Fundamental sets
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Regular polygons (Gx)
Kanitopes (Kfx)
Manifolds (Mx)
Toratopes (Tx)
Each fundamental set is infinite, but there are finitely many shapes of each dimension in each set. Thus the sets can be enumerated in dimension order.
Bricks
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A "brick" is a shape with brick symmetry, i.e. point (a,b,c,...) implies point (±a,±b,±c,...) for all sign permutations.
All toratopes are bricks
Any regular polygon Gx where x >= 4 and x is a natural power of 2 is a brick
All kanitopes that can be represented outside of a simplex family and aren't snubs are bricks
The dual of a brick is a brick
The only manifolds that are bricks are trivial so they already satisfy one of the above conditions
Any brick has an associated ''brick product'', including bracketopic ones: Cartesian = SQUARE = max, tegal = DIAMOND = sum, circular = CIRCLE = rss
The powertope A^P can be represented more generally as a brick product P{A,A,...} where there are n A's where n is the dimensionality of P
Shapes from the five sets can be combined using brick products, boolean operations (intersection, union, invertion), matrix transformations and the pyramid operation which tapers to a point.
Notation
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A regular polygon is denoted as Gx, where x is the number of sides on the polygon.
The dual of polygon Gx is written Hx.
A kanitope is denoted as Kfx, where f is the family and x is the Dx number.
Manifold notation is not yet defined but will start with M where used.
Toratopes are denoted as Tx, where x is the rotope notation. In addition, if x is a number, Tx represents a hypersphere of that dimension.
Square brackets [] are used to denote order of evaluation.
Brick products are of the form P{A,B,C,...} where P is the brick and the other letters are the shapes. There must be exactly as many shapes as the dimensionality of P. ^ or superscript integer inside a brick product represent repeats of a shape.
Boolean operations are written as A n B, A u B and !A.
Matrix transformations are written [Matrix][Object], including the square brackets.
The pyramid operation is written as a &A.
A x B, A o B and A v B can be used as shortcuts for the square, circle and diamond brick products respectively.
As special cases, G0 is the point and G1 is the digon i.e. line segment.
S and Rx can be used in the same sense as they are in SSC notation.
Summary of symbols used
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Set starters: G, H, K, M, T
Brackets: [] precedence and matrices, {} brick products, () toratopes
Prefix operators: & pyramid, ! invert
Binary operators: n intesection, u union, x square, o circle, v diamond
Comma: , used in brick products
Store and recall: S, Rx
Examples
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Cube: Ko1
Sphone: &T3
Torus: T((II)I)
Crind: T2{G4,G1} or G4oG1
Bicone: H4{T2,G1} or T2vG1
Cubic snub prism: G4{Ko0,G1} or Ko0xG1
Triangular octagoltriate: G8{G3^2}
Advantages over SSC
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Ambiguous and immeasurable rotopes cannot crop up.
The messy sets of Bracketopes and Rotopes are not enumerated.
Brick products and powertopes are generalized.
Cannot possibly try to use operations in the wrong place - e.g. vertex folds and truncations were specific to uniforms, torii to rotopes - now these are handled inside the fundamental sets and do not exist in general combinations.
Tigroids are representable again, as toratopes.
A definition is now unrelated to its orientation.
Disadvantages over SSC
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No longer compatible with bracketopic notation.