The cusphere

Abstract

A cusphere, portmanteau of cube and sphere, is the constant magnitude surface in imaginary scator algebra. The computer renderings exhibit a fascinating geometry with great aesthetic value. The scator space in 1 + 2 dimensions, is endowed with a scalar (real) and two hyperimaginary components that can be represented in the three orthogonal axes of Euclidean space. A myriad of plane curves are obtained on the cusphere surface: Families of ellipses, circles and lemniscatae are three of the familiar ones. There are also less conventional ones, like four pointed stars and squircles. Implicit as well as parametric equations of these curves are derived. The three dimensional geometrical object is explored from different perspectives.

GRAPHICAL ABSTRACT

Keywords:

• Scator algebra
• Isometric surfaces
• Hypercomplex magnitude
• 3D curves
• Metric sculpture

Acknowledgments

The author is greatly indebted to Labná Fernández-Eraña for her excellent suggestions and revision of several parts of the manuscript; and Alejandra Leonides-Zamora for penetrating comments from a designer's point of view. The editor and reviewers comments were also helpful to improve the quality of this work.

Nomenclature/Notation

Scator quantities are represented with an overhead oval, $\stackrel{o}{\phi }$.

Unit directors are represented with a check and the appropriate subindex ${\check{\mathbf{e}}}_x,{\check{\mathbf{e}}}_y$. The check instead of a hat (${\hat{\mathbf{e}}}_j{\hat{\mathbf{e}}}_j=1$), stresses the negative square relationships, ${\check{\mathbf{e}}}_x{\check{\mathbf{e}}}_x=-1$ and ${\check{\mathbf{e}}}_y{\check{\mathbf{e}}}_y=-1$.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 https://www.johndcook.com/blog/2022/10/27/variant-squircle/.

2 https://en.wikipedia.org/wiki/Squircle.

3 The name ‘cusfera’ has been used in the spanish translation (Marmolejo, 2017).

4 The notion of space here described may involve time. In physics the compound word space-time is often used, labelling time explicitly in addition to three dimensional space.

5 This file is included as supplementary material. It can also be provided on request.