Here you go...
Use . for a point and | for a line segment
For a regular polygon of n sides, write G followed by the number of sides. So G4 is a square, G7 is a heptagon, and G38 is a 38-gon.
For the regular polyhedra, use T, C, O, D, and I for tet, cube, oct, doe, and ike respectively.
For the polychora, use P, S, E, R, A, and X for pen, tes, hex, ico, hi, and ex respectively.
For higher dimensions, use Tn for n-simplex, Cn for n-cube, and On for n-cross.
Uniform convex polytopes
This is fairly simple. After a regular polytope, put "t" for truncation and then a "truncation code". The truncation code is easy to determine, simply write the diagram of the shape, and converts x to 1 and o to 0. So t1 is regular, t2 is rectified, t3 is truncated, t4 is birectified. etc. So Ot3 and Ct6 both represent the truncated octahedron.
You can also use Dn for the n-D demicube, and En, Jn, and Kn for the n-D E-polytope families. En is n_21, Jn is 2_n1, and Kn is 1_n2. In this case count from the branch of the ringed node, then go down the shorter branch before the longer one when counting truncates.
Use N, Q, and H for the triangular, square, and hexagonal tilings. Use U for the cubic honeybomc and Hn for the n-D cubic honeybomc. (I need to work out how to represent the rest of them.) You can use the truncation method described above on these too.
To represent an n/d star polygon, use Gn*d. So G7*3 is the great heptagram.
You can use Bn as the n-D hyperball and Sn as the n-D hypersphere.
The following linear operations are supported:
- | - makes a prism
- > - makes a pyramid
- X - makes a bipyramid
In addition you can use X*T for the prism product of X and Y, and X+Y for the tegum (dual prism) product of X and Y.
So B2| is a cylinder while S1| is the hollow tube.
Below is a list of shapes in the wiki expressed in this notation.