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The notation of Jonathon Bowers

Mr Bowers has done an extensive study to find all the uniform figures in four dimensions. The total currently stands at 8188. The notation that he uses for it is an open system of greekish names, following a three-dimensional patterns very closely. The longish names reduce to small abbreviations, which, when accomidated with vowels, become widely used acronyms that bear his name.

The notation appears to have very strong roots in three dimensions, being more a replacement of parts, then a new form. The thing ends up with some rather long greekish names. None the less, the notation is relatively extentable, there is less work required to extend this one to five dimensions, then some of the other ones.

The Notion

One takes a name like "truncated cube", and writes for it "TC". The next step is to make some simple word to represent "tc", such as "Tic". This is repeated over and again, with care taken to ensure that two diverse shapes do not end up with the same short name.

It is because of these short names, and the general coverage of them, that this notation finds widespread coverage.

The Notation

The notation closely mimmicks the standard three-dimensional forms. Figures that can be equally derived from a figure or its dual, are said to be derived from the middle-truncate. The rest are derived from the "nearest" regular polychoron.

The Regular figures

The names of the regular figures are simply the number of sides, coupled with the -choron element. The measure polychoron has its usual name of tesseract.

pentachoron: five [tetrahedral] facets.

hexadecachoron: 16 [tetrahedral] facets.

tesseract: tile: eight [cubical] facets.

icositetrachoron: 24 [octahedral] facets.

hexicosachoron: 600 [tetrahedral] facets.

hectonicosachoron: 120 [dodecahedral] facets.

The truncates

The truncates are a series of figures that have faces of two different types. In 4 dimensions, there are two of these, Bowers uses the standard names for these. Only the first one is used in three dimensions.

truncate: having just the vertex "mounts" removed.

rectified: having verticies in the mid-points of the edges.

The mesotruncates

The mesotruncate is the middle truncate, the point where the a truncate stops being a truncated icosahedron and becomes a truncated dodecahedron. As in three dimensions, a figure that identically derives from a regular figure, and its dual, are said to derive from the mesotruncate. The family in three dimensions is the cuboctahedron, and icosadodecahedron.

In four dimensions, the self duals {3,3,3} and {3,4,3} give rise to distinct mesotruncates as well, and from these are derived additional figures. Just as in three dimensions, the mesotruncate {3,3} is simply known as a long count of its sides (ie octahedron, rather than tetratetrahedron), this also happens here.

decachoron: or 10-choron

tesseractihexadecachoron: compare: cuboctahedron.

tetracontoctachoron: 48-choron

hexacosihectonicosachoron: or "icosadodecahedron"

Figures with three kinds of faces

There are two different operators that yield results with three kinds of facets, and hence two different figures in this class. The new kind of facet is a prism that appears along the edge of the original figure.

In three dimensions, a set of planes that contain the edges, and are tangental to a sphere centred on the centre, form a figure with rhombic faces. If the edges are worn down a small way, a longish hexagon appears. Removing the tips of this hexagon reveals the vertex figure and a rectangle. The three-dimensional figures derive equally from both sides, and so are named after the mesotruncate.

In four dimensions, the same sorts of planes yield r-gonal dipyramids (which are never rhombihedra), but the term has acquired a "second meaning" to be used here. The four-dimensional figures derive differently from the two duals, and thus are named after the figure which provides the edges.

small rhombi-Co: also, ID and icosadodecahedron.

great rhombi-Co: also called rhombitruncated, or truncated Co

small rhombated T: also: any other figure except the 16-cell. (small rhombated 16-cell = rectified 24-cell)

great rhombated T: also: any other figure except the 16-cell. (great rhombated 16-cell = truncated 24-cell)

Figures with four kinds of faces

There are three different kinds of figure here. These differ according to whether the original truncate faces share a common point, q-gon or 2q-gon. The first and last symetrically derive from the duals, and are thus allocated to the mesotruncates. The middle one does not, and is thus derived from the regular figures. This class does not exist in three dimensions.

The class that is derived from the regular figure is the prismatorhombiates. Although the class corresponds to the runcitruncates, the prismatorhombiate of X is the runcitruncate of the dual of X.

The other two classes corrospond to the runcinates and omnitruncates, which are now called small and great prismato- for self-duals, and small and great diprismato- for the non-self duals. These carry the name of the mesotruncate.

small diprismato-[mesotruncate]: The "di" is used when the mesotruncate is not of a self-dual. In these cases, there are two different kinds of prism.

rt{4D dual}
prismatorombiated [regular]: Note that this is applied to the dual of the figure referred to in the runcitruncate.

great diprismato-[mesotruncate]: The "di" is used only when the mesotruncate is not of a self-dual. In these cases, there are two kinds of prism.

The non-Wythoff-figures

The four non-wythoffian figures are named as follows.

snub cube:

snub dodecahedron:

snub disicoatetrachoron:

grand antiprism:


The following tables give the acronyms for the three and four dimensional figures, as well as the prisms based on the three-dimensional figures.

                    t{}      s{}          {}p       t{}p     s{}p
    {3,3}    Tet    Tut                   Tepe     Tuttip
    {3,4}    Oct    Toe                   Ope      Tope
    {4,3}    Cube   Tic      Snic                  Ticcip   Sniccip
    {3,5}    Ike    Ti                    Ipe      Tipe
    {5,3}    Doe    Tid      Snid         Dope     Tiddip   Sniddip

            r{}     rr{}     ot{}          r{}p      rr{}p   ot{}p
     CO      Co     Sirco    Grico        Cope     Sircope  Gircope
     ID      Id     Srid     Grid         Iddip    Sriddip  Griddip

These are the four dimensional ones. The Dx numbers are my short notation. In terms of the nodes and branches, the node values are 1-2-4-8. So Prahi is 1+o+4+8 surrounding {5,3,3} or x5o3x3x.

               {3,3,3}  {3,3,4}  {4,3,3}  {3,4,3}   {3,3,5}  {5,3,3}  Dx
                 Pen      Hex      Tes      Ico       Ex       Hi      1
   rectified     Rap     (Ico)     Rit      Rico     Rox      Rahi     2
   truncated     Tip      Thex     Tat      Tico     Tex      Tahi     3
   sm rhombi-   Srip     (Rico)   Srit      Srico    Srix     Srahi    5
   gr rhombi-   Grip     (Tico)   Grit      Grico    Grix     Grahi    7
   prismatorh   Prip      Proh    Prit      Prico    Prix     Prahi   13

                   bt{3,3,3}   bt{3,3,4}  bt{3,4,3}  bt{3,3,5}  Dx
                    Deca         Tah        Cont      Xhi        6
   small prismato-  Spid       Sidpith      Spic    Sidpixhi     9
   great prismato-  Gippid     Gidpith     Gippic   Gidpixhi    15

                   ss{3,4,3} = Sadi     ss{3,3,5} = Gap

Extending to 5 Dimensions

This notation is the easiest to extend to five dimensions, with a couple of adjustments. In order to follow what's going on, I use my short notation.

Copyright 2002 Wendy Krieger

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