Tapertope (EntityClass, 4)
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A '''tapertope''' is any combination of [[Cartesian product]] and [[pyramid]] operations on [[hypersphere]]s. These include the hyperspheres themselves, the [[hypercube]]s, the [[simplices]], varieties of [[cylinder]]s and [[cone]]s and other combinations of the above. | A '''tapertope''' is any combination of [[Cartesian product]] and [[pyramid]] operations on [[hypersphere]]s. These include the hyperspheres themselves, the [[hypercube]]s, the [[simplices]], varieties of [[cylinder]]s and [[cone]]s and other combinations of the above. | ||
| - | Tapertopes were first invented by [[Keiji]] and [[Paul Wright]] in summer 2006, as a subset of the growing set of [[rotope]]s at the time. However, combining them with the toratopes brought about a large number of invalid shapes, which plagued any analysis of rotopes. To overcome this problem, the set of rotopes | + | Tapertopes were first invented by [[Keiji]] and [[Paul Wright]] in summer 2006, as a subset of the growing set of [[rotope]]s at the time. However, combining them with the toratopes brought about a large number of invalid shapes, which plagued any analysis of rotopes. To overcome this problem, the set of rotopes was finally split up into two separate sets of tapertopes and toratopes in late 2009. In addition, the redefinition of "tapertope" allowed the set to include a few more shapes which were not rotopes but are worthy of inclusion. |
The intersections of any two sets out of the tapertopes, [[toratope]]s and [[bracketope]]s produces [[Garrett Jones]]' classic set of [[rotatope]]s, which occupies the first P(''n'') slots of the tapertopes in each dimension ''n'' where P is the [[partition function]]. | The intersections of any two sets out of the tapertopes, [[toratope]]s and [[bracketope]]s produces [[Garrett Jones]]' classic set of [[rotatope]]s, which occupies the first P(''n'') slots of the tapertopes in each dimension ''n'' where P is the [[partition function]]. | ||
| - | All tapertopes can be represented in the new [[tapertopic notation]], [[SSC2]], [[SSCN]] and even the ancient [[CSG notation]]. Rotopic digit | + | All tapertopes can be represented in the new [[tapertopic notation]], [[SSC2]], [[SSCN]] and even the ancient [[CSG notation]]. [[Rotopic digit and group notations]] can only represent those tapertopes that were originally rotopes. |
== Tapertopic statistics == | == Tapertopic statistics == | ||
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|style="background-color:#eeeeff; text-align:center;"|[[:Category:Tapergons|3 (100%)]] | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Tapergons|3 (100%)]] | ||
|style="background-color:#ddddff; text-align:center;"|[[:Category:Tapergons|3 (100%)]] | |style="background-color:#ddddff; text-align:center;"|[[:Category:Tapergons|3 (100%)]] | ||
| - | |style="background-color:#eeeeff; text-align:center;"|2 (67%) | + | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Rotagons|2 (67%)]] |
|- | |- | ||
|style="background-color:#eeeeff; text-align:center;"|3 | |style="background-color:#eeeeff; text-align:center;"|3 | ||
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|style="background-color:#eeeeff; text-align:center;"|[[:Category:Taperhedra|7 (100%)]] | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Taperhedra|7 (100%)]] | ||
|style="background-color:#ddddff; text-align:center;"|[[:Category:Taperhedra|7 (100%)]] | |style="background-color:#ddddff; text-align:center;"|[[:Category:Taperhedra|7 (100%)]] | ||
| - | |style="background-color:#eeeeff; text-align:center;"|3 (43%) | + | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Rotahedra|3 (43%)]] |
|- | |- | ||
|style="background-color:#eeeeff; text-align:center;"|4 | |style="background-color:#eeeeff; text-align:center;"|4 | ||
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|style="background-color:#eeeeff; text-align:center;"|17 (94%) | |style="background-color:#eeeeff; text-align:center;"|17 (94%) | ||
|style="background-color:#ddddff; text-align:center;"|15 (83%) | |style="background-color:#ddddff; text-align:center;"|15 (83%) | ||
| - | |style="background-color:#eeeeff; text-align:center;"|5 (28%) | + | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Rotachora|5 (28%)]] |
|- | |- | ||
|style="background-color:#eeeeff; text-align:center;"|5 | |style="background-color:#eeeeff; text-align:center;"|5 | ||
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|style="background-color:#eeeeff; text-align:center;"|40 (89%) | |style="background-color:#eeeeff; text-align:center;"|40 (89%) | ||
|style="background-color:#ddddff; text-align:center;"|31 (69%) | |style="background-color:#ddddff; text-align:center;"|31 (69%) | ||
| - | |style="background-color:#eeeeff; text-align:center;"|7 (16%) | + | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Rotatera|7 (16%)]] |
|- | |- | ||
|style="background-color:#eeeeff; text-align:center;"|6 | |style="background-color:#eeeeff; text-align:center;"|6 | ||
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|style="background-color:#eeeeff; text-align:center;"| | |style="background-color:#eeeeff; text-align:center;"| | ||
|style="background-color:#ddddff; text-align:center;"| | |style="background-color:#ddddff; text-align:center;"| | ||
| - | |style="background-color:#eeeeff; text-align:center;"| | + | |style="background-color:#eeeeff; text-align:center;"|[[:Category:Rotapeta|11 (9.3%)]] |
|- | |- | ||
|style="background-color:#ddddff; text-align:center;"|'''Trend''' | |style="background-color:#ddddff; text-align:center;"|'''Trend''' | ||
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There is currently one main method for finding tapertopes: | There is currently one main method for finding tapertopes: | ||
*[[List of tapertopes]] | *[[List of tapertopes]] | ||
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Latest revision as of 19:57, 9 March 2011
A tapertope is any combination of Cartesian product and pyramid operations on hyperspheres. These include the hyperspheres themselves, the hypercubes, the simplices, varieties of cylinders and cones and other combinations of the above.
Tapertopes were first invented by Keiji and Paul Wright in summer 2006, as a subset of the growing set of rotopes at the time. However, combining them with the toratopes brought about a large number of invalid shapes, which plagued any analysis of rotopes. To overcome this problem, the set of rotopes was finally split up into two separate sets of tapertopes and toratopes in late 2009. In addition, the redefinition of "tapertope" allowed the set to include a few more shapes which were not rotopes but are worthy of inclusion.
The intersections of any two sets out of the tapertopes, toratopes and bracketopes produces Garrett Jones' classic set of rotatopes, which occupies the first P(n) slots of the tapertopes in each dimension n where P is the partition function.
All tapertopes can be represented in the new tapertopic notation, SSC2, SSCN and even the ancient CSG notation. Rotopic digit and group notations can only represent those tapertopes that were originally rotopes.
Tapertopic statistics
Here is a table to show the number and percentage of various types of tapertopes in each dimension.
| Dimension | Tapertopes | Rotopic | Linear | Rotatopic |
| 1 | 1 | 1 (100%) | 1 (100%) | 1 (100%) |
| 2 | 3 | 3 (100%) | 3 (100%) | 2 (67%) |
| 3 | 7 | 7 (100%) | 7 (100%) | 3 (43%) |
| 4 | 18 | 17 (94%) | 15 (83%) | 5 (28%) |
| 5 | 45 | 40 (89%) | 31 (69%) | 7 (16%) |
| 6 | 118 | 11 (9.3%) | ||
| Trend | Increasing | Decreasing % | Decreasing % | Decreasing % |
Finding tapertopes
There is currently one main method for finding tapertopes:
