4}"
(vertices and {10/4}-edges both coincide by pairs)

p=5/4 q=2   r=5/2 -> n=5 m=2 k=5 g=120 µ=21
-> vf=6[4,10/2,10/4] -> "2ditdid+rhom"
(vertices coincide by 6, edges coincide by three)

p=5/4 q=2   r=3   -> n=5 m=2 k=3 g=120 µ=19
-> vf=2[4,6,10/4] -> "ri+12{10/4}"
(vertices and {10/4}-edges both coincide by pairs)

p=5/4 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=16
-> vf=4[6,10/2,10/4] -> "did+sidhei+gidhei"
(vertices coincide by 4, edges coincide by three, {6} coincide by pairs)

p=5/4 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=10
-> vf=2[6,10,10/4] -> "siddy+12{10/4}"
(vertices and {10/4}-edges both coincide by pairs)

p=4/3 q=3/2 r=2   -> n=4 m=3 k=2 g=48 µ=11
-> vf=2[4,6/2,8/3] -> "groh+8{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=4/3 q=2   r=3   -> n=4 m=2 k=3 g=48 µ=7
-> vf=[4,6,8/3] -> quitco

p=4/3 q=3   r=4   -> n=4 m=3 k=4 g=48 µ=4
-> vf=[6,8,8/3] -> cotco

p=3/2 q=5/3 r=2   -> n=3 m=5 k=2 g=120 µ=23
-> vf=2[4,6/2,10/3] -> "gird+20{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=3/2 q=5/3 r=3   -> n=3 m=5 k=3 g=120 µ=18
-> vf=2[6,6/2,10/3] -> "giddy+20{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=3/2 q=2   r=5/2 -> n=3 m=2 k=5 g=120 µ=17
-> vf=6[4,6/2,10/2] -> "sidtid+ditdid+gidtid"
(vertices coincide by 6, edges coincide by three)

p=3/2 q=2   r=3   -> n=3 m=2 k=3 g=24 µ=3
-> vf=2[4,6,6/2] -> "cho+4{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=3/2 q=2   r=4   -> n=3 m=2 k=4 g=48 µ=5
-> vf=2[4,6/2,8] -> "sroh+8{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=3/2 q=2   r=5   -> n=3 m=2 k=5 g=120 µ=11
-> vf=2[4,6/2,10] -> "sird+20{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=3/2 q=5/2 r=5   -> n=3 m=5 k=5 g=120 µ=8
-> vf=4[6/2,10,10/2] -> "id+seihid+sidhid"
(vertices coincide by 4, edges coincide by three, {10} coincide by pairs)

p=3/2 q=3   r=5   -> n=3 m=3 k=5 g=120 µ=6
-> vf=2[6,6/2,10] -> "siddy+20{6/2}"
(vertices and {6/2}-edges both coincide by pairs)

p=5/3 q=2   r=3   -> n=5 m=2 k=3 g=120 µ=13
-> vf=[4,6,10/3] -> gaquatid

p=5/3 q=2   r=5   -> n=5 m=2 k=5 g=120 µ=9
-> vf=[4,10,10/3] -> quitdid

p=5/3 q=5/2 r=3   -> n=5 m=5 k=3 g=120 µ=10
-> vf=2[6,10/2,10/3] -> "giddy+12{10/2}"
(vertices and {10/2}-edges both coincide by pairs)

p=5/3 q=3   r=5   -> n=5 m=3 k=5 g=120 µ=4
-> vf=[6,10,10/3] -> idtid

p=2   q=5/2 r=3   -> n=2 m=5 k=3 g=120 µ=7
-> vf=2[4,6,10/2] -> "ri+12{10/2}"
(vertices and {10/2}-edges both coincide by pairs)

p=2   q=5/2 r=5   -> n=2 m=5 k=5 g=120 µ=3
-> vf=2[4,10,10/2] -> "sird+12{10/2}"
(vertices and {10/2}-edges both coincide by pairs)

p=2   q=3   r=4   -> n=2 m=3 k=4 g=48 µ=1
-> vf=[4,6,8] -> girco

p=2   q=3   r=5   -> n=2 m=3 k=5 g=120 µ=1
-> vf=[4,6,10] -> grid

p=2   q=3   r=6   -> n=2 m=3 k=6 g=∞ µ=1
-> vf=[4,6,12] -> othat
```

#### xPxQxQ*a with p=n/d, q=m/b≠p, and x1=x2   (up)

(alike p=q for colored faces)

According to the symmetrical Dynkin symbol the total symmetry order is increased: g+ = 2g

```. . .              |  g |  2   1 |    1   2
-------------------+----+--------+---------
x . .     &  . x . |  2 |  g   * |    1   1
. . x              |  2 |  * g/2 |    0   2
-------------------+----+--------+---------
xPx .              | 2n | 2n   0 | g/2n   *
x . xQ*a  &  . xQx | 2m |  m   m |    * g/m

vf. is isoceles triangle: pt || pt || pt : #(2p,2q,2q)

special cases with x1=x2=x3:
p=5/4 q=3/2 -> n=5 m=3 g=120 µ=34
-> vf=10[6/2,6/2,10/4] -> "2sissid+4gike"
(vertices coincide by ten, edges coincide by six, {6/2} coincide by pairs)

p=5/4 q=3   -> n=5 m=3 g=120 µ=14
-> vf=2[6,6,10/4] -> "2tiggy"
(vertices, edges, and {6} all coincide by pairs)

p=5/4 q=5   -> n=5 m=5 g=120 µ=6
-> vf=2[10,10,10/4] -> "2tigid"
(vertices, edges, and {10} all coincide by pairs)

p=3/2 q=5/4 -> n=3 m=5 g=120 µ=38
-> vf=10[6/2,10/4,10/4] -> "4sissid+2gike"
(vertices coincide by ten, edges coincide by six, {10/4} coincide by pairs)

p=3/2 q=4/3 -> n=3 m=4 g=48 µ=14
-> vf=2[6/2,8/3,8/3] -> "2quith"
(vertices, edges, and {8/3} all coincide by pairs)

p=3/2 q=5/3 -> n=3 m=5 g=120 µ=26
-> vf=2[6/2,10/3,10/3] -> "2quitgissid"
(vertices, edges, and {10/3} all coincide by pairs)

p=3/2 q=5/2 -> n=3 m=5 g=120 µ=14
-> vf=10[6/2,10/2,10/2] -> "2ike+4gad"
(vertices coincide by ten, edges coincide by six, {10/2} coincide by pairs)

p=3/2 q=3   -> n=3 m=3 g=24 µ=2
-> vf=2[6,6,6/2] -> "2tut"
(vertices, edges, and {6} all coincide by pairs)

p=3/2 q=4   -> n=3 m=4 g=48 µ=2
-> vf=2[6/2,8,8] -> "2tic"
(vertices, edges, and {8} all coincide by pairs)

p=3/2 q=5   -> n=3 m=5 g=120 µ=2
-> vf=2[6/2,10,10] -> "2tid"
(vertices, edges, and {10} all coincide by pairs)

p=3/2 q=6   -> n=3 m=6 g=∞ µ=2
-> vf=2[6/2,12,12] -> "2toxat"
(vertices, edges, and {12} all coincide by pairs)

p=2   q=3/2 -> n=2 m=3 g=24 µ=5
-> vf=4[4,6/2,6/2] -> "2oct+6{4}"
(vertices coincide by four, edges coincide by three, {4} coincide by pairs)

p=2   q=3   -> n=2 m=3 g=24 µ=1
-> vf=[4,6,6] -> toe

p=2   q=4   -> n=2 m=4 g=∞ µ=1
-> vf=[4,8,8] -> tosquat

p=5/2 q=3/2 -> n=5 m=3 g=120 µ=22
-> vf=10[6/2,6/2,10/2] -> "4ike+2gad"
(vertices coincide by ten, edges coincide by six, {6/2} coincide by pairs)

p=5/2 q=5/3 -> n=5 m=5 g=120 µ=18
-> vf=2[10/2,10/3,10/3] -> "2quitsissid"
(vertices, edges, and {10/3} all coincide by pairs)

p=5/2 q=3   -> n=5 m=3 g=120 µ=2
-> vf=2[6,6,10/2] -> "2ti"
(vertices, edges, and {6} all coincide by pairs)

p=n/d q=2   -> n=n m=2 g=4n µ=1
-> vf=[4,4,2p] -> 2p-prism
(degeneracy for even d)
```

#### xPxPxP*a with p=n/d and x1=x2=x3   (up)

According to the symmetrical Dynkin symbol the total symmetry order is increased: g+ = 6g

