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This page tabulates all of the rotatopes from the first dimension to the fourth dimension. There are several methods that these shapes can be classified with. I show three of them here - the first by me (Garrett Jones), the second by Jonathan Bowers, and the third by Patrick Stein. The following is a list of the shapes that are classified below:
Linespace (first dimension): line
Planespace (second dimension): square, circle
Realmspace (third dimension): cube, cylinder, sphere
Tetraspace (fourth dimension): tetracube (tesseract or hypercube), cubinder, duocylinder, spherinder, glome (hypersphere)
The basic rotatopes of any dimension can be constructed from lower dimensional rotatopes by the use of rotations and extensions. All rotations are centered around axes that lie in the original dimension, so that the resulting shape is always a rotation into a higher dimension as opposed to a rotation within the original dimension. In the same way, all extensions are always into the next dimension higher, in a direction perpendicular to the directions taken so far to construct the object.
As an example, the circle and square from planespace can both be constructed from a line that originates in linespace. To construct a square, the line is extended in a direction perpendicular to itself. The space that the line sweeps through is a square. To construct a circle, this same line could be instead rotated around its central point. The space that this line sweeps through is a circle. These constructive methods generalize to higher dimensions. All possible rotatopes can thus be constructed in this manner.
In the tables below, the top row is the shape which is to be extended or rotated, and the left column is the operation to perform on it. The shapes are color coded in order to see their natural groupings easier. For rotations and extensions, the letter in parentheses denotes the axis that the object is either rotated around or extended along. For the shapes, the letter in parentheses denotes the orientation of the shape's symmetry axis.
rotate(v) = rotate around the v axis
extend(v) = extend along the v axis
cylinder(v) = cylinder with linear axis v
spherinder(v) = spherinder with linear axis v
cubinder(vv) = cubinder
with planar axis vv
duocylinder(vv:vv) = duocylinder with planar axises
vv and vv
line |
line | |
rotate | circle |
extend(y) | square |
circle | square | |
rotate(x) | sphere | cylinder(x) |
rotate(y) | sphere | cylinder(y) |
extend(z) | cylinder(z) | cube |
sphere | cylinder(x) | cylinder(y) | cylinder(z) | cube | |
rotate(xy) | glome | spherinder(x) | spherinder(y) | duocylinder(zw:xy) | cubinder(xy) |
rotate(xz) | glome | spherinder(x) | duocylinder(yw:xz) | spherinder(z) | cubinder(xz) |
rotate(yz) | glome | duocylinder(xw:yz) | spherinder(y) | spherinder(z) | cubinder(yz) |
extend(w) | spherinder(w) | cubinder(xw) | cubinder(yw) | cubinder(zw) | tetracube |
This method uses a single number to represent an n-dimensional sphere, and combinations of the numbers to represent the cross-products of those shapes. Thus, '1' is a line, '2' is a circle, and '3' is a sphere. A shape with only 1's in its sequence of digits is an n-dimensional hypercube. Thus, '1' is a line, '11' is a square, and '111' is a cube. The sum of the numbers in a shape is the dimensionality of the shape.
The digits in any particular sequence are interchangeable, so a sequence of digits refers to the same shape when any of its digits are transposed. For example, '21' and '12' both represent a cylinder. Another result of this invariance under transposition is that the same shape can result from cross products of different shapes. Thus, the sequences '1212' and '1122' both represent the same shape, which can be constructed from any of the following cross products - the cross product of two cylinders, the cross product of a square and a duocylinder, or the cross product of a cubinder and a circle, or any other combinations of its subshapes. If a sequence contains a digit greater than 1, it is capable of rolling. If all of a sequence's digits are greater than one, then it will always roll when placed on a surface. If it only has 1's then it only has flat sides and is incapable of rolling. All of these shapes can be refered to by their signatures.
Linespace | 1 = line segment | ||||
Planespace | 11 = square | 2 = circle | |||
Realmspace | 111 = cube | 12 = cylinder | 3 = sphere | ||
Tetraspace | 1111 = tesseract | 112 = cubinder | 22 = duocylinder | 13 = spherinder | 4 = glome |
This method uses a pair of numbers (k,n) to classify each of the rotatopes. The number n denotes the total number of dimensions of the object, and the number k determines how many dimensions of the object are "round". The set of round dimensions of an object are constrained by the formula
x_{1}^{2} + x_{2}^{2} + ... + x_{k}^{2} ≤ 1
The round dimensions are all constrained together within a single formula. The square of each of the terms all added together must be less than one. The remaining dimensions are restricted individually, each with a constraint formula of its own:
| x_{k+1} | ≤ 1
| x_{k+2} | ≤ 1
...
| x_{n} | ≤ 1
With the terms above, each value is individually restricted to being less than one, as opposed to all of them together. Each of these terms corresponds to a "flat" dimension of the shape. Since a^{2} ≤ 1 whenever | a | ≤ 1 , you could write the above like (x_{n})^{2} ≤ 1 instead of with the absolute value signs. They would then look like this:
(x_{k+1})^{2} ≤ 1
(x_{k+2})^{2} ≤ 1
...
(x_{n})^{2} ≤ 1
Using this scheme, (0,4) and (1,4) are both the tetracube. It's an unfortunate oddity in the naming scheme which goes away if you assert that k > 0. To show an example of this system, a spherinder is denoted (3,4) and has the following constraints:
x^{2} + y^{2} + z^{2} ≤ 1 and
w^{2} ≤ 1
It is possible for a shape to have more than one set of round constraints. There is only one such shape in the fourth dimension, which is the duocylinder. It is denoted (2,2,4) and has the following constraints:
x^{2} + y^{2} ≤ 1 and
z^{2} + w^{2} ≤ 1
This shows that the two round aspects of the duocylinder are independent. Below are all the shapes tabulated.
(k,n) | constraints | |
line segment | (1,1) | x^{2} ≤ 1 |
(k,n) | constraints | |
square | (1,2) |
x^{2} ≤ 1 y^{2} ≤ 1 |
circle | (2,2) | x^{2} + y^{2} ≤ 1 |
(k,n) | constraints | |
cube | (1,3) |
x^{2} ≤ 1 y^{2} ≤ 1 z^{2} ≤ 1 |
cylinder | (2,3) |
x^{2} + y^{2} ≤ 1 z^{2} ≤ 1 |
sphere | (3,3) | x^{2} + y^{2} + z^{2} ≤ 1 |
(k,n) | constraints | |
tetracube | (1,4) |
x^{2} ≤ 1 y^{2} ≤ 1 z^{2} ≤ 1 w^{2} ≤ 1 |
cubinder | (2,4) |
x^{2} + y^{2} ≤ 1 z^{2} ≤ 1 w^{2} ≤ 1 |
duocylinder | (2,2,4) |
x^{2} + y^{2} ≤ 1 z^{2} + w^{2} ≤ 1 |
spherinder | (3,4) |
x^{2} + y^{2} + z^{2} ≤ 1 w^{2} ≤ 1 |
glome | (4,4) | x^{2} + y^{2} + z^{2} + w^{2} ≤ 1 |