Extended toratopic notation (no ontology)

From Hi.gher. Space

Extended toratopic notation is a notation based on toratopic notation, but extended to allow representing multiple copies of toratopes as well as points and the empty set. The motivation for this notation is to explain how toratopic notation (and toratopes themselves) works more intuitively, and to concisely represent cross-sections (cuts) of toratopes. It was developed by Marek14.[1]

Definition

An expression written in extended toratopic notation is an unordered list of zero or more terms, where each term is either an I or a pair of parentheses () containing another expression. Any set of shapes which can be represented by such an expression is called a herd.

As such, in contrast to the traditional toratopic notation, each pair of parentheses may contain only a single term, or even be empty; the whole expression can also be empty. This allows us to stipulate the following rules:

1. The empty string "" represents the point.
2. The string I represents a pair of points (the "one-dimensional sphere" - all points a fixed distance from the origin in 1D).
3. Concatenating two expressions into one gives you the Cartesian product of the two original expressions.
4. Spherating an expression (adding parentheses around it) considers separately two parts of the original expression:
   1. any groups and their contents, representing a herd X in p dimensions (where p is the total number of Is in X's notation), and
   2. any Is not inside groups, representing q additional dimensions (where q is the number of Is not inside groups).
   • X is then spherated into n = p+q dimensions, by replacing each point in X with an n-ball (i.e. the 2-ball is the disc and the 1-sphere is a digon) and then taking the (n-1)-dimensional surface of the result.

Note that while closed toratopes are exactly the same whether written in extended notation or not, open toratopes are not the same. For example, in the traditional notation, (II)I is a cylinder and III is a cube, but in the extended notation, (II)I is two circles in parallel planes, and III is eight points at the vertices of a cube. However, one can easily convert the latter to the former, by splitting the outermost expression into terms, "solidifying" each term (the opposite of taking the surface of it), and then taking the Cartesian product of the "solidified" terms. The former can also be converted to the latter, by taking the (n-r)-dimensional surface, where n is the total number of dimensions, and r is the number of terms.

Cross-sections can be worked out trivially: simply remove any I from the expression, and you obtain a cross-section one dimension lower. If your cross-section ends up with a () term in it, then this means the "origin is empty": although the cross-section which includes the origin is the empty set, if you move the cutting hyperplane away from the origin, parts of the original shape will start to appear.

Examples

• As above, the empty string is the point.
• () is the empty set. In addition, any string containing an () is also the empty set (because the Cartesian product of the empty set with anything is the empty set, and the spheration of the empty set into any number of dimensions is also the empty set.
• Sphere species, (), contains two separated points on a line (I), circle (II), sphere (III), glome (IIII), pentasphere (IIIII) etc. This is a HERBIVORE species that grows nice and fat and falls prey to the beasts.
• Torus species, (()), contains torus ((II)I), spheritorus ((II)II), torisphere ((III)I), toratesserinder ((II)III), toracubspherinder ((III)II), toraglominder ((IIII)I) etc. Apart from these, it also contains pairs of spheres ((x)) and ((I)x) and quartet of points ((I)).
• Tiger species, (()()), contains tiger ((II)(II)), toraduocyldinder ((II)(II)I), cylspherintigroid ((III)(II)) etc. As for herds, it contains pairs of toruses like ((II)(I)), and quartets of spheres like ((I)(I)).