```. . .                        |  g |    3 |     3
-----------------------------+----+------+------
x . .  &  . x .     &  . . x |  2 | 3g/2 |     2
-----------------------------+----+------+------
xPx .  &  x . xP*a  &  . xPx | 2n |   2n | 3g/2n

vf. is regular triangle: pt || pt || pt : #(2p,2p,2p)

special cases:
p=5/4 -> n=5 g=120 µ=42
-> vf=6[(10/4)3] -> "6gissid"
(vertices and edges both coincide by six, {10/4} coincide by three)

p=3/2 -> n=3 g=24 µ=6
-> vf=6[(6/2)3] -> "6tet"
(vertices and edges both coincide by six, {6/2} coincide by three)

p=2   -> n=2 g=8 µ=1
-> vf=[43] -> cube

p=5/2 -> n=5 g=120 µ=6
-> vf=6[(10/2)3] -> "6doe"
(vertices and edges both coincide by six, {10/2} coincide by three)

p=3   -> n=3 g=∞ µ=1
-> vf=[63] -> hexat
```

#### βPoQoR*a with p=n/d, q=m/b, r=k/c, p<r, n≠2, n≠4, m≠2, k≠2, k≠4, n or k odd   (up)

(alike p=r for colored faces)

```both( . . .    ) | g/2m |  2m  2m |    m    m   2m
-----------------+------+---------+---------------
sefa( βPo .    ) |    2 | g/2   * |    1    0    1
sefa( β . oR*a ) |    2 |   * g/2 |    0    1    1
-----------------+------+---------+---------------
βPo .      ♦    n |   n   0 | g/2n    *    *
β . oR*a   ♦    k |   0   k |    * g/2k    *
sefa( βPoQoR*a ) |   2m |   m   m |    *    * g/2m

special cases (p<r):
```

#### sPoQoR*a with p=n/d, q=m/b, r=k/c, p<r, n≠2, n≠4, m≠2, k≠2, k≠4, n and k even   (up)

(alike p=r for colored faces)

```demi( . . .    ) | g/4m |  2m  2m |    m    m   2m
-----------------+------+---------+---------------
sefa( sPo .    ) |    2 | g/4   * |    1    0    1
sefa( s . oR*a ) |    2 |   * g/4 |    0    1    1
-----------------+------+---------+---------------
sPo .      ♦  n/2 | n/2   0 | g/2n    *    *
s . oR*a   ♦  k/2 |   0 k/2 |    * g/2k    *
sefa( sPoQoR*a ) |   2m |   m   m |    *    * g/4m

special cases (p<r):
```

#### βPoQoR*a with p=n/d, q=m/b, r=4/c, n≠2, n≠4, m≠2, n odd   (up)

```both( . . .    ) | g/2m |   m  2m |    m   2m
-----------------+------+---------+----------
both( s . oR*a ) |    2 | g/4   * |    0    2
sefa( βPo .    ) |    2 |   * g/2 |    1    1
-----------------+------+---------+----------
βPo .      ♦    n |   0   n | g/2n    *
sefa( βPoQoR*a ) |   2m |   m   m |    * g/2m

special cases:
```

#### sPoQoR*a with p=n/d, q=m/b, r=4/c, n≠2, n≠4, m≠2, n even   (up)

(alike p=r for colored faces)

```demi( . . .    ) | g/4m |   m  2m |    m   2m
-----------------+------+---------+----------
s . oR*a   |    2 | g/8   * |    0    2
sefa( sPo .    ) |    2 |   * g/4 |    1    1
-----------------+------+---------+----------
sPo .      ♦  n/2 |   0 n/2 | g/2n    *
sefa( sPoQoR*a ) |   2m |   m   m |    * g/4m

special cases (p<r):
```

#### sPoQoR*a with p=4/d, q=m/b, r=4/c, p<r, m≠2,   (up)

```demi( . . .    ) | g/4m |   m   m |   2m
-----------------+------+---------+-----
s . oR*a   |    2 | g/8   * |    2
sPo .      |    2 |   * g/8 |    2
-----------------+------+---------+-----
sefa( sPoQoR*a ) |   2m |   m   m | g/4m

single case:
```

#### βPoQoP*a with p=n/d, q=m/b, m≠2, n odd   (up)

```both( . . .    )                      | g/2m | 4m |  2m   2m
--------------------------------------+------+----+---------
sefa( βPo .    )  &  sefa( β . oP*a ) |    2 |  g |   1    1
--------------------------------------+------+----+---------
βPo .       &        β . oP*a   ♦    n |  n | g/n    *
sefa( βPoQoP*a )                      |   2m | 2m |   * g/2m

special cases:
```

#### sPoQoP*a with p=n/d, q=m/b, n≠2, n≠4, m≠2, n even   (up)

```demi( . . .    )                      | g/4m |  4m |  2m   2m
--------------------------------------+------+-----+---------
sefa( sPo .    )  &  sefa( s . oP*a ) |    2 | g/2 |   1    1
--------------------------------------+------+-----+---------
sPo .       &        s . oP*a   ♦    n |   n | g/n    *
sefa( sPoQoP*a )                      |   2m |  2m |   * g/4m

special cases:
```

#### sPoQoP*a with p=4/d, q=m/b, m≠2   (up)

```demi( . . .    )            | g/4m |  2m |   2m
----------------------------+------+-----+-----
sPo .     &  s . oP*a |    2 | g/4 |    2
----------------------------+------+-----+-----
sefa( sPoQoP*a )            |   2m |  2m | g/4m

special cases:
```

#### βPoQo with p=n/d, q=m/b, m≠2, n odd   (up)

```both( . . . ) | g/2m |  2m |    m    m
--------------+------+-----+----------
sefa( βPo . ) |    2 | g/2 |    1    1
--------------+------+-----+----------
βPo .   ♦    n |   n | g/2n    *
sefa( βPoQo ) |    m |   m |    * g/2m

special cases:
p=5/2 q=3   -> n=5 m=3 g=120 µ=7 -> vf=[(3,5)3]/2 -> gidtid
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=[510]/4 -> "2gad"
p=3   q=5/2 -> n=3 m=5 g=120 µ=7 -> vf=[(5/2,3)5]/3 -> gacid
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=[36]/2 -> "2tet"
p=3   q=4   -> n=3 m=4 g=48  µ=1 -> vf=[(3/2,4)4] = [(3,4)4]/3 -> "oct+6{4}"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=[(3/2,5)5] -> cid
p=5   q=3   -> n=3 m=5 g=120 µ=1 -> vf=[(5/2,3)3] -> sidtid
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=[(5/2)10]/2 -> "2sissid"
```

#### sPoQo with p=n/d, q=m/b, n≠2, n≠4, m≠2, n even   (up)

```demi( . . . ) | g/4m |  2m |    m    m
--------------+------+-----+----------
sefa( sPo . ) |    2 | g/4 |    1    1
--------------+------+-----+----------
sPo .   ♦  n/2 | n/2 | g/2n    *
sefa( sPoQo ) |    m |   m |    * g/4m

special cases:
```

#### sPoQo with p=4/d, q=m/b, m≠2   (up)

```demi( . . . ) | g/4m |   m |    m
--------------+------+-----+-----
sPo .   |    2 | g/8 |    2
--------------+------+-----+-----
sefa( sPoQo ) |    m |   m | g/4m

special cases:
p=4 q=3 -> n=4 m=3 g=48 µ=1 -> vf=[34] -> tet
```

#### oPβQo with p=n/d, q=m/b, n≠2, n≠4, m≠2, m≠4, p<q, n or m odd   (up)

(alike p=q for colored faces)

```both( . . . ) | g/4 |   4   4 |    2    2   4
--------------+-----+---------+--------------
sefa( oPβ . ) |   2 | g/2   * |    1    0   1
sefa( . βQo ) |   2 |   * g/2 |    0    1   1
--------------+-----+---------+--------------
oPβ .   ♦   n |   n   0 | g/2n    *   *
. βQo   ♦   m |   0   m |    * g/2m   *
sefa( oPβQo ) |   4 |   2   2 |    *    * g/4

special cases (p<q):
p=5/2 q=5 -> n=5 m=5 g=120 µ=3 -> (not uniform) -> "o5/2β5o"
p=3   q=5 -> n=3 m=5 g=120 µ=1 -> (not uniform) -> "o3β5o"
```

#### oPsQo with p=n/d, q=m/b, n≠2, n≠4, m≠2, m≠4, p<q, n and m even   (up)

(alike p=q for colored faces)

