Rotope (EntityClass, 3)
From Hi.gher. Space
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|valign="top" width="20%" style="background-color:#ddddff; text-align:center;"|'''Name''' | |valign="top" width="20%" style="background-color:#ddddff; text-align:center;"|'''Name''' | ||
- | |valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[ | + | |valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[Rotopic index]]''' |
|valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[Group notation]]''' | |valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[Group notation]]''' | ||
|valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[Digit notation]]''' | |valign="top" width="16%" style="background-color:#eeeeff; text-align:center;"|'''[[Digit notation]]''' |
Revision as of 22:21, 16 June 2007
Sets of rotopes
Rotopes are combinations of rotatopes, toratopes and tapertopes. A rotope may be any combination of these, with the exception that a toratope may never be a tapertope and vice versa. There are also rotopes that are none of these. An example is the torinder. The number of non-tapertopes in any dimension is always twice the number of toratopes. In the table below, 'x' denotes the cartesian product, '#' denotes the torus product and '~' denotes tapering. Note that the CSG Notation column shows the notation for a completely solid form of the object.
Rotatopes
A rotatope, invented by Garrett Jones is an object formed by linear extensions or rotations about the origin.
Toratopes
Toratopes were coined by Paul Wright, and invented by him and Marek14. A toratope is an object formed by spheration, i.e. putting a new k-sphere at every point in an object. This is also called the "torus product", #. A#B is the result of replacing every point in A with a copy of B, oriented along the normal space of A.
Tapertopes
Tapertopes were coined by Keiji, and invented by him and Paul Wright. A tapertope is an object formed by tapering another object to a point. It has been suggested that tapertopes be limited to only include the line and objects formed by extruding or tapering other objects.
Table of rotopes
Name | Rotopic index | Group notation | Digit notation | Product notation | CSG notation |
0D rotopes | |||||
Point | 0 | Empty string | Empty string | 0 | Empty string |
1D rotopes | |||||
Line | 1 | x | 1 | 1 | E |
2D rotopes | |||||
Square | 2 | xy | 11 | 1x1 | EE |
Triangle | 3 | xy | 11 | 1~0 | ET |
Circle | 4 | (xy) | 2 | 2 | EL |
3D rotopes | |||||
Cube | 5 | xyz | 111 | 1x1x1 | EEE |
Square pyramid | 6 | xyz | 111 | (1x1)~0 | EET |
Sphere | 7 | (xyz) | 3 | 3 | ELL |
Triangular prism | 8 | xyz | 111 | (1~0)x1 | ETE |
Tetrahedron | 9 | xyz | 12 | 1~0~0 | ETT |
Triangular torus | 10 | (xyz) | (111) | 2#(1~0) | ETQ |
Cylinder | 11 | (xy)z | 21 | 2x1 | ELE |
Cone | 12 | (xy)z | 21 | 2~0 | ELT |
Torus | 13 | ((xy)z) | (21) | 2#2 | ELQ |
4D rotopes | |||||
Tesseract | 14 | xyzw | 1111 | 1x1x1x1 | EEEE |
Cubic pyramid | 15 | xyzw | 1111 | (1x1x1)~0 | EEET |
Glome | 16 | (xyzw) | 4 | 4 | ELLL |
Square pyramid prism | 17 | xyzw | 1111 | ((1x1)~0)x1 | EETE |
Square dipyramid | 18 | xyzw | 112 | (1x1)~0~0 | EETT |
Square pyramid torus | 19 | (xyzw) | (1111) | ((1x1)~0)#2 | EETQ |
Spherinder | 20 | (xyz)w | 31 | 3x1 | ELLE |
Sphone | 21 | (xyz)w | 31 | 3~0 | ELLT |
Toraspherinder | 22 | ((xyz)w) | (31) | 3#2 | ELLQ |
Triangular diprism | 23 | xyzw | 1111 | (1~0)x1x1 | ETEE |
Triangular prismidal pyramid | 24 | xyzw | 1111 | ((1~0) x1)~0 | ETET |
Triangular diprismidal torus | 25 | (xyzw) | (1111) | 2#((1~0) x1) | ETEQ |
Tetrahedral prism | 26 | xyzw | 121 | (1~0~0) x1 | ETTE |
Pentachoron | 27 | xyzw | 13 | 1~0~0~0 | ETTT |
Tetrahedral torus | 28 | (xyzw) | (121) | 2#(1~0~0) | ETTQ |
Triangular toroidal prism | 29 | (xyz)w | (111)1 | 2#(1~0) x1 | ETQE |
Triangular toroidal pyramid | 30 | (xyz)w | (111)1 | (2#(1~0)) ~0 | ETQT |
Triangular ditorus | 31 | ((xyz)w) | ((111)1) | 2#(2#(1~0)) | ETQQ |
Unknown shape | 32 | xy(zw) | 112 | Unknown | Unknown |
Unknown shape | 33 | (xy(zw)) | (112) | Unknown | Unknown |
Cubinder | 34 | (xy)zw | 211 | 2x1x1 | ELEE |
Cylindrical pyramid | 35 | (xy)zw | 211 | (2x1)~0 | ELET |
Toracubinder | 36 | ((xy)zw) | (211) | 2#3 | ELEQ |
Coninder | 37 | (xy)zw | 211 | (2~0)x1 | ELTE |
Circular dipyramid | 38 | (xy)zw | 22 | 2~0~0 | ELTT |
Conindral torus | 39 | ((xy)zw) | (211) | 2#(2~0) | ELTQ |
Torinder | 40 | ((xy)z)w | (21)1 | (2#2)x1 | ELQE |
Toroidal pyramid | 41 | ((xy)z)w | (21)1 | (2#2)~0 | ELQT |
Tetratorus | 42 | (((xy)z)w) | ((21)1) | (2#2)#2 | ELQQ |
Duocylinder | 43 | (xy)(zw) | 22 | 2x2 | EL*EL |
Tiger | 44 | ((xy)(zw)) | (22) | (2x2)#2 | Unknown |