This is a device to turn non-regular groups into regular groups. The device is to adjust one or more branches of the Dynkin symbol so that the thing is a simple chain. This chain then orders the nodes for other uses.
Once a non-regular figure has been written as a pseudoregular figure, it can then have all the usual things that regular figures have done to them. For example, we can write a Schlafli symbol for it, eg {3,3,A}.
The trace refers to the sequence of verticies of the figure, the idea being much as one might trace something into a more convenient form, keeping the important features, and discarding the rest.
--------- 3
/ \ 2_11 {3,3, }
o----o----o-----e e 3
A
o----o----o-----e------e 3A {3,3,3,A}
4
o----o----o-----e------o 3Q {3,3,3,4}
The above picture shows the symmetry of the 5D half-cube presented as a Dynkin symbol, and under it, is written the pseudo-regular representation. Note that the "A" branch connects the two nodes "e", but actually refers to the node that connects the second e to the node before the first one.
Nodes are provided for the branches to allow k11, k21 and k31 to be presented as regular figures. In these cases, the branches are "A", "B", and "C".
The following shows the eight-node forms in all the different forms they occur in Euclidean and Spherical geometry. In hyperbolic geometry, there is also "E5B" = "G5A".
---------
/ \ 5_11
o-----o-----o-----o-----o-----o-----o o
3 3 3 3 3 3 A 6A
---------------
/ \ 4_21
o-----o-----o-----o-----o-----o-----o o
3 3 3 3 3 3 B 6B
---------------------
/ \ 3_31
o-----o-----o-----o-----o-----o-----o o
3 3 3 3 3 3 C 6C
--------- ---------
/ \ / \ 3_31
o o-----o-----o-----o-----o-----o o
E 3 3 3 3 3 A E5A
---------
/ \ 5_11
o o-----o-----o-----o-----o-----o-----o
E 3 3 3 3 3 3 E6
---------------
/ \ 4_21
o o-----o-----o-----o-----o-----o-----o
G 3 3 3 3 3 3 G6
Although the intent was to have the loose node at the end, this is not always possible. None the less we minimise it. Although the 4B1 does not strike one as having three symetric arms, we can populate the arms separately, even showing relative orientations. /4B1, 4/B1 and 4B1/ are all 222, but in different orientations.
---------------
/ \ 2_22
o-----o-----o-----o-----o o-----o
3 3 3 3 B 3 4B1
There are also rings and rings with branches on them. These can be traced as well, if one has a device to designate a node as being "revisited".
Copyright 2002 Wendy Krieger