```demi( . . . ) | g/8 |   4   4 |    2    2   4
--------------+-----+---------+--------------
sefa( oPs . ) |   2 | g/4   * |    1    0   1
sefa( . sQo ) |   2 |   * g/4 |    0    1   1
--------------+-----+---------+--------------
oPs .   ♦ n/2 | n/2   0 | g/2n    *   *
. sQo   ♦ m/2 |   0 m/2 |    * g/2m   *
sefa( oPsQo ) |   4 |   2   2 |    *    * g/8

special cases (p<q):
```

#### oPβQo with p=n/d, q=4/b, n≠2, n≠4, n odd   (up)

```both( . . . ) | g/4 |   2   4 |    2   4
--------------+-----+---------+---------
both( . sQo ) |   2 | g/4   * |    0   2
sefa( oPβ . ) |   2 |   * g/2 |    1   1
--------------+-----+---------+---------
oPβ .   ♦   n |   0   n | g/2n   *
sefa( oPβQo ) |   4 |   2   2 |    * g/4

special cases:
p=3 q=4 -> n=3 m=4 g=48 µ=1 -> (not uniform) -> "o3β4o"
```

#### oPsQo with p=n/d, q=4/b, n≠2, n≠4, n even   (up)

(alike p=q for colored faces)

```demi( . . . ) | g/8 |   2   4 |    2   4
--------------+-----+---------+---------
. sQo   |   2 | g/8   * |    0   2
sefa( oPs . ) |   2 |   * g/4 |    1   1
--------------+-----+---------+---------
oPs .   ♦ n/2 | n/2   0 | g/2n   *
sefa( oPsQo ) |   4 |   2   2 |    * g/8

special cases (p≠q):
```

#### oPβPo with p=n/d, n odd   (up)

```both( . . . )                   | g/4 | 8 |   4   4
--------------------------------+-----+---+--------
sefa( oPβ . )  &  sefa( . βPo ) |   2 | g |   1   1
--------------------------------+-----+---+--------
oPβ .    &        . βPo   ♦   n | n | g/n   *
sefa( oPβPo )                   |   4 | 4 |   * g/4

special cases:
p=3 -> n=3 g=24 µ=1 -> vf=[(3,4)4]/3 = [(3/2,4)4] -> "oct+6{4}"
```

#### oPsPo with p=n/d, n≠2, n≠4, n even   (up)

```demi( . . . )                   | g/8 |   8 |   4   4
--------------------------------+-----+-----+--------
sefa( oPs . )  &  sefa( . sPo ) |   2 | g/2 |   1   1
--------------------------------+-----+-----+--------
oPs .    &        . sPo   ♦ n/2 | n/2 | g/n   *
sefa( oPsPo )                   |   4 |   4 |   * g/8

special cases:
```

#### oPsPo with p=4/d   (up)

```demi( . . . )         | g/8 |   4 |   4
----------------------+-----+-----+----
oPs .  &  . sPo |   2 | g/4 |   2
----------------------+-----+-----+----
sefa( oPsPo )         |   4 |   4 | g/8

single case:
```

#### βPxQoR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, k≠2, k≠4, n or k odd   (up)

```both( . . .    ) | g/2 |   2   2   2 |    1    2    1    2
-----------------+-----+-------------+--------------------
both( . x .    ) |   2 | g/2   *   * |    1    1    0    0
sefa( βPx .    ) |   2 |   * g/2   * |    0    1    0    1
sefa( β . oR*a ) |   2 |   *   * g/2 |    0    0    1    1
-----------------+-----+-------------+--------------------
both( . xQo    ) |   m |   m   0   0 | g/2m    *    *    *
βPx .      ♦  2n |   n   n   0 |    * g/2n    *    *
β . oR*a   ♦   k |   0   0   k |    *    * g/2k    *
sefa( βPxQoR*a ) |  2m |   0   m   m |    *    *    * g/2m

special cases:
```

#### sPxQoR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, k≠2, k≠4, n and k even   (up)

```demi( . . .    ) | g/4 |   2   2   2 |    1    2    1    2
-----------------+-----+-------------+--------------------
demi( . x .    ) |   2 | g/4   *   * |    1    1    0    0
sefa( sPx .    ) |   2 |   * g/4   * |    0    1    0    1
sefa( s . oR*a ) |   2 |   *   * g/4 |    0    0    1    1
-----------------+-----+-------------+--------------------
demi( . xQo    ) |   m |   m   0   0 | g/4m    *    *    *
sPx .      ♦   n | n/2 n/2   0 |    * g/2n    *    *
s . oR*a   ♦ k/2 |   0   0 k/2 |    *    * g/2k    *
sefa( sPxQoR*a ) |  2m |   0   m   m |    *    *    * g/4m

special cases:
```

#### βPxQoR*a with p=n/d, q=m/b, r=4/c, n≠2, m≠2, n odd   (up)

```both( . . .    ) | g/2 |   2   1   2 |    1    2    2
-----------------+-----+-------------+---------------
both( . x .    ) |   2 | g/2   *   * |    1    1    0
both( s . oR*a ) |   2 |   * g/4   * |    0    0    2
sefa( βPx .    ) |   2 |   *   * g/2 |    0    1    1
-----------------+-----+-------------+---------------
both( . xQo    ) |   m |   m   0   0 | g/2m    *    *
βPx .      ♦  2n |   n   0   n |    * g/2n    *
sefa( βPxQoR*a ) |  2m |   0   m   m |    *    * g/2m

special cases:
```

#### sPxQoR*a with p=n/d, q=m/b, r=4/c, n≠2, m≠2, n even   (up)

```demi( . . .    ) | g/4 |   2   1   2 |    1    2    2
-----------------+-----+-------------+---------------
demi( . x .    ) |   2 | g/4   *   * |    1    1    0
s . oR*a   |   2 |   * g/8   * |    0    0    2
sefa( sPx .    ) |   2 |   *   * g/4 |    0    1    1
-----------------+-----+-------------+---------------
demi( . xQo    ) |   m |   m   0   0 | g/4m    *    *
sPx .      ♦   n | n/2   0 n/2 |    * g/2n    *
sefa( sPxQoR*a ) |  2m |   0   m   m |    *    * g/4m

special cases:
```

#### βPxQo with p=n/d, q=m/b, m≠2, n odd   (up)

```both( . . . ) | g/2 |   2   2 |    1    2    1
--------------+-----+---------+---------------
both( . x . ) |   2 | g/2   * |    1    1    0
sefa( βPx . ) |   2 |   * g/2 |    0    1    1
--------------+-----+---------+---------------
both( . xQo ) |   m |   m   0 | g/2m    *    *
βPx .   ♦  2n |   n   n |    * g/2n    *
sefa( βPxQo ) |   m |   0   m |    *    * g/2m

special cases:
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=2[(5,10/4)2] -> "2did"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=2[(3,6/2)2] -> "2oct"
p=3   q=4   -> n=3 m=4 g=48  µ=1 -> vf=2[(6/2,4)2] -> "2co"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=2[(6/2,5)2] -> "2id"
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=2[(5/2,10/2)2] -> "2did"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> vf=2[(10/2,3)2] -> "2id"
```

#### sPxQo with p=n/d, q=m/b, n≠2, m≠2, n even   (up)

```demi( . . . ) | g/4 |   2   2 |    1    2    1
--------------+-----+---------+---------------
demi( . x . ) |   2 | g/4   * |    1    1    0
sefa( sPx . ) |   2 |   * g/4 |    0    1    1
--------------+-----+---------+---------------
demi( . xQo ) |   m |   m   0 | g/4m    *    *
sPx .   ♦   n | n/2 n/2 |    * g/2n    *
sefa( sPxQo ) |   m |   0   m |    *    * g/4m

special cases:
p=4 q=3 -> n=4 m=3 g=48 µ=1 -> vf=[(3,4)2] -> co
```

#### βPoQx with p=n/d, q=m/b, m≠2, n odd   (up)

```both( . . . ) | g/2 |   2   2 |    1    1    2
--------------+-----+---------+---------------
both( . . x ) |   2 | g/2   * |    1    0    1
sefa( βPo . ) |   2 |   * g/2 |    0    1    1
--------------+-----+---------+---------------
both( . oQx ) |   m |   m   0 | g/2m    *    *
βPo .   ♦   n |   0   n |    * g/2n    *
sefa( βPoQx ) |  2m |   m   m |    *    * g/2m

special cases:
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=2[5/4,10,5,10] -> "2sidhid"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=[3/2,6,3,6] -> oho
p=3   q=4   -> n=3 m=4 g=48  µ=1 -> vf=[3/2,8,4,8] -> socco
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=[3/2,10,5,10] -> saddid
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=2[(5/2,10/2)2] -> "2did"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> vf=[5/2,6,3,6] -> siid
```

#### sPoQx with p=n/d, q=m/b, n≠4, m≠2, n even   (up)

