Rotope (EntityClass, 3)
From Hi.gher. Space
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 | Group Notation | Digit Notation | Product Notation | CSG Notation |
1D Rotopes | ||||
Line | x | 1 | 1 | E |
2D Rotopes | ||||
Square | xy | 11 | 1x1 | EE |
Circle | (xy) | 2 | 2 | EL |
Triangle | xy | 11 | 1~0 | ET |
3D Rotopes | ||||
Cube | xyz | 111 | 1x1x1 | EEE |
Sphere | (xyz) | 3 | 3 | ELL |
Square pyramid | xyz | 111 | (1x1)~0 | EET |
Cylinder | (xy)z | 21 | 2x1 | ELE |
Torus | ((xy)z) | (21) | 2#2 | ELQ |
Cone | (xy)z | 21 | 2~0 | ELT |
Triangular prism | xyz | 111 | (1~0)x1 | ETE |
Triangular torus | (xyz) | (111) | 2#(1~0) | ETQ |
Tetrahedron | xyz | 12 | 1~0~0 | ETT |
4D Rotopes | ||||
Tesseract | xyzw | 1111 | 1x1x1x1 | EEEE |
Glome | (xyzw) | 4 | 4 | ELLL |
Cubic pyramid | xyzw | 1111 | (1x1x1)~0 | EEET |
Spherinder | (xyz)w | 31 | 3x1 | ELLE |
Toraspherinder | ((xyz)w) | (31) | 3#2 | ELLQ |
Sphone | (xyz)w | 31 | 3~0 | ELLT |
Square pyramid prism | xyzw | 1111 | ((1x1)~0)x1 | EETE |
Square pyramid torus | (xyzw) | (1111) | ((1x1)~0)#2 | EETQ |
Square dipyramid | xyzw | 112 | (1x1)~0~0 | EETT |
Cubinder | (xy)zw | 211 | 2x1x1 | ELEE |
Toracubinder | ((xy)zw) | (211) | 2#3 | ELEQ |
Cylindrical pyramid | (xy)zw | 211 | (2x1)~0 | ELET |
Torinder | ((xy)z)w | (21)1 | (2#2)x1 | ELQE |
Tetratorus | (((xy)z)w) | ((21)1) | (2#2)#2 | ELQQ |
Toroidal pyramid | ((xy)z)w | (21)1 | (2#2)~0 | ELQT |
Coninder | (xy)zw | 211 | (2~0)x1 | ELTE |
Conindral torus | ((xy)zw) | (211) | 2#(2~0) | ELTQ |
Circular dipyramid | (xy)zw | 22 | 2~0~0 | ELTT |
Duocylinder | (xy)(zw) | 22 | 2x2 | EL*EL |
Tiger | ((xy)(zw)) | (22) | (2x2)#2 | Unknown |
Triangular diprism | xyzw | 1111 | (1~0)x1x1 | ETEE |
Triangular diprismidal torus | (xyzw) | (1111) | 2#((1~0) x1) | ETEQ |
Triangular prismidal pyramid | xyzw | 1111 | ((1~0) x1)~0 | ETET |
Triangular toroidal prism | (xyz)w | (111)1 | 2#(1~0) x1 | ETQE |
Triangular ditorus | ((xyz)w) | ((111)1) | 2#(2#(1~0)) | ETQQ |
Triangular toroidal pyramid | (xyz)w | (111)1 | (2#(1~0)) ~0 | ETQT |
Tetrahedral prism | xyzw | 121 | (1~0~0) x1 | ETTE |
Tetrahedral torus | (xyzw) | (121) | 2#(1~0~0) | ETTQ |
Pentachoron | xyzw | 13 | 1~0~0~0 | ETTT |