Rotope (EntityClass, 3)
From Hi.gher. Space
In early 2003, Garrett Jones discovered and coined the rotatopes. Rotatopes are Cartesian products of hyperspheres, and the number of n-dimensional rotatopes is equal to P(n), where P is the partition function. Alternatively, one could say that rotatopes are formed by combinations of extrusion and lathing operations.
In late 2005, the notion of toratopes was invented by Paul Wright, Marek14 and Wendy Krieger. The set of toratopes then became a superset of the rotatopes by generalizing the "lathe" operation into a "spherate" operation.
However, in summer 2006, after working on tapertopes with Wright, Keiji simply decided to add the "taper" operation to the mix. This produced the superset of rotopes as it's currently known, but this caused three problems. Firstly, it took away the "commutativity" of the rotopic notations, meaning they could no longer be reordered in some of the previously possible ways. Secondly, a number of shapes (which grows as the dimension increases; there is only one in 4D) were excluded from the set of rotopes for no other reason than they could not be written in the rotopic notations (hence the "almost correct" at the top of this page). And perhaps most importantly, mixing tapering and spheration operations created a large number of invalid shapes, which plagued much of the later analysis into rotopes.
In late 2009, Keiji decided it was time to sort out this mess once and for all, and began separating the toratopes from the tapertopes once again. Of course, that won't uninvent the two now defunct notations he created based on the rotopes, firstly CSG notation and secondly SSC notation, but he did create a far more sensible SSC2 about a year before this split, which acknowledges the innocence of strange rotopes and invalidates the immeasurable rotopes. There's also a rumour of a third version of SSC to invalidate the ambiguous ones too, but that's in the future, not the past :)
Subsets of rotopes
Strange rotopes are rotopes which have more than one group inside any group in the group notation definition of the rotope, or if they have a group following a superscript letter inside any group. These are marked by red lines on the rotope construction chart. The original SSC notation disallowed the spheration operation to get rid of strange rotopes, but after the first version was finalized, a proper definition for all strange (and otherwise valid) rotopes was found. For this reason, SSC2 reintroduced spheration in the context of toratopes alone. This removed the set of immeasurable rotopes but leaves the slightly less inconvenient set of ambiguous rotopes definable.
Ambiguous and immeasurable rotopes
Ambiguous rotopes are rotopes which are tapered after attaining a nonzero genus. In other words, if, in the group notation definition of the rotope, there is a superscript letter after a level 2 or higher nested group, the rotope is ambiguous. On the other hand, immeasurable rotopes are rotopes which have a superscript letter inside a group in their group notation definition.
A is an ambiguous rotope iff there is some n for which the number of n-dimensional elements in A cannot be counted; it is an immeasurable rotope iff there is some n for which the n-dimensional volume of A cannot be calculated.
The rotope construction chart highlights the ambiguous and immeasurable rotopes with brown and blue dots respectively.
Two notations were originally developed to represent rotopes: digit notation and group notation. Digit notation was originally created by Jonathan Bowers, and is better suited to the tapertopes. Group notation was derived from digit notation by Wright (I think), and is better suited to the toratopes. The two notations were eventually adapted into use for these two sets, unsurprisingly being called tapertopic notation and toratopic notation respectively.
Digit notation uses a single number to represent a hypersphere, and combinations of the numbers to represent the cartesian products of those shapes. Thus, '1' is a line, '2' is a circle, and '3' is a sphere. A shape with only 1s in its sequence of digits is a hypercube. Thus, '1' is a line, '11' is a square, and '111' is a cube. The sum of the numbers in a shape is the dimensionality of the shape.
In group notation, however, a letter is used to represent each dimension, and a "group" of letters in parentheses is used to represent spheration, so nested parentheses produce torii. This nesting property was carried back to digit notation, but this extension makes the digits macros rather than atoms.
