Rotope (EntityClass, 3)

From Hi.gher. Space

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:''Pure rotopes'' are rotopes which are not ambiguous, immeasurable or strange.
:''Pure rotopes'' are rotopes which are not ambiguous, immeasurable or strange.
*Note that impure rotopes only exist because spheration is allowed. [[SSC notation]] disallows the spheration operation for this reason.
*Note that impure rotopes only exist because spheration is allowed. [[SSC notation]] disallows the spheration operation for this reason.
 +
*After SSC notation was finalized, a proper definition for all strange rotopes except ones including taper operations has been found. For this reason, [[SSC2]] reintroduces spheration in the context of toratopes alone. This removes the set of immeasurable rotopes but leaves the slightly less inconvenient set of ambiguous rotopes definable.
== Rotopic statistics ==
== Rotopic statistics ==

Revision as of 15:51, 6 November 2008

The set of rotopes is the 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 Hayate, 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.
  • Note that impure rotopes only exist because spheration is allowed. SSC notation disallows the spheration operation for this reason.
  • After SSC notation was finalized, a proper definition for all strange rotopes except ones including taper operations has been found. For this reason, SSC2 reintroduces spheration in the context of toratopes alone. This removes the set of immeasurable rotopes but leaves the slightly less inconvenient set of ambiguous rotopes definable.

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:

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