multi-dimensional numbers Questions:
I am trying to derive a multi-dimensional number system.
I know it is not the traditional way of doing things; my questions are:


*

*Is this valid? If not where are my mistakes?

*Has someone else done any thing similar? (The closest thing I am
aware of is Silviu Olariu, Complex Numbers in n Dimensions,
available at Google books)
Informal description
Instead of looking at i as a square root of minus one, we view multiplication by i as a unary function that takes 4 applications before returning to the initial argument, or, if you like, we view i as the solution of $i^4-1=0$ rather than $i^2+1 = 0$. 
We do not allow multiplication of rotational numbers with each other.
We will generalise this to define one rotational constant $r_n$ per loop order n, such that $(r_n)^n=1$. Multiplying any non-zero number by $r_n$ any number of times creates a set of n distinct numbers.
This system is conceptually similar to cyclotomic fields, expect that here the unity roots are separate dimensions rather than complex numbers.
Formal definition
Open functions
An open function is conceptually similar a unary function, except that functions are defined as a mapping with a domain and a range. We use open functions to define new sets of numbers, and the domain is the set S of number objects already defined. This means we are able to apply any of our unary open functions to any number object in S. The result will either be a number object in S or not in S. In the latter case we can use this new number object to extend S.
We denote repeated applications of an open function to an object with a superscript. For open function f and all objects p:
$ 
\begin{array}{l}
f^{0}(p)=p \\
f^{n+1}(p)=f(f^{n}(p)) \\
\end{array}
$
We will define some number sets using one single initial number object 0 and two types of open functions, loops and chains.
Axiom 1: Zero
There is a number 0.
Loops and chains:
We define 2 types of open functions, chains and loops, which conform to certain rules:
Chain:


*

*keeps generating new objects 

*is commutative


Loop of order n:


*

*forms an n element cycle with any number object other than 0.

