Can coxeter notation enumerate all symmetry groups?

Higher-dimensional geometry (previously "Polyshapes").

Can coxeter notation enumerate all symmetry groups?

Postby ubersketch » Tue Feb 26, 2019 5:15 pm

Coxeter notation and coxeter diagrams with nodes representing rotational symmetry are byfar, the most versatile notation for symmetry I've seen. But can they enumerate all symmetry groups? (excluding the complex symmetries (except maybe with coxeter diagrams.))
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Re: Can coxeter notation enumerate all symmetry groups?

Postby wendy » Wed Feb 27, 2019 9:34 am

They are not very good at enumerating symmetries. You need other methods of symmetry.

I've been playing around with trying to write a polytope notation for the Conway-Thurston notation, which is a much greater discription of 2d groups, especially in hyperbolic groups. The notation is a complete description of 2D groups, but trying to arrange the symbols to build polyhedra, for example, is quite hard.

The Coxeter groups describe all but one of the uniform polyhedra in all dimensions. It fails on the grand antiprism, which can not be written in CD, as it is a laminate. In terms of uniform tilings, the CD can not describe the laminate tilings, such as the band of alternating triangles and squares, which is LPC1. There are five different tilings in 3D: LB2, LC2, LPA2, LPB2, LPC2, and a larger number in 4D. Laminates exist in all dimensions. There are some hyperbolic laminates as well, but only a handful are known.

The group "2 3 5" is the rotational icosahedral group. It can be used when you inscribe a rotational sense to each face, such as a dodecahedron with a clockwise circuit on each face. You can construct polyhedra by putting the symbols x, o and % after each of these. x and o have their usual meanings of 1 and 0, while % is a swallow-node. The effect of these is to convert the snub faces (ie the triangles of the snub dodecahedron, that border two triangles and a pentagon), into 1:1:1, 1:1:0, and 1:1:% respectively.

From smallest to largest, we get when we put the 'o' against the larger to smaller values, ie 2x 3x 5o = icosahedron, 2x 3o 5x = dodecahedron, 2o 3x 5x = icosadodecahedron. Then 2x 3x 5x snub dodecahedron. In the first two, the edges are actually two triangles and the edge, in a Z-shape (the triangles fit into the gaps of the Z).

The next three in size are 2% 3x 5x = rhombo- icosadodecahedron, 2x 3% 5x = truncated icosahedron, and 2x 3x 5% = truncated dodecahedron.

Here, the snub triangles get 'eaten' (swallowed) by one of the three adjacent polygons, so the triangles around a p-gon, and the p-gon, become a 2p gon, with alternating neighbours. When the neighbour is a '2', the same polygon is on the opposite side, so there is a 2p gon touching the 2p-gon.

So just as we can create these six figures from the x3x5x, we can create polytopes from the snub s3s5s, by altering the edges to 0, 1, %.

A group like "3 * 2" is the pyritohedral group. It occurs in nature as pyrite. It is the shared symmetry between the cube and icosahedron, of order 24. The '3' means there is an order-3 rotation group. It is separate from a chain of mirrors that forms a right-angle. This rightangle is rotated three times to form an icosahedron.

The orbifold notation has things like "cones" (number by itself = rotation), mirror-chains with angles (each chain starts *), wanders and miracles (@, ×) which are retained and inversions of a vertex. The last two are rather hard to convert into something that we can decorate, like we do with x and o in the CD diagram.

The problem here is 'wrap'. A generalised tri-snub is written as '2 2 2'. This forms a tetrahedron, or in general, a tetrahedron formed out of irregular triangles. You can make one out of s2s2s (a rectangular block, with alternate vertices removed). We can replace 2 with any other number, to get 'p 2 2' which is an antiprism. Of course, infinity u, and the transinfinites w, are also polygons, so we can have u 2 2 (a strip of triangles), along with u * [a strip of squares, literally with a polygon not on the equator, and an equatorial mirror. Joining these along the 'u' edge gives 2 2 * (the symmetry of alternating bands of triangles and squaes.

An open polygon allows for different halves to be adjoined. We can then insert different matching halves into 'layers' of essentially unrelated symmetries.
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