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 cantellated 120-cell is 4D {5,3,3} with x-o-x-o [ie 1+o+4+o] or Dd5.
• A cuboctahedral prism is 1D * 3D {3,4} o+2+o = ACo2.

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".

• A is a 1D segment
• B is a 2D polygon, Bp(n) is the polygon, eg B3 or Bp3 is the triangle.
• C is a 3D polyhedron. Cp(n) is an antiprism, eg Cp5 is the pentagonal antiprism.
• D is a 4D polychoron.
• E,F,G,H represent 5, 6, 7 and 8 dimensions
• Nn represents higher dimensions, eg N9 is nine dimensions.

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

• p is a polygon or antiprism: it is followed by the sides on the base.
• t is the tetrahedron, and by extention, all simplexes.
• o is the octahedron, and by extention, all cross polytopes.
• c is the cube, and by extention, all measure polytopes.
• q is the {3,4,3}, which has no analogues above or below.
• i is the icosahedron {3,5}, and the {3,3,5}
• d is the dodecahedron {5,3}, and the {5,3,3}
• e is the half-cube in 5 or more dimensions: see below.
• g is gosset's figures, in the "icosahedral" form.
• h is gosset's figures, in "dodecahedral" form.
• x is any figure, or none in particular.

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.

• Cpp: The p-gonal antiprism
• Cc8 or Co8: Snub cube
• Cd8 or Ci8: Snub dodecahedron
• Dq16: Snub {3,4,3}
• Di16: Grand Antiprism

Prisms

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.

• Prisms are to be ordered in increasing dimensions of elements, so the hexagonal prism is AB6, not B6A.
• A figure is to be assigned the lesser operator. So a truncated cube could be written as Cc3 (ie x4x3o), or Co6 (ie o3x4x). Here Cc3 would be the preferred form.
• Where the same form gives the same shape, eg Cc2 and Co2, it is correct to write the "icosahedral" form (ie leading-3 form), so Co2 would be correct.
• The measure polytope is properly read as a prism-power of a line, and so the correct style for it is An. However, the form Xc1 is also correct, but should not be used in prisms. It is correct to sort it as if it were An, and make a notation to it at Nc1.

Non-regular mirror-groups

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

• e = {E,3,3,3,...} for the half-cubic group.
• g = {3,3,3...,3,B} for the "icosahedral" view of the gosset family
• h = {G,3,3,3....3} for the "dodecahedral" view of the gosset family.

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...
       g
     o---o---o---o---o---o....

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