Tapertope (EntityClass, 4)
From Hi.gher. Space
(Tapertope moved to List of tapertopes) |
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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]], various 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 <!--was finally-->is being 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]]. | ||
| + | |||
| + | All tapertopes can be represented in the new [[tapertopic notation]], [[SSC2]], [[SSCN]] and even the ancient [[CSG notation]]. Rotopic digit or 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. | ||
| + | |||
| + | {|style="border: 1px solid; border-color:#808080; border-collapse: collapse;" cellpadding="2" width="100%" | ||
| + | |width="20%" style="background-color:#ddddff; text-align:center;"|'''Dimension''' | ||
| + | |width="20%" style="background-color:#ccccff; text-align:center;"|'''Tapertopes''' | ||
| + | |width="20%" style="background-color:#ddddff; text-align:center;"|'''Rotopic''' | ||
| + | |width="20%" style="background-color:#ccccff; text-align:center;"|'''Linear''' | ||
| + | |width="20%" style="background-color:#ddddff; text-align:center;"|'''Rotatopic''' | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|1 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[Digon|1]] | ||
| + | |style="background-color:#eeeeff; text-align:center;"|[[Digon|1 (100%)]] | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[Digon|1 (100%)]] | ||
| + | |style="background-color:#eeeeff; text-align:center;"|[[Digon|1 (100%)]] | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|2 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[:Category:Tapergons|3]] | ||
| + | |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:#eeeeff; text-align:center;"|2 (67%) | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|3 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[:Category:Taperhedra|7]] | ||
| + | |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:#eeeeff; text-align:center;"|3 (43%) | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|4 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[:Category:Taperchora|18]] | ||
| + | |style="background-color:#eeeeff; text-align:center;"|17 | ||
| + | |style="background-color:#ddddff; text-align:center;"|15 | ||
| + | |style="background-color:#eeeeff; text-align:center;"|5 | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|5 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[:Category:Tapertera|45]] | ||
| + | |style="background-color:#eeeeff; text-align:center;"|40 | ||
| + | |style="background-color:#ddddff; text-align:center;"|31 | ||
| + | |style="background-color:#eeeeff; text-align:center;"|7 | ||
| + | |- | ||
| + | |style="background-color:#eeeeff; text-align:center;"|6 | ||
| + | |style="background-color:#ddddff; text-align:center;"|[[:Category:Taperpeta|118]] | ||
| + | |style="background-color:#eeeeff; text-align:center;"| | ||
| + | |style="background-color:#ddddff; text-align:center;"| | ||
| + | |style="background-color:#eeeeff; text-align:center;"| | ||
| + | |- | ||
| + | |style="background-color:#ddddff; text-align:center;"|'''Trend''' | ||
| + | |style="background-color:#ccccff; text-align:center;"|Increasing | ||
| + | |style="background-color:#ddddff; text-align:center;"| | ||
| + | |style="background-color:#ccccff; text-align:center;"| | ||
| + | |style="background-color:#ddddff; text-align:center;"| | ||
| + | |} | ||
| + | |||
| + | == Finding tapertopes == | ||
| + | There is currently one main method for finding tapertopes: | ||
| + | *[[List of tapertopes]] | ||
| + | |||
| + | [[Category:Tapertopes| ]] | ||
Revision as of 18:39, 22 November 2009
A tapertope is any combination of Cartesian product and pyramid operations on hyperspheres. These include the hyperspheres themselves, the hypercubes, the simplices, various 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 is being split up into two separate sets of tapertopes and toratopes. 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 or 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 | 15 | 5 |
| 5 | 45 | 40 | 31 | 7 |
| 6 | 118 | |||
| Trend | Increasing |
Finding tapertopes
There is currently one main method for finding tapertopes:
