Rotope (EntityClass, 3)

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Revision as of 21:27, 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.

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%)
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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+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 ==
-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 for<span>m</span> of the object.
+=== By operations ===
+;Rotatopes
-=== Rotatopes ===
+:A rotatope, invented by [[Garrett Jones]] is an object formed by [[extrusion|linear extensions]] or [[lathing|rotations]] about the origin.
-A rotatope, invented by [[Garrett Jones]] is an object formed by [[extrusion|linear extensions]] or [[lathing|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 ===
+;Toratopes
-Tapertopes were coined by [[Keiji]], and invented by him and [[Paul Wright]].
+: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 rotope]]s and [[tigroid]]s, of which very little is known.
-Tapertopes are the line and shapes formed by [[tapering]] other tapertopes to a point.
+;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 rotope]]s and [[ambiguous rotope]]s.
 == Sets of rotopes by complexity ==