Expanded rotatope (InstanceTopic, 3)
From Hi.gher. Space
(New page: The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. The expanded rotatope is always embedded in a higher dimension than the toratope. T...) |
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The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. The expanded rotatope is always embedded in a higher dimension than the toratope. | The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. The expanded rotatope is always embedded in a higher dimension than the toratope. | ||
| - | 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. | + | 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. <br> |
| - | Examples: | + | Examples:<br> |
| - | (21) ~ 22 since there are two brackets each with 2 elements. | + | (21) ~ 22 since there are two brackets each with 2 elements.<br> |
| - | (31) ~ 32 | + | (31) ~ 32<br> |
| - | (211) ~ 32 | + | (211) ~ 32<br> |
| - | (22) ~ 222 | + | (22) ~ 222<br> |
| - | ((211)2) ~ 3222 | + | ((211)2) ~ 3222<br> |
== Finding homeomorphisms == | == Finding homeomorphisms == | ||
Revision as of 10:18, 30 November 2009
The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope. The expanded rotatope is always embedded in a higher dimension than the toratope.
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.
