Rotatope

In elementary geometry, a rotatope is a geometric object which is the Cartesian product of a set of hyperballs. The facets are either all flat, curved, or a combination of both. Rotatopes may exist in any general number of dimensions $n$ , as an $n$ -dimensional rotatope or $n$ -rotatope. For example, a two-dimensional circle is a 2-rotatope, and a three-dimensional sphere is a 3-rotatope. . A curved facet is specified by equations of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{k} - a_{k})^{2} = r^{2}$ , whereas two parallel flat facets are specified by equations of the form $(y - b)^{2} = r^{2}$ .

The concept of a rotatope was invented by Jonathan Bowers and the name was coined by Garret Jones in 2003.

Characteristics

A rotatope can be constructed from the Cartesian product of a set of hyperballs. It can also be defined as the set of all shapes that exist in dimension $n$ that lie in the $n$ -dimensional space between regions that are specified by $k$ equations of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{p} - a_{p})^{2} = r^{2}$ where $\sum_{i = 1}^{k} p_{i} = n$ , $1 \leq k \leq n$ , and $1 \leq p \leq n$ . For example, in 3-dimensions a cylinder would be considered a rotatope, since the points on the edges lie in the regions between subsets of $ℝ^{3}$ that are specified by the equations $x^{2} + y^{2} = r^{2}$ and $z^{2} = d^{2}$ , whereas a torus would not since the equations that specify a torus are not of that form. In all dimensions less than $n = 3$ there are $n$ rotatopes, whereas in higher dimensions the number of rotatopes is the partition function of $n$ .

Classes of Rotatopes

Rotatopes can be put into several different classes based on the equations that they are specified by. A regular classification, which ascribes $n$ rotatopes to a $n$ dimensional space, can be defined as one in which there are $n - k$ equations of the form $(y - b)^{2} = h^{2}$ and one equation of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{k} - a_{k})^{2} = r^{2}$ specifying the facets where $1 \leq k \leq n$ . In a 3-dimensional space and all lower dimensions all rotatopes are regular, whereas in higher dimensions there are non-regular rotatopes. The first example of a non-regular rotatope would be the duocylinder whereas a spherinder would be considered regular by the definition listed above.

Another class of rotatopes is the composite rotatope classification. A composite rotatope exists in dimension $d$ where $d = n * k$ . There are $k$ equations of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{n} - a_{n})^{2} = r^{2}$ specifying the facets of these rotatopes, where $n$ and $k$ take on the values of all possible factors of dimension $d$ , meaning that $d = n * k$ . The composite rotatopes have composite dimensionality (dimension it exists in is a composite number). The duocylinder is the first example of a composite rotatope. The exception to this rule is the line segment since the dimension it lives in is not a composite number.

For Power-2 rotatopes, the facets are specified by $n$ equations of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{n} - a_{n})^{2} = r^{2}$ where $n^{2} = d$ and $d$ is the dimensionality of the rotatope. The first example of a Power-2 rotatope is the line segment and the second is the duocylinder.

More generally, Power-N rotatopes are specified by $\frac{d}{k}$ equations of the form $(x_{1} - a_{1})^{2} + (x_{2} - a_{2})^{2} + (x_{3} - a_{3})^{2} + . . . + (x_{k} - a_{k})^{2} = r^{2}$ where $k^{n} = d$ and $d$ is the dimensionality of the rotatope.

Toratopic Notation

Toratopic notation can be used to classify toratopes. One vertical line represented by $|$ represents the a digon whereas parentheses $(|_{1} |_{2} |_{3} . . . |_{n})$ represent a spheration that lives in dimension $n$ . For example, a circle would be represented by the symbol $(| |)$ whereas a hypersphere would be represented by the symbol $(| | | |)$ .

References

Citations

Bibliography

The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms and duocylinders (double cylinders)
The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces, Chris McMullen, 2008, ISBN 978-1438298924 Search this book on .
"Introduction to the Fourth Dimension". 23 December 2003. Retrieved 13 April 2025.

External links

This article "Rotatope" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Rotatope. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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Rotatope

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