Rotope (EntityClass, 3)

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=== Rotatopes ===
=== Rotatopes ===
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A [[Rotatope]], invented by [[Garrett Jones]] is an object formed by [[extrude|linear extensions]] or [[lathe|rotations]] about the origin.
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A rotatope, invented by [[Garrett Jones]] is an object formed by [[extrusion|linear extensions]] or [[lathing|rotations]] about the origin.
=== Toratopes ===
=== Toratopes ===
Toratopes were coined by [[Paul Wright]], and invented by him and [[Marek14]].
Toratopes were coined by [[Paul Wright]], and invented by him and [[Marek14]].
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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.
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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 ===
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[[Category:Shapes]]
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{{Shapes}}
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Revision as of 20:14, 15 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 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

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