```demi( . . . ) | g/4 |   2   2 |    1    1    2
--------------+-----+---------+---------------
. . x   |   2 | g/4   * |    1    0    1
sefa( sPo . ) |   2 |   * g/4 |    0    1    1
--------------+-----+---------+---------------
demi( . oQx ) |   m |   m   0 | g/4m    *    *
sPo .   ♦ n/2 |   0 n/2 |    * g/2n    *
sefa( sPoQx ) |  2m |   m   m |    *    * g/4m

special cases:
```

#### sPoQx with p=4/d, q=m/b, m≠2   (up)

```demi( . . . ) | g/4 |   2   1 |    1    2
--------------+-----+---------+----------
. . x   |   2 | g/4   * |    1    1
sPo .   |   2 |   * g/8 |    0    2
--------------+-----+---------+----------
demi( . oQx ) |   m |   m   0 | g/4m    *
sefa( sPoQx ) |  2m |   m   m |    * g/4m

special cases:
p=4 q=3 -> n=4 m=3 g=48 µ=1 -> vf=[3,62] -> tut
```

#### xPβQo with p=n/d, q=m/b, n≠2, m≠4, n or m odd   (up)

```both( . . . ) | g/2 |   1   2   2 |    2    1   2
--------------+-----+-------------+--------------
both( x . . ) |   2 | g/4   *   * |    2    0   0
sefa( xPβ . ) |   2 |   * g/2   * |    1    0   1
sefa( . βQo ) |   2 |   *   * g/2 |    0    1   1
--------------+-----+-------------+--------------
xPβ .   ♦  2n |   n   n   0 | g/2n    *   *
. βQo   ♦   m |   0   0   m |    * g/2m   *
sefa( xPβQo ) |   4 |   0   2   2 |    *    * g/4

special cases:
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> (not uniform) -> "x5/2β5o"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> (not uniform) -> "x3β3o"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> (not uniform) -> "x3β5o"
p=4   q=3   -> n=4 m=3 g=48  µ=1 -> (not uniform) -> "x4β3o"
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> (not uniform) -> "x5β5/2o"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> (not uniform) -> "x5β3o"
```

#### xPsQo with p=n/d, q=m/b, n≠2, m≠4, n and m even   (up)

```demi( . . . ) | g/4 |   1   2   2 |    2    1   2
--------------+-----+-------------+--------------
demi( x . . ) |   2 | g/8   *   * |    2    0   0
sefa( xPs . ) |   2 |   * g/4   * |    1    0   1
sefa( . sQo ) |   2 |   *   * g/4 |    0    1   1
--------------+-----+-------------+--------------
xPs .   ♦   n | n/2 n/2   0 | g/2n    *   *
. sQo   ♦ m/2 |   0   0 m/2 |    * g/2m   *
sefa( xPsQo ) |   4 |   0   2   2 |    *    * g/8

special cases:
```

#### xPβQo with p=n/d, q=4/b, n odd   (up)

```both( . . . ) | g/2 |   1   1   2 |    2   2
--------------+-----+-------------+---------
both( x . . ) |   2 | g/4   *   * |    2   0
both( . sQo ) |   2 |   * g/4   * |    0   2
sefa( xPβ . ) |   2 |   *   * g/2 |    1   1
--------------+-----+-------------+---------
xPβ .   ♦  2n |   n   0   n | g/2n   *
sefa( xPβQo ) |   4 |   0   2   2 |    * g/4

special cases:
p=3 q=4 -> n=3 m=4 g=48 µ=1 -> (not uniform) -> "x3β4o"
```

#### xPsQo with p=n/d, q=4/b, n≠2, n even   (up)

```demi( . . . ) | g/4 |   1   1   2 |    2   2
--------------+-----+-------------+---------
demi( x . . ) |   2 | g/8   *   * |    2   0
. sQo   |   2 |   * g/8   * |    0   2
sefa( xPs . ) |   2 |   *   * g/4 |    1   1
--------------+-----+-------------+---------
xPs .   ♦   n | n/2   0 n/2 | g/2n   *
sefa( xPsQo ) |   4 |   0   2   2 |    * g/8

special cases:
```

#### x2βPo with p=n/d, n odd   (up)

```both( . . . ) | g/2 |   1   2 |    1   2
--------------+-----+---------+---------
both( x . . ) |   2 | g/4   * |    0   2
sefa( . βPo ) |   2 |   * g/2 |    1   1
--------------+-----+---------+---------
. βPo   ♦   n |   0   n | g/2n   *
sefa( x2βPo ) |   4 |   2   2 |    * g/4

special cases:
```

#### x2sPo with p=n/d, n≠4, n even   (up)

```demi( . . . ) | g/4 |   1   2 |    1   2
--------------+-----+---------+---------
demi( x . . ) |   2 | g/8   * |    0   2
sefa( . sPo ) |   2 |   * g/4 |    1   1
--------------+-----+---------+---------
. sPo   ♦ n/2 |   0 n/2 | g/2n   *
sefa( x2sPo ) |   4 |   2   2 |    * g/8

general case:
p=2m/d m≠2 -> n=2m g=4n/d=8m/d µ=d -> vf=[42,m/d] -> m/d-p
```

#### βPβQoR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, m≠4, k≠2, k≠4, q<r, m or k odd   (up)

(alike q=r for colored faces)

```both( . . .    ) | g/2 | 4   2   2 |   2    1    1   4
-----------------+-----+-----------+------------------
sefa( sPs .    ) |   2 | g   *   * |   1    0    0   1
sefa( β . oR*a ) |   2 | * g/2   * |   0    1    0   1
sefa( . βQo    ) |   2 | *   * g/2 |   0    0    1   1
-----------------+-----+-----------+------------------
both( sPs .    ) ♦   n | n   0   0 | g/n    *    *   *
β . oR*a   ♦   k | 0   k   0 |   * g/2k    *   *
. βQo      ♦   m | 0   0   m |   *    * g/2m   *
sefa( βPβQoR*a ) |   4 | 2   1   1 |   *    *    * g/2

special cases (q<r):
```

#### sPsQoR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, m≠4, k≠2, k≠4, q<r, m and k even   (up)

(alike q=r for colored faces)

```demi( . . .    ) | g/4 |   4   2   2 |    2    1    1   4
-----------------+-----+-------------+-------------------
sefa( sPs .    ) |   2 | g/2   *   * |    1    0    0   1
sefa( s . oR*a ) |   2 |   * g/4   * |    0    1    0   1
sefa( . sQo    ) |   2 |   *   * g/4 |    0    0    1   1
-----------------+-----+-------------+-------------------
sPs .      ♦   n |   n   0   0 | g/2n    *    *   *
s . oR*a   ♦   k |   0   k   0 |    * g/4k    *   *
. sQo      ♦   m |   0   0   m |    *    * g/4m   *
sefa( sPsQoR*a ) |   4 |   2   1   1 |    *    *    * g/4

special cases (q<r):
```

#### βPβQoR*a with p=n/d, q=m/b, r=4/c, n≠2, m≠2, m≠4, m odd   (up)

```both( . . .    ) | g/2 |   1 4   2 |   2    1   4
-----------------+-----+-----------+-------------
both( s . oR*a ) |   2 | g/4 *   * |   0    0   2
sefa( sPs .    ) |   2 |   * g   * |   1    0   1
sefa( . βQo    ) |   2 | *   * g/2 |   0    1   1
-----------------+-----+-----------+-------------
both( sPs .    ) ♦   n |   0 n   0 | g/n    *   *
. βQo      ♦   m |   0 0   m |   * g/2m   *
sefa( βPβQoR*a ) |   4 |   1 2   1 |   *    * g/2

special cases:
```

#### sPsQoR*a with p=n/d, q=m/b, r=4/c, n≠2, m≠2, m≠4, m even   (up)

```demi( . . .    ) | g/4 |   1   4   2 |    2    1   4
-----------------+-----+-------------+--------------
s . oR*a   |   2 | g/8   *   * |    0    0   2
sefa( sPs .    ) |   2 |   * g/2   * |    1    0   1
sefa( . sQo    ) |   2 |   *   * g/4 |    0    1   1
-----------------+-----+-------------+--------------
sPs .      ♦   n |   0   n   0 | g/2n    *   *
. sQo      ♦   m |   0   0   m |    * g/4m   *
sefa( sPsQoR*a ) |   4 |   1   2   1 |    *    * g/4

special cases:
```

#### sPsQoR*a with p=n/d, q=4/b, r=4/c, n≠2, q<r   (up)

```demi( . . .    ) | g/4 |   1   1   4 |    2   4
-----------------+-----+-------------+---------
s . oR*a   |   2 | g/8   *   * |    0   2
. sQo      |   2 |   * g/8   * |    0   2
sefa( sPs .    ) |   2 |   *   * g/2 |    1   1
-----------------+-----+-------------+---------
sPs .      ♦   n |   0   0   n | g/2n   *
sefa( sPsQoR*a ) |   4 |   1   1   2 |    * g/4

single case:
```

#### βPβQoQ*a with p=n/d, q=m/b, n≠2, m odd   (up)

