Draft:Rotatope

• Comment: Referencing issue not fixed. Nobody (talk) 06:14, 14 April 2025 (UTC)

• Comment: Wikis are not reliable sources —pythoncoder (talk | contribs) 23:34, 13 April 2025 (UTC)

---

A cylinder is a 3-dimensional 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 $\mathbb {R} ^{3}$ 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 $(||||)$.

See also

• Duocylinder
• Spherinder
• Sphere
• Cylinder
• Hypersphere
• Circle

References

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
• "Introduction to the Fourth Dimension". 23 December 2003. Retrieved 13 April 2025.

External links

• Rotachora (4-dimensional objects with circular surfaces)
• Classification of rotatopes

This page will be placed in the following categories if it is moved to the article namespace.

Categories:

• Dimension
• Polytope