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The short notation

The short notation is designed to give the multitude of processes and figures simple numeric position-names. Unlike the "named" systems, this one is relevant to eight or more dimensions.

The Notion

Given that x5o3x3o is the "cantellated 120-cell", the segments "cantellated" and "120-cell" are "x-o-x-o" and "-5-3-3-". Further, since the 120-cell is the "4D dodecahedron", one can build a name for this figure as the "4D dodecahedron modified by x-o-x-o. This last bit can be read as a reversed binary number, so it would be read against the key 1-2-4-8. Prisms are made by apposition, eg pentagon-prism is line:pentagon.

In the small dimensions, A, B, C and D are 1, 2, 3, and 4 dimensions. Cp are the antiprisms, and t,o,c,i,d are the three and four dimenional analogs of figures that start with that letter in 3D. The reamining figure is q for {3,4,3}. Examples:

The non-Wythoffian figures are allocated to Cx8 and Dx16. Cc8 and Cd8 are the snub cube and dodecahedron, Dq16 the snub 24-cell, and Di16 the grand antiprism, made by removing equators from the Di1. See below for higher dimensions.

The Notation

The figures are allocated a dimension letter, a form letter, and an operator number. The closest expression to it is saying something like the X is a "t-0,2 4D dodecahedron". 4D is the dimension, "dodecahedron" is the form, and t-0,2 is the operator. The same shape is written as Dd5, meaning 4dimension dodecahedron, modified by operators "1" and "4".

The second letter represents the form in question. This is usually refered to its three-dimensional form, and describes the order of nodes.

The number represents which of the nodes is to be marked: this follows a "reverse binary" approach: ie the nodes carry values of 1,2,4,8...., rather than as if the marked and unmarked nodes represented the 1's and 0's of a binary number.

Non-Wythoffian figures

The value 2^n is used for non-wythoffian figures: there are five of these. The antiprisms have their own class, but are listed here for completeness.


Any prism is simply made by putting the elements of the product together. The class of prisms is simply the capital letters: a 1D*3D prism is an AC prism. A polygon-polygon prism is a BB prism.

For example, a dodecahedron is "Cd1". A prism with this as a base is ACd1. A pentagonal Dodecahedral prism is B5Cd1.

One may write A2, A3, A4 for the square, cube and tesseract, when these are held as the second, third and fourth power of the line. However, any operator applied to these must be done to B4, Cc1 and Dc1. One may regard A2, A3 and A4 as meaning square, cube and tesseract, and B4, Cc1 and Dc1 as tetragon, hexahedron and octachoron.

Correct Style

The "correct style" denotes the proper form of the figure. Just as "cube" is more "correct" than "square prism", so also here there is a preferred forms.

Non-regular mirror-groups

For the groups in 5 and more dimensions, the pseudoregular representations are named. The relevant cases are:

Here is how the Dynkin graph maps onto each of these. The branches marked E, G and B are treated as if they fell between the pair of nodes that the letter spans, not a different pair.

       /         \                         Half-cubic
      o     o-----o-----o-----o.....       "Dodecahedral view"
      {  E     3     3     3    .... }
         e                                     Xe...
      o-----o-----o-----o-----o .....

                  /            \            Gosset figures
    ....o---o----o----o----o    o           "Icosahedral view"
     {... 3,   3,  3  ,  3 ,  B }              Xg...

      /         \                           Gosset figures
     o   o---o---o---o---o......            "Dodecahedral view"
     { G , 3 , 3 , 3 , 3 , ...}               Xh...

So something like 231 is read as if it were a oGx3o3o3o3o3o3o or a Gh2. G here means "7D", "h" means dodecahedral form of gosset's figure and 2 means only the second node is marked.

For the half-cubics, the only "distinct" figures are 4x+2. Xe(4n)=Xc(4n), Xe(4n+1)=Xe(4n+2) and Xe(4n+3)=Xc(4n+2).

Copyright 2002 Wendy Krieger

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