Rotope (EntityClass, 3)
From Hi.gher. Space
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- | A '''rotope''' is a [[set]] of [[ | + | A '''rotope''' is a [[set]] of [[shape]]s formed by [[extruding]], [[tapering]] and [[spherating]] other rotopes. The [[point]] is a rotope; all other rotopes are formed from it. |
== Sets of rotopes == | == Sets of rotopes == |
Revision as of 21:29, 22 September 2007
A rotope is a set of shapes formed by extruding, tapering and spherating other rotopes. The point is a rotope; all other rotopes are formed from it.
Sets of rotopes
By operations
- Rotatopes
- A rotatope, invented by Garrett Jones is an object formed by linear extensions or rotations about the origin.
- Toratopes
- Toratopes were coined by Paul Wright, and invented by him and Marek14. A toratope is an object formed by extrusion or spheration, which involves putting a new n-sphere at every point in an object. This is also called the "torus product", #. A#B is the result of replacing every point in A with a copy of B, oriented along the normal space of A. With the introduction of toratopes came the set of strange rotopes and tigroids, of which very little is known.
- Tapertopes
- Tapertopes were coined by Keiji, and invented by him and Paul Wright. Tapertopes are rotopes involving tapering to a point. With the introduction of tapertopes came the sets of immeasurable rotopes and ambiguous rotopes.
By complexity
- Ambiguous 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.
- Immeasurable rotopes
- Immeasurable rotopes are rotopes which have a superscript letter inside a group in their group notation definition. Some hypervolumes cannot be calculated for immeasurable rotopes.
- 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.
- Pure rotopes
- Pure rotopes are rotopes which are not ambiguous, immeasurable or strange.
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 % |
- The data in the table above was calculated with a computer program. For dimensions 4 and 5 which previously had human-counted data, the previous data was incorrect.
Finding rotopes
There are currently two main methods for finding rotopes: