Expanded rotatope (InstanceTopic, 3)
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| - | 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. | + | <[#ontology [kind topic] [cats Property]]> |
| + | 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 [[ | + | To find the expanded rotatope of a toratope written in [[toratopic notation]], look at each group, including nested groups, and write down the number of elements in the group. Then append any outer digons to the result. |
| - | Examples: | + | Examples: |
| - | ( | + | *((II)I) => 22, since there are two groups each with two elements. |
| - | ( | + | *((III)I) => 32, since there is one group with three elements, and another group with two elements (the sub-group counts as one element). |
| - | ( | + | *((II)II) => 32 |
| - | ( | + | *((II)I)I => 221, since there are two groups with two elements, and an extra digon on the outside. |
| - | (( | + | *((II)(II)) => 222, since there are two (II) groups, and another group containing these two groups. |
| + | *(((II)II)(II)) => 3222 | ||
== Finding homeomorphisms == | == Finding homeomorphisms == | ||
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== Homology groups == | == 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. | 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. | ||
| + | |||
| + | == See also == | ||
| + | *[[Table of expanded rotatopes]] | ||
| + | *[[List of toratopes by expanded rotatope]] | ||
Latest revision as of 22:50, 11 February 2014
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 toratopic notation, look at each group, including nested groups, and write down the number of elements in the group. Then append any outer digons to the result.
Examples:
- ((II)I) => 22, since there are two groups each with two elements.
- ((III)I) => 32, since there is one group with three elements, and another group with two elements (the sub-group counts as one element).
- ((II)II) => 32
- ((II)I)I => 221, since there are two groups with two elements, and an extra digon on the outside.
- ((II)(II)) => 222, since there are two (II) groups, and another group containing these two groups.
- (((II)II)(II)) => 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.
