Expanded rotatope (InstanceTopic, 3)

From Hi.gher. Space

Revision as of 10:25, 30 November 2009

The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. For example, the torus is homeomorphic to the duocylinder, so the duocylinder is the expanded rotatope of the torus. The expanded rotatope is always embedded in a higher dimension than the toratope, but they have the same minimal frame.

To find the expanded rotatope of a toratope written in group notation, replace each set of brackets with k elements with a separate sphere in k-dimensions. Remember that 3 = (111) is a set with three elements.

Examples:
(21) ~ 22 since there are two brackets each with 2 elements.
(31) ~ 32
(211) ~ 32
(22) ~ 222
((211)2) ~ 3222

Finding homeomorphisms

Expanded rotatopes can be used to find homeomorphisms between toratopes. For example, since (22) and ((21)1) have the same expanded rotatope, 222, we have (22) ~ 222 ~ ((21)1)

Homology groups

If two shapes are homeomorphic then they have the same homology groups. It follows that to calculate homology groups for all toratopes, we need only calculate them for all rotatopes.