```both( . . .    )                   | g/2 | 4 4 |   2   2   4
-----------------------------------+-----+-----+------------
sefa( sPs .    )                   |   2 | g * |   1   0   1
sefa( β . oQ*a )  &  sefa( . βQo ) |   2 | * g |   0   1   1
-----------------------------------+-----+-----+------------
both( sPs .    )                   ♦   n | n 0 | g/n   *   *
β . oQ*a    &        . βQo   ♦   m | 0 m |   * g/m   *
sefa( βPβQoQ*a )                   |   4 | 2 2 |   *   * g/2

special cases:
```

#### sPsQoQ*a with p=n/d, q=m/b, n≠2, m≠2, m≠4, m even   (up)

```demi( . . .    )                   | g/4 |   4   4 |    2    2   4
-----------------------------------+-----+---------+--------------
sefa( sPs .    )                   |   2 | g/2   * |    1    0   1
sefa( s . oQ*a )  &  sefa( . sQo ) |   2 |   * g/2 |    0    1   1
-----------------------------------+-----+---------+--------------
sPs .                        ♦   n |   n   0 | g/2n    *   *
s . oQ*a    &        . sQo   ♦   m |   0   m |    * g/2m   *
sefa( sPsQoQ*a )                   |   4 |   2   2 |    *    * g/4

special cases:
```

#### sPsQoQ*a with p=n/d, q=4/b, n≠2   (up)

```demi( . . .    )         | g/4 |   2   4 |    2   4
-------------------------+-----+---------+---------
s . oQ*a  &  . sQo |   2 | g/4   * |    0   2
sefa( sPs .    )         |   2 |   * g/2 |    1   1
-------------------------+-----+---------+---------
sPs .              ♦   n |   0   n | g/2n   *
sefa( sPsQoQ*a )         |   4 |   2   2 |    * g/4

special cases:
```

#### βPβQo with p=n/d, q=m/b, n≠2, m odd   (up)

```both( . . . ) | g/2 | 4   2 |   2    1   3
--------------+-----+-------+-------------
sefa( sPs . ) |   2 | g   * |   1    0   1
sefa( . βQo ) |   2 | * g/2 |   0    1   1
--------------+-----+-------+-------------
both( sPs . ) ♦   n | n   0 | g/n    *   *
. βQo   ♦   m | 0   m |   * g/2m   *
sefa( βPβQo ) |   3 | 2   1 |   *    * g/2

special cases:
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=3[(5/2,3)3] -> "3sidtid"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=2[3/2,35] -> "2oct+8{3}"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=[5/2,35] -> seside
p=4   q=3   -> n=4 m=3 g=48  µ=1 -> vf=2[3/2,3,4,3,4,3] -> "2co+16{3}"
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=3[(3,5)3] -> "3gidtid"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> vf=2[3/2,3,5,3,5,3] -> "2id+40{3}"
```

#### sPsQo with p=n/d, q=m/b, n≠2, m≠4, m even   (up)

```demi( . . . ) | g/4 |   4   2 |    2    1   3
--------------+-----+---------+--------------
sefa( sPs . ) |   2 | g/2   * |    1    0   1
sefa( . sQo ) |   2 |   * g/4 |    0    1   1
--------------+-----+---------+--------------
sPs .   ♦   n |   n   0 | g/2n    *   *
. sQo   ♦ m/2 |   0 m/2 |    * g/2m   *
sefa( sPsQo ) |   3 |   2   1 |    *    * g/4

special cases:
```

#### sPsQo with p=n/d, q=4/b, n≠2   (up)

```demi( . . . ) | g/4 |   1   4 |    2   3
--------------+-----+---------+---------
. sQo   |   2 | g/8   * |    0   2
sefa( sPs . ) |   2 |   * g/2 |    1   1
--------------+-----+---------+---------
sPs .   ♦   n |   0   n | g/2n   *
sefa( sPsQo ) |   3 |   1   2 |    * g/4

special cases:
p=3   q=4   -> n=3 m=4 g=48 µ=1  -> vf=[35] -> ike
p=3   q=4/3 -> n=3 m=4 g=48 µ=7  -> vf=[35] -> ike
p=3/2 q=4   -> n=3 m=4 g=48 µ=5  -> vf=[35]/2 -> gike
p=3/2 q=4/3 -> n=3 m=4 g=48 µ=11 -> vf=[35]/2 -> gike
```

#### β2βPo with p=n/d, n odd   (up)

```both( . . . ) | g/2 |   2   2 |    1   3
--------------+-----+---------+---------
both( s2s . ) |   2 | g/2   * |    0   2
sefa( . βPo ) |   2 |   * g/2 |    1   1
--------------+-----+---------+---------
. βPo   ♦   n |   0   n | g/2n   *
sefa( β2βPo ) |   3 |   2   1 |    * g/2

general case:
p=n/d -> g=4n/d µ=d -> vf=[33,n/2d] -> n/2d-ap
```

#### s2sPo with p=n/d, n≠4, n even   (up)

```demi( . . . ) | g/4 |   2   2 |    1   3
--------------+-----+---------+---------
s2s .   |   2 | g/4   * |    0   2
sefa( . sPo ) |   2 |   * g/4 |    1   1
--------------+-----+---------+---------
. sPo   ♦ n/2 |   0 n/2 | g/2n   *
sefa( s2sPo ) |   3 |   2   1 |    * g/4

general case:
p=2m/d m≠2 -> n=2m g=4n/d=8m/d µ=d -> vf=[33,m/d] -> m/d-ap
```

#### s2sPo with p=4/d   (up)

```demi( . . . ) | g/4 |   2   1 |   3
--------------+-----+---------+----
s2s .   |   2 | g/4   * |   2
. sPo   |   2 |   * g/8 |   2
--------------+-----+---------+----
sefa( s2sPo ) |   3 |   2   1 | g/4

single case:
p=4 -> n=4 g=16 µ=1 -> vf=[34] -> tet
```

#### βPoQβ with p=n/d, q=m/b, n≠2, n≠4, m≠2, m≠4, p<q, n or m odd   (up)

(alike p=q for colored faces)

```both( . . .    ) | g/2 |   2   2   2 |    1    1   4
-----------------+-----+-------------+--------------
both( s . s2*a ) |   2 | g/2   *   * |    0    0   2
sefa( βPo .    ) |   2 |   * g/2   * |    1    0   1
sefa( . oQβ    ) |   2 |   *   * g/2 |    0    1   1
-----------------+-----+-------------+--------------
βPo .      ♦   n |   0   n   0 | g/2n    *   *
. oQβ      ♦   m |   0   0   m |    * g/2m   *
sefa( βPoQβ    ) |   4 |   2   1   1 |    *    * g/2

special cases (p<q):
p=5/2 q=5 -> n=5 m=5 g=120 µ=3 -> (not uniform) -> "β5/2o5β"
p=3   q=5 -> n=3 m=5 g=120 µ=1 -> (not uniform) -> "β3o5β"
```

#### sPoQs with p=n/d, q=m/b, n≠2, n≠4, m≠2, m≠4, p<q, n and m even   (up)

(alike p=q for colored faces)

```demi( . . .    ) | g/4 |   2   2   2 |    1    1   4
-----------------+-----+-------------+--------------
s . s2*a   |   2 | g/4   *   * |    0    0   2
sefa( sPo .    ) |   2 |   * g/4   * |    1    0   1
sefa( . oQs    ) |   2 |   *   * g/4 |    0    1   1
-----------------+-----+-------------+--------------
sPo .      ♦ n/2 |   0 n/2   0 | g/2n    *   *
. oQs      ♦ m/2 |   0   0 m/2 |    * g/2m   *
sefa( sPoQs    ) |   4 |   2   1   1 |    *    * g/4

special cases (p<q):
```

#### βPoQβ with p=n/d, q=4/b, n≠2, n≠4, n odd   (up)

(alike p=q for colored faces)

```both( . . .    ) | g/2 |   2   1   2 |    1   4
-----------------+-----+-------------+---------
both( s . s2*a ) |   2 | g/2   *   * |    0   2
both( . oQs    ) |   2 |   * g/4   * |    0   2
sefa( βPo .    ) |   2 |   *   * g/2 |    1   1
-----------------+-----+-------------+---------
βPo .      ♦   n |   0   0   n | g/2n   *
sefa( βPoQβ    ) |   4 |   2   1   1 |    * g/2

special cases (p<q):
p=3 q=4 -> n=3 m=4 g=48 µ=1 -> (not uniform) -> "β3o4β"
```

#### sPoQs with p=n/d, q=4/b, n≠2, n≠4, n even   (up)

(alike p=q for colored faces)

