The Coxeter-Dynkin symbol, and its meaning

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

The Coxeter-Dynkin symbol, and its meaning

Postby wendy » Sat Mar 30, 2019 11:15 am

This is how i learnt what the CD symbol meant. Since this was not in Coxeter's books (Regular Polytopes and Regular Complex Polytopes), it was more a matter of finding my own way through the fog.

A graph like @----o--5--o represents a particular solid, here an icosahedron. Coxeter never developed it much past that, save for some oblique references in RCP. In any case, it became $----o----o--5-o, with a vertex-node $, and three mirror-nodes o. The lines between the nodes are called 'branches'. The four nodes make together a tetrahedron here, but generally, 4 = tetrahedron = 3D, 3 = triangle = 2D, and so on.

A mirror-edge connects a vertex-node to a mirror-node, represents the fact that the mirror | cuts the edge v----|----v in half. If there is no connection between a mirror and vertex, it means there is no edge, and hence the vertex is on that mirror. Different lengths can be made by putting a number on these, as $--4--o (q), $--5--o (f) etc.

Any subset of nodes represents a surtope. If any node does not have a path to a vertex-node, then the surtope is a zero-height prism. So $---o o is a zero-height rectangle, or edge.

We can call a polygon {p} a 'di-polygon', if the edges are alternately a,b,a,b,.. a polygon if b=0, and a nul-polygon if a=b=0. When in a given figure like ax---b--5-c, we can progressively cover up single nodes to find the figures. Covering up node a, gives a vertex-node connected to b-c 0 times, a nul-pentagon. Covering up b gives a single connection to a-c (digon), makes a digon or edge. Covering c gives a-b makes a triangle.

Covering up pairs of nodes gives us what lies between the polygons. Although here we have faces at node c only, note that c only connects to b, and b is a zeroheight prism, so we get c |b|c = c||c. This is the only way like faces can connect to each other in a Wythoff polytope. c is at the tail of a chain, and is not connected to a vertex-figure.

In x3x5o, we see now that node c is not connected to the vertex-node a, but the vertex node connects to branch-ab at two points, making a di-triangle (or hexagon). In general, the hexagon edges can alternate without destroying the symmetry. Node c is an unconnected tail, and b is a zero-height prism, so c-b-c reduces to c-c, the hexagons have a direct connection to each other, being whatever $a reduces to. Here $a is an edge.

In o3x5o, both a and c connect to faces of the same kind, but removing either gives $--a or $--c as zero-length points (they are unconnected).

You just work up through the dimensions.

Counts, rooms and SAW.

The mirror-group is divided into 'cells', bounded by sets of mirrors. To a given surtope, these mirrors are of several types, according to how the surtope is affected.

An S-mirror is a surround-mirror, is one of the symmetries of the surtope. It reflects the vertex onto a different vertex, or an edge to a different edge. These nodes are the ones that are connected to the vertex-node.

The A-mirrors are 'around-mirrors'. These are mirrors which the surtope lies in completely, and thus the surtope is reflected onto itself as a unit operation relative to the symmetry of the surtope.

The W-mirrors are 'wall' mirrors. These reflect the surtope onto a different copy of itself. The wall-mirrors enclose a 'room' whose symmetries are those of S and A types. These mirrors are at an angle to both S and A mirrors (not right angle), and so are connected to both.

If we look at the icosahedron as $---a---b-5-c we can identify the surtopes, their wall nodes and the group/room = number.

vertex: S $, W a A b-c. The value of G/SA is 120/1/10 = 12, so there are 12 vertices.
edge S $a W b A c The value of G/SA = 120/2/2 = 30 make 30 edges.
face S $ab W c A - The value of G/SA = 120/6/1 = 20 makes 20 faces
body $ $sabc W = A - The value of G/SA is 120/120/1 = 1 makes one body.

Note that the edge has one kind of each mirror. Here the room contains SA=4 cells, being the face of a rhombic tricontahedron. The edge runs along the long diagonal of this, the vertices are at the sharp corners. The S mirror is the short diagonal, these are vertical to selected edges of the figure. The A mirror is the long diagonal, contains the entire figure, but without transform. The wall-mirrors are the edges of the tricontahedron, leads one to a different edge of the same figure.
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