Rotope (EntityClass, 3)

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Revision as of 10:22, 21 June 2007 by Keiji (Talk | contribs)

Sets of rotopes

Rotopes are combinations of rotatopes, toratopes and tapertopes. A rotope may be any combination of these, with the exception that a toratope may never be a tapertope and vice versa. There are also rotopes that are none of these. An example is the torinder. The number of non-tapertopes in any dimension is always twice the number of toratopes. In the table below, 'x' denotes the cartesian product, '#' denotes the torus product and '~' denotes tapering. Note that the CSG Notation column shows the notation for a completely solid form of the object.

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 spheration, i.e. 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.

Tapertopes

Tapertopes were coined by Keiji, and invented by him and Paul Wright.

Tapertopes are the line and shapes formed by tapering other tapertopes to a point.

Sets of 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%)
6 413 111 (26%) 227 (54 %) 170 (41%) 89 (21%) 11 (2%)
7 1549 576 (37%) 1027 (66%) 820 (52%) 184 (11%) 15 (0%)
8 5849 2713 (46%) 4401 (75%) 3662 (62%) 375 (6%) 20 (0%)
9 22151 12044 (54%) 18184 (82%) 15590 (70%) 758 (3%) 25 (0%)
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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