```demi( . . .    ) | g/4 |   2   1   2 |    1   4
-----------------+-----+-------------+---------
s . s2*a   |   2 | g/4   *   * |    0   2
. oQs      |   2 |   * g/8   * |    0   2
sefa( sPo .    ) |   2 |   *   * g/4 |    1   1
-----------------+-----+-------------+---------
sPo .      ♦ n/2 |   0   0 n/2 | g/2n   *
sefa( sPoQs    ) |   4 |   2   1   1 |    * g/4

special cases (p<q):
```

#### βPoPβ with p=n/d, n odd   (up)

```both( . . .    )                   | g/2 |   2 4 |   2   4
-----------------------------------+-----+-------+--------
both( s . s2*a )                   |   2 | g/2 * |   0   2
sefa( βPo .    )  &  sefa( . oPβ ) |   2 |   * g |   1   1
-----------------------------------+-----+-------+--------
βPo .       &        . oPβ   ♦   n |   0 n | g/n   *
sefa( βPoPβ    )                   |   4 |   2 2 |   * g/2

special cases:
p=3 -> n=3 g=24 µ=1 -> (not uniform) -> "β3o3β"
```

#### sPoPs with p=n/d, n≠2, n≠4, n even   (up)

```demi( . . .    )                   | g/4 |   2   4 |   2   4
-----------------------------------+-----+---------+--------
s . s2*a                     |   2 | g/4   * |   0   2
sefa( sPo .    )  &  sefa( . oPs ) |   2 |   * g/2 |   1   1
-----------------------------------+-----+---------+--------
sPo .       &        . oPs   ♦ n/2 |   0 n/2 | g/n   *
sefa( sPoPs    )                   |   4 |   2   2 |   * g/4

special cases:
```

#### sPoPs with p=4/d   (up)

```demi( . . .    )         | g/4 |   2   2 |   4
-------------------------+-----+---------+----
sPo .     &  . oPs |   2 | g/4   * |   2
s . s2*a           |   2 |   * g/4 |   2
-------------------------+-----+---------+----
sefa( sPoPs    )         |   4 |   2   2 | g/4

single case:
```

#### βPxQxR*a with p=n/d, q=m/b, r=k/c, n≠2, k≠2, p<r, n or k odd   (up)

(alike p=r for colored faces or if node x2<x3)

```both( . . .    ) |  g |   1   1   1   1 |    1    1    1    1
-----------------+----+-----------------+--------------------
both( . x .    ) |  2 | g/2   *   *   * |    1    1    0    0
both( . . x    ) |  2 |   * g/2   *   * |    1    0    1    0
sefa( βPx .    ) |  2 |   *   * g/2   * |    0    1    0    1
sefa( β . xR*a ) |  2 |   *   *   * g/2 |    0    0    1    1
-----------------+----+-----------------+--------------------
both( . xQx    ) | 2m |   m   m   0   0 | g/2m    *    *    *
βPx .      ♦ 2n |   n   0   n   0 |    * g/2n    *    *
β . xR*a   ♦ 2k |   0   k   0   k |    *    * g/2k    *
sefa( βPxQxR*a ) | 2m |   0   0   m   m |    *    *    * g/2m

special cases (p<r):
p=5/2 q=2 r=5 -> n=5 m=2 k=5 g=120 µ=3 -> vf=2[4,10/4,4,10/2] -> "2raded"
p=3   q=2 r=4 -> n=3 m=2 k=4 g=48  µ=1 -> vf=2[6/2,43] -> "2sirco"
p=3   q=2 r=5 -> n=3 m=2 k=5 g=120 µ=1 -> vf=2[6/2,4,5,4] -> "2srid"
```

#### sPxQxR*a with p=n/d, q=m/b, r=k/c, n≠2, k≠2, p<r, n and k even   (up)

(alike p=r for colored faces or if node x2<x3)

```demi( . . .    ) | g/2 |   1   1   1   1 |    1    1    1    1
-----------------+-----+-----------------+--------------------
demi( . x .    ) |   2 | g/4   *   *   * |    1    1    0    0
demi( . . x    ) |   2 |   * g/4   *   * |    1    0    1    0
sefa( sPx .    ) |   2 |   *   * g/4   * |    0    1    0    1
sefa( s . xR*a ) |   2 |   *   *   * g/4 |    0    0    1    1
-----------------+-----+-----------------+--------------------
demi( . xQx    ) |  2m |   m   m   0   0 | g/4m    *    *    *
sPx .      ♦   n | n/2   0 n/2   0 |    * g/2n    *    *
s . xR*a   ♦   k |   0 k/2   0 k/2 |    *    * g/2k    *
sefa( sPxQxR*a ) |  2m |   0   0   m   m |    *    *    * g/4m

special cases (p<r):
```

#### βPxQx with p=n/d, q=m/b, n odd   (up)

```both( . . . ) |  g |   1   1   1 |    1    1    1
--------------+----+-------------+---------------
both( . x . ) |  2 | g/2   *   * |    1    1    0
both( . . x ) |  2 |   * g/2   * |    1    0    1
sefa( βPx . ) |  2 |   *   * g/2 |    0    1    1
--------------+----+-------------+---------------
both( . xQx ) | 2m |   m   m   0 | g/2m    *    *
βPx .   ♦ 2n |   n   0   n |    * g/2n    *
sefa( βPxQx ) | 2m |   0   m   m |    *    * g/2m

special cases:
p=n/d q=2   -> m=2 g=4n/d  µ=b -> vf=2[2n/2d,42] -> 2n/2d-p
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=2[5/2,102] -> "2tigid"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=2[6/2,62] -> "2tut"
p=3   q=4   -> n=3 m=4 g=48  µ=1 -> vf=2[6/2,82] -> "2tic"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=2[6/2,102] -> "2tid"
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=6[(10/2)3] -> "6doe"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> vf=2[10/2,62] -> "2ti"
```

#### sPxQx with p=n/d, q=m/b, n≠2, n even   (up)

```demi( . . . ) | g/2 |   1   1   1 |    1    1    1
--------------+-----+-------------+---------------
demi( . x . ) |   2 | g/4   *   * |    1    1    0
demi( . . x ) |   2 |   * g/4   * |    1    0    1
sefa( sPx . ) |   2 |   *   * g/4 |    0    1    1
--------------+-----+-------------+---------------
demi( . xQx ) |  2m |   m   m   0 | g/4m    *    *
sPx .   ♦   n | n/2   0 n/2 |    * g/2n    *
sefa( sPxQx ) |  2m |   0   m   m |    *    * g/4m

special cases:
p=4    q=3 -> n=4  m=3 g=48 µ=1 -> vf=[4,62] -> toe
p=2k/d q=2 -> n=2k m=2 g=4n/d=8k/d µ=d -> vf=[42,2k/d] -> 2k/d-p
```

#### βPxQxP*a with p=n/d, q=m/b, n odd   (up)

```both( . . .    )                      |  g |  2  2 |    1   2    1
--------------------------------------+----+-------+--------------
both( . x .    )  &  both( . . x    ) |  2 |  g  * |    1   1    0
sefa( βPx .    )  &  sefa( β . xP*a ) |  2 |  *  g |    0   1    1
--------------------------------------+----+-------+--------------
both( . xQx    )                      | 2m | 2m  0 | g/2m   *    *
βPx .       &        β . xP*a   ♦ 2n |  n  n |    * g/n    *
sefa( βPxQxP*a )                      | 2m |  0 2m |    *   * g/2m

special cases:
p=3 q=2 -> n=3 m=2 g=24 µ=1 -> vf=2[(4,6/2)2] -> "2co"
```

#### sPxQxP*a with p=n/d, q=m/b, n≠2, n even   (up)

```demi( . . .    )                      | g/2 |   2   2 |    1   2    1
--------------------------------------+-----+---------+--------------
demi( . x .    )  &  demi( . . x    ) |   2 | g/2   * |    1   1    0
sefa( sPx .    )  &  sefa( s . xP*a ) |   2 |   * g/2 |    0   1    1
--------------------------------------+-----+---------+--------------
demi( . xQx    )                      |  2m |  2m   0 | g/4m   *    *
sPx .       &        s . xP*a   ♦   n | n/2 n/2 |    * g/n    *
sefa( sPxQxP*a )                      |  2m |   0  2m |    *   * g/4m

special cases:
```

#### βPβQxR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, k≠2, q<r, m or k odd   (up)

(alike q=r for colored faces)

