Rotope (EntityClass, 3)
From Hi.gher. Space
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| - | + | A '''rotope''' is a shape that can be written in the ''rotopic notations''; it is almost correct to say that it is any combination of [[extruding]], [[tapering]] and [[spherating]] operations on [[point]]s and other rotopes. | |
| - | == | + | == History == |
| - | + | In early 2003, [[Garrett Jones]] discovered and coined the ''rotatopes''. Rotatopes are [[Cartesian products]] of [[hypersphere]]s, 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. | |
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| - | + | In late 2005, the notion of [[toratope]]s was invented by [[Paul Wright]], [[Marek14]] and [[Wendy Krieger]]. The set of toratopes then became a superset of the [[rotatope]]s by generalizing the "lathe" operation into a "spherate" operation. | |
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| - | + | However, in summer 2006, after working on [[tapertope]]s 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 [[Duotrianglinder|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. | |
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| - | + | 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]] not too long 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 :) | |
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| - | + | === Strange 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. | |
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| - | + | === 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. | |
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| - | + | 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. | |
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| - | + | The rotope construction chart highlights the ambiguous and immeasurable rotopes with brown and blue dots respectively. | |
== Rotopic statistics == | == Rotopic statistics == | ||
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|style="background-color:#ddddff; text-align:center;"|Decreasing % | |style="background-color:#ddddff; text-align:center;"|Decreasing % | ||
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== Finding rotopes == | == Finding rotopes == | ||
| - | + | 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. | |
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| + | (shapes to be moved here) | ||
[[Category:Rotopes|*]] | [[Category:Rotopes|*]] | ||
Revision as of 17:12, 24 November 2009
A rotope is a shape that can be written in the rotopic notations; it is almost correct to say that it is any combination of extruding, tapering and spherating operations on points and other rotopes.
History
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 not too long 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 :)
Strange 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.
Rotopic statistics
Here is a table to show the number and percentage of various types of rotopes in each dimension.
| Dimension | Rotopes | Ambiguous | Immeasurable | Strange | Pure | Bracketopes |
| 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 % |
Finding rotopes
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.
(shapes to be moved here)
