New uniform bollochora.

[wendy]

This thread is for the non-Wythoffian ones, ie the ones that do not have a coxeter-dynkin diagram.

1. The laminates, based on o4o3o8o, o4o8oAo, and o3o4o3o8o.

2. The families based on snub vertex-figures
s3s4s, s3s5s, the diminished and irregular snubs.

3. The families based on the pyritohedral groups
3/a * /b * /c

4, The partial diminishings of regular groups, eg pd{3,5,3}

[wendy]

Laminatruncates.

In hyperbolic space, the polytope t{4,3,8} = x4x3o8o, has the same curvature as one of its faces {3,8}. This means that this cell can be used to reflect the truncated cubes, to fill all-space, by itself. The dual is of octagon-prisms, these being the opposite faces of the truncated cube, so is not uniform. The symmetry is crossed {4,3,8}'s, with sections {3,8} and {8,4}. 16 tC at a vertex.

A second uniform is made from the octahedral ball. The verf is oxqxo8ooooo&#qt. The polar caps belong to triangular prisms, the lattitudes above and below the equator belong to 16 cells, each x4o3x. The diagonals of this verf is an octagon-verf, and the 24 non-polar vertices fall into two CO of this size.

In H4, the octahedral family is represented by ~4,3,8,2~, the symmetry forms a hexagon group with the mirror running through the diagonal. Using standard node positions, we note 1 and 4 are a dual pair of octagonny ~4x3o8o2~, and of bioctagon-prisms, ~4o3o8x2x~, which forms the only uniform ME/MM pair that is not wythoffian. ¬4x3o8o2~ which is the runcinate, also exists, along with Klitzing's discovery of ~4x3x8o2~. It is not known whether #2, #6 or #7 exist,

[wendy]

Families derived from Snubs

The general snub is an orbifold, in the form 2 3 P. This admits an edge after each number, in the style 2/a 3/b P/c. By varying these edges to 0 or precalc (%), the triangles can either disappear, or be absorbed into adjacent faces, to double the edge-count of the face (the edge of the triangle and polygon is absorbed.)

Currently, there are three snub groups, s3s4s, s3s5s, and s3s3s4o. Each of these can be used as a vertex-figure, and the rectates and truncates also exist. These are designated by x3o:vf , o3x:vf and x3x:vf, where vf is the vertex-figure.

The user GAP on the discord, presented a version made entirely of snub cubes and octahedra. Two polar squares of s3s4s are kept, the four equatorial squares are divided to keep the four-fold axial rotation. These half-squares unite with the four triangles at the right-angle to form the verf of a snub cube. In essence, it's a partitioning division of the x3o:s3s4s. Note that the snub cubes form endless open yickles, where squares fall between sC of the same yickle. No loops form, because the angles are too open.

The corresponding version on s3s5s is much more complex. The tewlve pentagons are each divided into 2-edges and 3-edges. The 3-edge part forms a pentagonal antiprism, the 2-edges form the long edge of the verf of a snub dodecahedron. This process consumes 36 snub triangles, and 12 icosahedral faces, leaving 24 snub and 8 icosahedral triangles. The snub dodecahedra + pentagonal antiprisms, form a yickle that is open, and twelve radials at each node. The tetrahedra in the original tiling are all alike, so we suppose the only solution is for each tetrahedron to have a base and three attached, the base pointing to a dodecahedron. In this way, it may well be the 'glue' to hold the dodecahedra-pent-antiprisms together.

[wendy]

The pyritohedral groups.

The notation used to describe this group is of the form x: 3/a * /b 2 /c. The x: denotes that it's being used as a vertex figure, rather much as if it were x(Y) where Y is some cd.
.
The pyritohedral group is \( 3\ *\ 2 \). It's a subgroup of the octahedral group, implemented by EPAC (even permutations, all changes of sign).

There are three free edge variables in this group, these edges are represented by a forward slash, so 3/a * /b 2 /c forms a general pyritohedral polytope. The general faces consist of a triangle of edge 'a', a rectangle of edge b × c, and a trapezium of bc &# a (b opposite c, sloping edges a). The edges can either go to zero, or the edges can go to such as to force the faces to merge together. When this happens, the edge is replaced by a % sign.

There are seven uniform polyhedra with pyritohedral symmetry, as

• 3 /x * 2 This forms an octahedron
• 3 /x * 2 % This is a cuboctahedron. The % equates to the squares split into a 0||r2 trapezium and a r2 * 0 rectangle, joined over r2.
• 3 * /x 2 The cube.
• 3 /x * /x 2 . The icosdodecahedron
• 3/x * /x 2 % a truncated cube. The additional edge merges 1:c3 rectabgle, and 1 || c3 trapezia over c3 (chord 3 of octagon), forming an octagon.
• 3 /x * /x 2 /x The rhombocuboctahedron. The squares adjacent to triangles are the trapezia

For the general verf, we can abbreviate these to abc, being the shortchords of three polygons, as 3/a*/b2/c.

• qrp The borrochemean series. For p = 3, 5, ,,, U
• qsq cells: cubes, cuboctahedra, and rCO
• qsf cells: cubes, icosadodeca and rID
• ssq cells: tetrahedra, square antiprisms and cuboctahedra
• ssf cells: tetrahedra, pentagonal antiprisms, and icosadodeca.
• fss cells: dodecahedra, icosadodecahedra, and octahedra = x5o3o4o3z

[wendy]

Partial Diminishings.

These arise when the vertex-figure and adjacent cells can be rearranged. Examples are known of x3o5o3o.

The yickle here is formed by a chain of dodecahedra and pentagonal antiprisms, the dodecahedra form 12-way nodes, the result is faced with triangles only.

The semi-partial diminishing removes two opposite vertices of the dodecahedron, the vertex is surrounded by six icosahedra (the untouch pentagons), six pentagonal antiprisms (three at each end, the result of diminishing), and two (opposite) dodecahedra (the diminished vertex).

The partial diminishing removes four vertices in tetrahedral arrangement. This leave 12 pentagonal antiprisms and 4 dodecahedra, at a vertex, the symmetry is rotational, and the reduced element is small enough to be convex fractional in Conway's meaning.