Re: 4D Klein bottle with 4-complex number

On Jun 6, 2024, Kuan Peng wrote
(in article <9mYgdlsPlETq54-_M1lMeZonlg4@jntp>):

« 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.
>
***********
>
See « N-complex number, N-dimensional polar coordinate and 4D Klein
bottle with 4-complex number» for more detail.
https://www.academia.edu/120524016/N_complex_number_N_dimensional_polar_coordi
nate_and_4D_Klein_bottle_with_4_complex_number
>
https://pengkuanonmaths.blogspot.com/2024/06/n-complex-number-n-dimensional-po
lar.html
>
Kuan Peng

Are the 4-complex numbers related to quaternions?

And is there any way to apply them to a spacetime in
which time is complex, perhaps with a small imaginary
part that is more important near the Big Bang, if any?

--
David Dalton dalton@nfld.com https://www.nfld.com/~dalton (home page)
https://www.nfld.com/~dalton/dtales.html Salmon on the Thorns (mystic page)
“And the cart is on a wheel; And the wheel is on a hill; And the
hill is shifting sand; And inside these laws we stand" (Ferron)