Rotopic group notation (EntityClass, 3)

From Hi.gher. Space

In group notation, letters are used to represent dimensions.

• A normal letter represents that the object is extruded in that dimension.
• A superscript letter represents that the object is tapered in that dimension.
• A pair of parentheses represents that the object is spherated.

Ordering

If there are no superscripts in a group-notationally defined rotope, the order of groups (that is, letters and bracketed series of letters) may be changed in any way. However, if there are superscripts, groups may not be moved into a different "part" of the definition, where a "part" is a series of consecutive non-superscript groups. Ordering inside a particular group works the same way as for the whole thing.

For example, the ten-dimensional shape (a(bc)(de))f^g(hi)j may be reordered as f((ab)c(de))^gh(ij), and does not change the shape at all (only the orientation).

Conversions

To digit notation

1. Change every letter (superscript or not) to a number 1.
2. If there is a sequence (111...) with n 1s and nothing else inside the parentheses, change the entire sequence, including parentheses, to the number n.
3. If there is a sequence of superscript 1s, change the sequence to the number of 1s in the sequence, retaining the superscript.

To surface equation

Marek14 found a way to convert group notation to a surface equation, edited here for ease of use:

Using the example (((xy)z)w):

1. Make a square of each variable and add terms within parenthesis: (((x²+y²)+z²)+w²)
2. Replace each parenthesis with a square root function: sqrt(sqrt(sqrt(x²+y²)+z²)+w²)
3. Immediately outside of each square root function, subtract a parameter and square this: (sqrt((sqrt((sqrt(x²+y²)-A)²+z²)-B)²+w²)-C)²
4. Remove the outermost square and parentheses, and form an equation with this expression as the LHS and zero as the RHS: sqrt((sqrt((sqrt(x²+y²)-A)²+z²)-B)²+w²)-C = 0