Rotope (EntityClass, 3)

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Revision as of 21:38, 16 June 2007 by Keiji (Talk | contribs)

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

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