```both( . . .    ) |  g |   1 2   1   1 |   1    1    1   2
-----------------+----+---------------+------------------
both( . . x    ) |  2 | g/2 *   *   * |   0    1    1   0
sefa( sPs .    ) |  2 |   * g   *   * |   1    0    0   1
sefa( β . xR*a ) |  2 |   * * g/2   * |   0    1    0   1
sefa( . βQx    ) |  2 |   * *   * g/2 |   0    0    1   1
-----------------+----+---------------+------------------
both( sPs .    ) ♦  n |   0 n   0   0 | g/n    *    *   *
β . xR*a   ♦ 2k |   k 0   k   0 |   * g/2k    *   *
. βQx      ♦ 2m |   m 0   0   m |   *    * g/2m   *
sefa( βPβQxR*a ) |  4 |   0 2   1   1 |   *    *    * g/2

special cases (q<r):
```

#### sPsQxR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, k≠2, q<r, m and k even   (up)

(alike q=r for colored faces)

```demi( . . .    ) | g/2 |   1   2   1   1 |    1    1    1   2
-----------------+-----+-----------------+-------------------
demi( . . x    ) |   2 | g/4   *   *   * |    0    1    1   0
sefa( sPs .    ) |   2 |   * g/2   *   * |    1    0    0   1
sefa( s . xR*a ) |   2 |   *   * g/4   * |    0    1    0   1
sefa( . sQx    ) |   2 |   *   *   * g/4 |    0    0    1   1
-----------------+-----+-----------------+-------------------
sPs .      ♦   n |   0   n   0   0 | g/2n    *    *   *
s . xR*a   ♦   k | k/2   0 k/2   0 |    * g/2k    *   *
. sQx      ♦   m | m/2   0   0 m/2 |    *    * g/2m   *
sefa( sPsQxR*a ) |   4 |   0   2   1   1 |    *    *    * g/4

special cases (q<r):
```

#### βPβQxQ*a with p=n/d, q=m/b, n≠2, m odd   (up)

```both( . . .    )                   |  g |   1 2 2 |   1   2   2
-----------------------------------+----+---------+------------
both( . . x    )                   |  2 | g/2 * * |   0   2   0
sefa( sPs .    )                   |  2 |   * g * |   1   0   1
sefa( β . xQ*a )  &  sefa( . βQx ) |  2 |   * * g |   0   1   1
-----------------------------------+----+---------+------------
both( sPs .    )                   ♦  n |   0 n 0 | g/n   *   *
β . xQ*a    &        . βQx   ♦ 2m |   m 0 m |   * g/m   *
sefa( βPβQxQ*a )                   |  4 |   0 2 2 |   *   * g/2

special cases:
```

#### sPsQxQ*a with p=n/d, q=m/b, n≠2, m≠2, m even   (up)

```demi( . . .    )                   | g/2 |   1   2   2 |    1   2   2
-----------------------------------+-----+-------------+-------------
demi( . . x    )                   |   2 | g/4   *   * |    0   2   0
sefa( sPs .    )                   |   2 |   * g/2   * |    1   0   1
sefa( s . xQ*a )  &  sefa( . sQx ) |   2 |   *   * g/2 |    0   1   1
-----------------------------------+-----+-------------+-------------
sPs .                        ♦   n |   0   n   0 | g/2n   *   *
s . xQ*a    &        . sQx   ♦   m | m/2   0 m/2 |    * g/m   *
sefa( sPsQxQ*a )                   |   4 |   0   2   2 |    *   * g/4

special cases:
```

#### βPβQx with p=n/d, q=m/b, n≠2, m odd   (up)

```both( . . . ) |  g |   1 2   1 |   1    1   2
--------------+----+-----------+-------------
both( . . x ) |  2 | g/2 *   * |   0    1   1
sefa( sPs . ) |  2 |   * g   * |   1    0   1
sefa( . βQx ) |  2 |   * * g/2 |   0    1   1
--------------+----+-----------+-------------
both( sPs . ) ♦  n |   0 n   0 | g/n    *   *
. βQx   ♦ 2m |   m 0   m |   * g/2m   *
sefa( βPβQx ) |  4 |   1 2   1 |   *    * g/2

special cases:
p=5/2 q=5   -> n=5 m=5 g=120 µ=3 -> vf=2[5/2,4,10/2,4] -> "2raded"
p=3   q=3   -> n=3 m=3 g=24  µ=1 -> vf=2[3,4,6/2,4] -> "2co"
p=3   q=5   -> n=3 m=5 g=120 µ=1 -> vf=2[3,4,10/2,4] -> "2srid"
p=4   q=3   -> n=4 m=3 g=48  µ=1 -> vf=2[6/2,43] -> "2sirco"
p=5   q=5/2 -> n=5 m=5 g=120 µ=3 -> vf=2[10/4,4,5,4] -> "2raded"
p=5   q=3   -> n=5 m=3 g=120 µ=1 -> vf=2[6/2,4,5,4] -> "2srid"
```

#### sPsQx with p=n/d, q=m/b, n≠2, m≠2, m even   (up)

```demi( . . . ) | g/2 |   1   2   1 |    1    1   2
--------------+-----+-------------+--------------
demi( . . x ) |   2 | g/4   *   * |    0    1   1
sefa( sPs . ) |   2 |   * g/2   * |    1    0   1
sefa( . sQx ) |   2 |   *   * g/4 |    0    1   1
--------------+-----+-------------+--------------
sPs .   ♦   n |   0   n   0 | g/2n    *   *
. sQx   ♦   m | m/2   0 m/2 |    * g/2m   *
sefa( sPsQx ) |   4 |   1   2   1 |    *    * g/4

special cases:
p=3 q=4 -> n=3 m=4 g=48 µ=1 -> vf=[3,43] -> sirco
```

#### sPs2x with p=n/d, n≠2   (up)

```demi( . . . ) | g/2 |   1   2 |    1   2
--------------+-----+---------+---------
demi( . . x ) |   2 | g/4   * |    0   2
sefa( sPs . ) |   2 |   * g/2 |    1   1
--------------+-----+---------+---------
sPs .   ♦   n |   0   n | g/2n   *
sefa( sPs2x ) |   4 |   2   2 |    * g/4

general case:
p=n/d n≠2 -> g=4n/d µ=d -> [42,n/d] -> n/d-p
```

#### βPxQβ with p=n/d, q=m/b, n≠2, m≠2, p<q, n or m odd   (up)

(alike p=q for colored faces)

```both( . . .    ) |  g |   1   1   1   1 |    1    1   2
-----------------+----+-----------------+--------------
both( . x .    ) |  2 | g/2   *   *   * |    1    1   0
both( s . s2*a ) |  2 |   * g/2   *   * |    0    0   2
sefa( βPx .    ) |  2 |   *   * g/2   * |    1    0   1
sefa( . xQβ    ) |  2 |   *   *   * g/2 |    0    1   1
-----------------+----+-----------------+--------------
βPx .      ♦ 2n |   n   0   n   0 | g/2n    *   *
. xQβ      ♦ 2m |   m   0   0   m |    * g/2m   *
sefa( βPxQβ    ) |  4 |   0   2   1   1 |    *    * g/2

special cases (p<q):
p=5/2 q=5 -> n=5 m=5 g=120 µ=3 -> (not uniform) -> "β5/2x5β"
p=3   q=4 -> n=3 m=4 g=48  µ=1 -> (not uniform) -> "β3x4β"
p=3   q=5 -> n=3 m=5 g=120 µ=1 -> (not uniform) -> "β3x5β"
```

#### sPxQs with p=n/d, q=m/b, n≠2, m≠2, p<q, n and m even   (up)

(alike p=q for colored faces)

```demi( . . .    ) | g/2 |   1   1   1   1 |    1    1   2
-----------------+-----+-----------------+--------------
demi( . x .    ) |   2 | g/4   *   *   * |    1    1   0
s . s2*a   |   2 |   * g/4   *   * |    0    0   2
sefa( sPx .    ) |   2 |   *   * g/4   * |    1    0   1
sefa( . xQs    ) |   2 |   *   *   * g/4 |    0    1   1
-----------------+-----+-----------------+--------------
sPx .      ♦   n | n/2   0 n/2   0 | g/2n    *   *
. xQs      ♦   m | m/2   0   0 m/2 |    * g/2m   *
sefa( sPxQs    ) |   4 |   0   2   1   1 |    *    * g/4

special cases (p<q):
```

#### β2xPβ with p=n/d, n odd   (up)