In either notation so far, the digits in any particular sequence are interchangeable, so a sequence of digits refers to the same shape when any of its digits are transposed. For example, 21 and 12 both represent a cylinder. Another result of this invariance under transposition is that the same shape can result from cross products of different shapes. Thus, the sequences 1212 and 1122 both represent the same shape, which can be constructed from any of the following cartesian products - the product of two cylinders, the product of a square and a duocylinder, or the product of a cubinder and a circle, or any other combinations of its subshapes.
Another extension was later added to the rotopic notations: tapering. A superscript number 1 means that the shape formed so far will now be tapered to a point. A superscript number greater than 1 means that it will be tapered that many times. In group notation, this was written using consecutive superscript letters instead. Groups separated by parentheses are independent from anything outside those parentheses.
Unfortunately, this latest extension introduced a problem with ordering the notation. 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 sequence (a(bc)(de))fg(hi)j may be reordered as a((bc)d(ef))gh(ij), and does not change the shape at all (only the orientation).
Three methods of conversion were found involving rotopic notations: one to convert between the two types of rotopic notation, and more importantly, ways to convert rotopic notation into product notation or a surface equation. Unsurprisingly, the second two don't work with superscripts.
To convert from group notation to digit notation, follow these steps:
- Change every letter (superscript or not) to a number 1.
- 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.
- If there is a sequence of superscript 1s, change the sequence to the number of 1s in the sequence, retaining the superscript.
Similarly, to do the reverse, follow these steps:
- Where there is a non-superscript digit n greater than 1, change the digit to a sequence of n 1s, and surround them with parentheses.
- Where there is a superscript digit n greater than 1, change the digit to a sequence of n 1s, keeping the sequence in superscript, but do not surround them with parentheses.
- Replace every digit 1 with a letter.
To product notation
- Ensure the sequence is in digit notation by converting it as above if necessary.
- Parentheses evaluate from inside out.
- Parentheses of form a1111... with b 1's evaluate to a#(b+1)
- Other parentheses containing any 1's evaluate to a#(b+k) where a#b is evaluation of the same parentheses without 1's, and k is the number of 1's.
- Parentheses containing terms a,b,c etc. with at least two terms and none of them equal to 1 evaluate to (a x b x c x ...)#n where n is the number of terms.
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):
- Ensure the sequence is in group notation by converting it as above if necessary.
- Make a square of each variable and add terms within parenthesis: (((x2+y2)+z2)+w2)
- Replace each parenthesis with a square root function: sqrt(sqrt(sqrt(x2+y2)+z2)+w2)
- 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
- 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
Here is a table to show the number and percentage of various types of rotopes in each dimension.
|1||1||0 (0%)||0 (0%)||0 (0%)||1 (100%)||1 (100%)|
|2||3||0 (0%)||0 (0%)||0 (0%)||3 (100%)||2 (67%)|
|3||9||0 (0%)||1 (11%)||0 (0%)||8 (89%)||3 (33%)|
|4||31||2 (6%)||8 (25%)||4 (13%)||19 (61%)||5 (16%)|
|5||111||18 (16%)||46 (41%)||30 (27%)||42 (37%)||7 (6%)|
|Trend||Increasing||Increasing %||Increasing %||Increasing %||Decreasing %||Decreasing %|
Generally, if you want to find a rotope, you should look at the list of tapertopes or the list of toratopes. If the shape isn't in either of these, it's either ambiguous or immeasurable, a select few of which appear below for comparisons' sake, the rest being summarized. However, the old list of rotopes and rotope construction chart are still present.
The triangular torus, or torapyramid, pictured right, can be defined as circle # triangle. Since the torus product is not uniquely defined in this case, this makes it an immeasurable rotope. However, CSG Notation defines the triangular torus written as "ETQ" as a triangle lathed in such a way that the bases of all the triangular radial slices lie in the same plane. It clearly has three curved faces and three circular edges. Its SSCN representation is G3T.
The toric pyramid is the first ambiguous rotope, and is four-dimensional. Its apex is both included and excluded from the shape at the same time.
The cyltrianglintigroidal pyramid is the first rotope which is strange, ambiguous and immeasurable. It is five-dimensional.