Are quaternions useful as coordinate systems for two-dimensional manifolds?

Question

I am learning how quaternions work. For that reason I have prepared a coordinate system based on quaternions applied to two-dimensional topological manifolds of the shape of a Klein bottle (it can be adapted easily to others like a Möbius strip, or a torus for instance).

Basically we can represent the Klein bottle by its gluing diagram:

And we will add extra information to the points that "inhabit" the surface of the manifold. A point of the surface will be represented by a quaternion as follows:

$$a+bi+cj+dk$$

Where:

1. $a$ represents the side. Any point of the surface of the bottle has an equivalent point in the internal side, so $a = 0,1$ ($0$ outer side, $1$ inner side). A point can be in some of the both inner or outer sides.

2. $i$ represents the horizontal position $x$ in the gluing diagram ($i \in [0,I]$ where $I$ is the length of the Klein bottle diagram)

3. $j$ represents the vertical position $y$ in the gluing diagram ($i \in [0,J]$ where $J$ is the height of the Klein bottle diagram)

4. Finally, $k$ will be an extra property of the point, in my case I named it the "spin", it has $4$ positions, ($k \in [0,3]$). I could be seen as a kind of local orientation associated to the gluing diagram (North,South,East,West).

This provides a coordinate system with sum and multiplication. As we are using quaternions, the standard rules of quaternions are applied:

$$i^2=j^2=k^2=ijk=-1$$

Addition, scalar multiplication and quaternion multiplication are available and the quaternion multiplication rotation laws hold:

$$ij=k, ji=-k, jk=i, kj=-i, ki=j, ik=-j$$

The Hamilton product is defined as the standard one:

$$(a_1+b_1i+c_1j+d_1k)\cdot(a_2+b_2i+c_2j+d_2k) = (a1a2-b1b2-c1c2-d1d2)+(a1b2+b1a2+c1d2-d1c2)i+(a1c2-b1d2+c1a2+d1b2)j+(a1d2+b1c2-c1b2+d1a2)k$$

To visualize how the points move along the Klein bottle, let us define a collision game. We put randomly $7000$ points in a Klein Bottle, visualized as seen in the gluing diagram. The yellow points are in the outer side and blue points in the inner side. The points are represented depending on the spin ($k$) as little points, circles, stars and "plus" (+) symbols ($k=0,1,2,3$ respectively). They are moving following random walks.

Collision rules:

1. When two points $P_1,P_2$ of located in the same side and with the same spin collide (they have exactly the same values $a,i,j,k$), they disappear and generate a new point $P_1+P_2$ (quaternion sum).

2. When two points $P_1,P_2$ of the different side are in the same position and spin collide (they have exactly the same values $i,j,k$), they disappear and generate a new point $P_1 \cdot P_2$ (quaternion Hamilton product).

This is how it looks like when the width and height of the diagram is $100$ units and we put $7000$ points on the surface of the manifold:

And this is the animation of the evolution of the first $500$ steps of a test:

(*) I cannot add a bigger version of the animation due to the weight of the image, a bigger version can be seen here. The Python code is also available, please use it and modify it freely.

Due to the properties of the Klein bottle:

1. When a point goes out from the right side, is going to the other side of the Klein bottle, so it will appear from the left side of the diagram with the opposite side color and the $j$ position is inverted.

2. When a point goes out from the left side, is going to the other side of the Klein bottle, so it will appear from the right side of the diagram with the opposite side color and the $j$ position is inverted.

3. When a point goes out from the top or bottom, it keeps in the same current side (inner or outer), so it will appear at the opposite region of the current side (Top $\to$ Bottom, or Bottom $\to$ Top).

As time evolves, collisions happen and the number of points decreases.

I would like to ask the following questions:

1. Are the properties of quaternions good for this kind of coordinate systems?

2. Are there documented references of this kind of application? (quaternions applied as coordinate systems to topological manifolds). Any insights are very appreciated.

Answer 1

This coordinate system does not seem natural at all.

The first issue is that you are using $4$ continuous dimensions to represent two continuous and two discrete dimensions if I understood correctly.

Second, neither addition nor multiplication are well defined on this set. You can define addition modulo some lattice, but multiplication will not be well defined modulo such a lattice.

For instance, if you take the coefficient of $i$ mod $7$ and the coefficient of $j$ mod $5$, then when you multiply $bi$ with $cj$, you get either $bck$ or $-bck$. But it doesn't make sense to multiply a mod $7$ number with a mod $5$ number (and order matters, and you don't get something mod $3$ which is what it seems you wanted).