```both( . . .    ) |  g |   1   1   1 |    1   2
-----------------+----+-------------+---------
both( . x .    ) |  2 | g/2   *   * |    1   1
both( s . s2*a ) |  2 |   * g/2   * |    0   2
sefa( . xPβ    ) |  2 |   *   * g/2 |    1   1
-----------------+----+-------------+---------
. xPβ      ♦ 2n |   n   0   n | g/2n   *
sefa( β2xPβ    ) |  4 |   1   2   1 |    * g/2

general case:
p=n/d -> g=4n/d µ=d -> vf=[2n/2d,42] -> 2n/2d-p
```

#### s2xPs with p=n/d, n≠2, n even   (up)

```demi( . . .    ) | g/2 |   1   1   1 |    1   2
-----------------+-----+-------------+---------
demi( . x .    ) |   2 | g/4   *   * |    1   1
s . s2*a   |   2 |   * g/4   * |    0   2
sefa( . xPs    ) |   2 |   *   * g/4 |    1   1
-----------------+-----+-------------+---------
. xPs      ♦   n | n/2   0 n/2 | g/2n   *
sefa( s2xPs    ) |   4 |   1   2   1 |    * g/4

general case:
p=2m/d m≠1 -> n=2m g=4n/d=8m/d µ=d -> vf=[42,2m/d] -> 2m/d-p
```

#### βPxPβ with p=n/d, n odd   (up)

```both( . . .    )                   |  g |   1   1 2 |   2   2
-----------------------------------+----+-----------+--------
both( . x .    )                   |  2 | g/2   * * |   2   0
both( s . s2*a )                   |  2 |   * g/2 * |   0   2
sefa( βPx .    )  &  sefa( . xPβ ) |  2 |   *   * g |   1   1
-----------------------------------+----+-----------+--------
βPx .       &        . xPβ   ♦ 2n |   n   0 n | g/n   *
sefa( βPxPβ    )                   |  4 |   0   2 2 |   * g/2

special cases:
p=3 -> n=3 g=24 µ=1 -> (not uniform) -> "β3x3β"
```

#### sPxPs with p=n/d, n≠2, n even   (up)

```demi( . . .    )                   | g/2 |   1   1   2 |   2   2
-----------------------------------+-----+-------------+--------
demi( . x .    )                   |   2 | g/4   *   * |   2   0
s . s2*a                     |   2 |   * g/4   * |   0   2
sefa( sPx .    )  &  sefa( . xPs ) |   2 |   *   * g/2 |   1   1
-----------------------------------+-----+-------------+--------
sPx .       &        . xPs   ♦   n | n/2   0 n/2 | g/n   *
sefa( sPxPs    )                   |   4 |   0   2   2 |   * g/4

special cases:
```

#### sPsQsR*a with p=n/d, q=m/b, r=k/c, n≠2, m≠2, k≠2, p<q<r   (up)

(alike p<q=r for colored faces)
(alike p=q<r for colored faces)
(alike p=q=r for colored faces)

```demi( . . .    ) | g/2 |   2   2   2 |    1    1    1   3
-----------------+-----+-------------+-------------------
sefa( sPs .    ) |   2 | g/2   *   * |    1    0    0   1
sefa( s . sR*a ) |   2 |   * g/2   * |    0    1    0   1
sefa( . sQs    ) |   2 |   *   * g/2 |    0    0    1   1
-----------------+-----+-------------+-------------------
sPs .      ♦   n |   n   0   0 | g/2n    *    *   *
s . sR*a   ♦   k |   0   k   0 |    * g/2k    *   *
. sQs      ♦   m |   0   0   m |    *    * g/2m   *
sefa( sPsQsR*a ) |   3 |   1   1   1 |    *    *    * g/2

special cases (p<q<r):
p=5/3 q=5/2 r=3 -> n=5 m=5 k=3 g=120 µ=10 -> vf=[5/3,33,5/2,3] -> gisdid
p=5/3 q=3   r=5 -> n=5 m=3 k=5 g=120 µ=4  -> vf=[5/3,33,5,3] -> sided
```

#### sPsQsQ*a with p=n/d, q=m/b, n≠2, m≠2, p≠q   (up)

(alike p=q for colored faces)

```demi( . . .    )                   | g/2 |   2 4 |    1   2   3
-----------------------------------+-----+-------+-------------
sefa( sPs .    )                   |   2 | g/2 * |    1   0   1
sefa( s . sQ*a )  &  sefa( . sQs ) |   2 |   * g |    0   1   1
-----------------------------------+-----+-------+-------------
sPs .                        ♦   n |   n 0 | g/2n   *   *
s . sQ*a    &        . sQs   ♦   m |   0 m |    * g/m   *
sefa( sPsQsQ*a )                   |   3 |   1 2 |    *   * g/2

special cases (p≠q):
p=5/2 q=3/2 -> n=5 m=3 g=120 µ=22 -> vf=[(3/2,3)2,5/2,3] -> sirsid
p=5/2 q=3   -> n=5 m=3 g=120 µ=2  -> vf=[(5/2,3)3] -> seside
```

#### sPsPsP*a with p=n/d, n≠2   (up)

```demi( . . .    )                                        | g/2 |    6 |     3   3
--------------------------------------------------------+-----+------+----------
sefa( sPs .    )  &  sefa( s . sP*a )  &  sefa( . sPs ) |   2 | 3g/2 |     1   1
--------------------------------------------------------+-----+------+----------
sPs .       &        s . sP*a    &        . sPs   ♦   n |    n | 3g/2n   *
sefa( sPsPsP*a )                                        |   3 |    3 |     * g/2

special cases:
```

#### sPsQs with p=n/d, q=m/b, n≠2, m≠2, p<q   (up)

(alike p=q for colored faces)

```demi( . . . )  | g/2 |   1   2   2 |    1    1   3
---------------+-----+-------------+--------------
s . s2*a |   2 | g/4   *   * |    0    0   2
sefa( sPs . )  |   2 |   * g/2   * |    1    0   1
sefa( . sQs )  |   2 |   *   * g/2 |    0    1   1
---------------+-----+-------------+--------------
sPs .    ♦   n |   0   n   0 | g/2n    *   *
. sQs    ♦   m |   0   0   m |    * g/2m   *
sefa( sPsQs )  |   3 |   1   1   1 |    *    * g/2

special cases (p<q):
p=3/2 q=5/3 -> n=3 m=5 g=120 µ=23 -> vf=[3/2,32,5/3,3] -> girsid
p=5/3 q=3   -> n=5 m=3 g=120 µ=13 -> vf=[5/3,34] -> gisid
p=5/3 q=5   -> n=5 m=5 g=120 µ=9  -> vf=[5/3,32,5,3] -> isdid
p=5/2 q=3   -> n=5 m=3 g=120 µ=7  -> vf=[5/2,34] -> gosid
p=5/2 q=5   -> n=5 m=5 g=120 µ=3  -> vf=[5/2,32,5,3] -> siddid
p=3   q=4   -> n=3 m=4 g=48  µ=1  -> vf=[34,4] -> snic
p=3   q=5   -> n=3 m=5 g=120 µ=1  -> vf=[34,5] -> snid
```

#### sPsPs with p=n/d, n≠2   (up)

```demi( . . . )                   | g/2 |   1 4 |   2   3
--------------------------------+-----+-------+--------
s . s2*a                  |   2 | g/4 * |   0   2
sefa( sPs . )  &  sefa( . sPs ) |   2 |   * g |   1   1
--------------------------------+-----+-------+--------
sPs .    &        . sPs   ♦   n |   0 n | g/n   *
sefa( sPsPs )                   |   3 |   1 2 |   * g/2

special cases:
p=3   -> n=3 g=24 µ=1 -> vf=[35] -> ike
p=3/2 -> n=3 g=24 µ=5 -> vf=[35]/2 -> gike
```

#### s2sPs with p=n/d, n≠2   (up)

```demi( . . . )              | g/2 |   2   2 |    1   3
---------------------------+-----+-------------+---------
s2s .    &  s . s2*a |   2 | g/2   * |    0   2
sefa( . sPs )              |   2 |   * g/2 |    1   1
---------------------------+-----+-------------+---------
. sPs                ♦   n |   0   n | g/2n   *
sefa( s2sPs )              |   3 |   2   1 |    * g/2

general case:
g=4n/d µ=d -> vf=[33,n/d] -> n/d-ap
(uniform only possible for 2n/3>d)

special case:
p=3 -> n=3 g=12 µ=1 -> vf=[33,4] -> oct
```

#### s2s2s   (up)

```demi( . . . )  | g/2 |   1   1   1 |   3
---------------+-----+-------------+----
s2s .    |   2 | g/4   *   * |   2
s . s2*a |   2 |   * g/4   * |   2
. s2s    |   2 |   *   * g/4 |   2
---------------+-----+-------------+----
sefa( s2s2s )  |   3 |   1   1   1 | g/2

single case:
g=8 µ=1 -> vf=[33] -> tet
```

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