SSC2 (InstanceTopic, 3)

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Revision as of 22:15, 28 October 2008

SSC2, standing for Standard Shape Construction version 2, is a notation for defining shapes. It is the successor to SSCN (also known as SSC1 or just SSC). SSC2 is the newest and preferred method for writing the definition of a shape.

Scope

SSC2, like its predecessor, is concerned only with the structure of a shape, not its position, size, orientation, frame or indeed stretching of any particular hypercell. Both versions do include a matrix transformation operation, but while the first version defined orientation either explicitly or implicitly, SSC2 includes this operation only so that boolean operations can become useful.

Fundamental sets

SSC2 has five articles called the fundamental sets. Each of these sets is infinite, but there are a finite number of shapes in each dimension in each set. Each set provides a method of enumeration which can later be used to enumerate any shape based on its SSC2 notation. Shapes are defined by referencing elements of the fundamental sets and combining them with various operators.

Manifolds

Manifolds are written in the form Mx, where x is the index of the manifold. This set is chosen to be the first in the list of sets because of all the SSC2 fundamental sets, it is the only one which contains the point and the line as the first two elements.

Regular polygons

Regular polygons are written in the form Gx, where x is the number of vertices in the polygon. G0, G1 and G2 are not valid.

Regular polygon duals

Regular polygon duals are exactly the same as regular polygons except they are written as Hx rather than Gx. Functionally, they are mainly used as brick products, for example the diamond, H4, is the tegal product.

Kanitopes

Kanitopes are written in one of two forms: Kfx for three- or four-dimensional shapes or Knfx for higher-dimensional shapes. Here, n is the shape's dimension, f is the family and x is the Dx number. Families are written with one letter, using t, o, i; p, e, k, s for Schlaefli symbols {3,3}, {3,4}, {3,5}; {3,3,3}, {3,3,4}, {3,4,3}, {3,3,5} respectively. For higher-dimensional shapes, the families are x for simplex and c for cross polytope or hypercube.

Toratopes

Toratopes are written in the form Tx. Here, x can either be an integer, in which case the toratope is an x-dimensional hypersphere, or a string in rotopic group notation restricted to having no superscripts and an outer pair of parentheses (thus making the string always represent a toratope, not just a more general rotope).

Bricks

A brick is a shape with brick symmetry, i.e. the existence of a point (a,b,c,...) implies point (±a,±b,±c,...) for all sign permutations.

The following statements make clear what shapes in the fundamental sets are considered bricks:

• 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 and already satisfy one of the above conditions; therefore, for the purposes of SSC2, manifolds are not considered bricks.

Operations

Firstly, any brick has an associated brick product. This is a generalization of and thus includes the three bracketopic products:

• The Cartesian or square product with function MAX is the brick square, or G4
• The tegal product with function SUM is the brick diamond, or H4
• The crindal product with function RSS is the brick circle, or T2.

Notice also that the powertope A^P can be represented more generally as a brick product P{A,A,...} where the number of A's is equal to the dimensionality of P.

For these reasons there are no operations representing any other products, nor is there a powertope operation. However, all of these have shortcuts - see later on in this page.

There are only five other operations:

• the pyramid operation, which tapers to a point
• the matrix tranformation operation, which should only be used prior to a boolean operation
• the three boolean operations: intersection, union and inversion.

Note that almost all combinations will be done by using brick products or the pyramid operation.

Notation

• Any shape in a fundamental set is written as already described.
• Precedence can be explicitly specified using square brackets (not parentheses or braces). No convention is set on implicit precedence; where ambiguous it simply remains ambiguous.
• The brick product is written P{A,A,A,...} where P is the operating shape and each A is an argument. The number of arguments must be equal to the bounding space of P.
• The pyramid operation is written as a prefix &.
• The matrix transformation operation is also written using square brackets. The matrix must be written first and the operand second, each enclosed in a pair of brackets.
• The binary boolean operations are written as infix n for intersection and u for union.
• The unary boolean operation, inversion, is written as a prefix !.

Shortcuts

In addition to the notation described above, several more definitions are added. These are shortcuts rather than fundamental operations and should be considered as mere string replacements.

• Infix A x B, A o B and A v B represent the square, circle and diamond brick products respectively, otherwise written as G4{A,B}, T2{A,B} and H4{A,B}.
• Prefix +A can also be used as a secondary shortcut for A x M1 (i.e. prism, or the Cartesian product with a line).
• S and Rx can be used in the same sense as they are in SSCN.

Summary of symbols used

Any italic symbols represent shortcuts rather than core notation.

Set starters

M, G, H, K, T

Brackets

[] precedence and matrices, {} brick products, () toratopes

Prefix operators

& pyramid, ! invert, + prism

Infix operators

n intesection, u union, x square, o circle, v diamond

Comma

, used in brick products

Store and recall

S, Rx

Examples

• Cube: Ko1
• Sphone: &T3
• Torus: T((II)I) or T((2)1)
• Crind: T2{G4,M1} or G4oM1
• Bicone: H4{T2,M1} or T2vM1
• Cubic snub prism: G4{Ko0,M1} or Ko0xM1 or +Ko0
• Triangular octagoltriate: G8{G3^2}

Advantages over SSCN

• Immeasurable rotopes cannot crop up.
• The messy sets of bracketopes and rotopes are not enumerated.
• Brick products and powertopes are generalized.
• It is impossible to use an operation in the wrong place - e.g. vertex folds and truncations were specific to uniform polytopes, 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 SSCN

• Incompatible with bracketopic notation or SSCN.