# N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number

“A concrete representation of a 4D Klein bottle has been desired by many but has never been presented. So, I decided to dive into the Klein bottle. Working with the Klein bottle was my first opportunity to practice with this system. To my surprise, the ease with which it allowed me to create 4D Klein bottles was remarkable. The 4D Klein bottles were generated smoothly without the slightest hitch. My video animations of the rotating 4D Klein bottle in 4D space, as well as the 3D slices ascending in the 4D space, were also computed effortlessly.”

Abstract: While a 3D complex number would be useful, it does not exist. Recently, I have constructed the N-complex number, which has demonstrated high efficiency in computations involving high-dimensional geometry. The N-complex number provides arithmetic operations and polar coordinates for N-dimensional spaces, akin to the classic complex number. In this paper, we will explain how these systems work and present studies on 4D Klein bottles and hyperspheres to illustrate the advantages of these systems.

The classic complex number system is a remarkable mathematical tool because it allows for the addition and rotation of vectors in two-dimensional space, following the same rules as real numbers for addition and multiplication. However, in three-dimensional space, it is impossible to manipulate vectors with similarly intuitive arithmetic operations because such a system does not currently exist. The development of a three-dimensional complex number system, analogous to the two-dimensional one, would represent a significant advancement in mathematics.

In 2022, I constructed a system of complex numbers for spaces with any number of dimensions, which I call the “N-complex number system.” Edgar Malinovsky used this system to create many beautiful 3D objects (see «Rendering of 3D Mandelbrot, Lambda and other sets using 3D complex number system»[4]). Figure 1 shows the 3D Mandelbrot set he created. Computing 3D fractal objects is very time-consuming; he would not have succeeded in this work without the 3-complex number system. His work demonstrates that the 3-complex number system significantly accelerates computations in 3D space.

I have worked on 4D Klein bottles by extending a 3D Klein bottle (see Figure 2) into 4D space. I rotated the 4D Klein bottles in 4D space and showcased the rotation in my video animation “Observing a 4D Klein Bottle in 4-Dimension” [5]. This work would have been impossible without the 4-complex number system. In addition to N-complex numbers, the new system provides a polar coordinate system for N-dimensional spaces, which was previously missing in mathematics.

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