*two objects define the loop between them

*one loop per order

*order 2 special case relating inverse chains


Definition Chains:
An open function inc is called a chain if for any object p we have $inc^n (p) \ne inc^m (p)$ for all natural numbers $n \ne m$ and for all objects p.
Definition Loops:
An open function $rot_n$ is called a loop of order n (with respect to 0) if $rot_n(0)=0$ and for all objects $p \ne 0$ we have $rot_n^m(p) = p$ if and only if n divides m.
Chains and loops are both bijective. The inverse open function of loop or chain $f(p)$ is $f^*(p)$ such that
$f(f^*(p)) = f^*(f(p)) = p$  for all p.
Axiom 2: Two non-zero objects define the loop between them
For any number object $p \ne 0$ if $rot_n(p) = rot_m(p)$ then n=m.
Axiom 3: only one loop per order
For any $p \ne 0$ and n>0 there is only one unique set of n number objects generated by $rot_n^m(p)$.
What this axiom states is best illustrated with an example. Here we have five objects a,b,c,d and e that are related with order 5 loops.
http://thewaytheworldworksdotcom.files.wordpress.com/2013/07/axiom3.png
There are 4 different order 5 loops, shown as solid and dotted, red and blue arrows. They are interchangeable in the sense that there isn’t anything special about one particular one. If we pick one randomly to be called $rot_5$, then the other three are $(rot_5)^2$, $(rot_5)^3$ and $(rot_5)^4$.
Axiom 3 states that there are no other order 5 loops containing either a,b,c,d or e.
Axiom 4: $rot_2$ is a special loop open function that relates any inc chain to its inverse.
Let $x=inc_n^m(0)$ and let $y=inc*_n^m(0)$ for any positive integer n and m. Then x and y are related by the $rot_2$ loop:
$ \begin{array}{l}
x=rot_{2} (y) \\
y=rot_{2} (x) \\
\end{array} $
Axiom 5: Relating loops and chains
For all n>0 there exist n distinct chains $inc_0, inc_1,…,inc_{n-1}$ and a loop $rot_n$ of order n such that $rot_y(inc_0^x(0))=inc_y^x(0)$ for all x>0 and all 0≤y≤n-1.
We have now defined the axes of our number system. All the inc chains are only connected at 0 but we can still get new number objects if we apply an inc chain to objects other than 0 on another inc chain.
For n >0 we define $M_n$ to be set of all objects that can be obtained from 0 by applying the composition of a finite sequence of the open functions $inc_0, inc_1,…,inc_{n-1}$.
The set $M_4$ is thus the set of gaussian integers.
We can extend this to complex numbers, but I want to stop here before it gets too long.
Comments:
Rotational numbers are not an advanced concept at the cutting edge of mathematical research, but a basic number set that, in my view, was passed over in the relentless drive towards abstraction. In fact it is telling that the professional mathematicians I discussed this with often seemed to prefer to derive this as a special case of a higher level concept, like a module over a ring, rather than accept a derivation from first principles, which I find more elegant.
This is nothing new, just a different perspective. All the rules for rotational numbers have already been explored for cylcotomic fields. I prefer to have complex numbers defined as a special order 4 case rather than the traditional way that defines all unity roots as complex numbers, in order to maintain general multiplication.
The rotational numbers do not include the quaternions, although the motivation for defining them is similar. Axiom 3 ensures only one loop per order. A quaternion contains 4 constants; one is 1 and the other three are all loop-4 numbers. Furthermore they are not commutative. Ditto about Clifford Algebras. 
More about open functions:
Open function is short for "open domain and open range function". Traditional functions have a well defined domain. You can say for example that function $f(x)=1/x$ is not valid for $x=0$.  An open function can take any object as an argument. The range can be used to define a set using one starting element. Example: unary open function inc is a loop. We start with element s which we put into set S. S={s}. inc(s) produces a new value, so we need to extend S. S=[s, inc(s)}. We can carry this on indefinitely so we get S={s, inc(s), inc(inc(s)), ...}.
Examples for inc would be +1, or +i, or +1+i. The generated sets are isomorphic to the positive integers.
Loops are different. Starting with the order 3 loop function $rot_3$ and any non-zero starting value s we get only three elements in our domain set. S={s, rot(s), rot(rot(s))}.
Example Set:
We can use the Gaussian integers which is set $M_4$ as an example set to illustrate the definition.
link thewaytheworldworksdotcom.files.wordpress.com/2013/07/grid.png
The grey objects are on the axes. The open functions shown as light and dark blue arrows can be viewed as the functions +1 and +i.
Multiplication:
A set $M_n$ can represent n-dimensional numbers if n is odd, and n/2 dimensional numbers if n is even.
These numbers can be added and loop functions can be applied to them, but there is no general multiplication function. The only set where multi-dimensional number multiplication works with the usual associative and distributive laws are the complex numbers, as stated by the fundamental theorem of algebra.
Example: Let $a + bj + cj^2$ be a 3-dimensional number element of $M_3$. $j=r_3$ in the same way that $i=r_4$. That means that $1$, $j$ and $j^2$ are all distinct dimensions and different from each other.
$((1+j+j^2) * -1) + (1+j+j^2) * j = (1+j+j^2) - (1+j+j^2) = 0$
If there was general multiplication for numbers in $M_3$ then we could factorise:
$(1+j+j^2)(j-1) = 0$
This means either $1+j+j^2=0$ which is not correct, or $j=1$ which is also not correct.
Not having a multiplication function also means that the dimensions of a particular order are all interchangeable. There is no special "1" direction.
Link:
Thank for you very much for your help and for reading this long question.
In case anyone is interested, a longer derivation with more diagrams is available here:
http://thewaytheworldworks.com/formal-derivation/
 A: For me, since I am new on this website, this post is a funny read!
In the first place congratulations to the author of this post and in the second place let's look at the end where it is remarked that
$(1 + j + j^2) \cdot (1 - j) = 0$
This simply says that $1 + j + j^2$ and $1 - j$ are a pair of divisors of zero.
Without going into details: Only the complex plane $\mathbb{C}$ does not contain divisors of zero (ok except for the quaternions but the quaternions make lousy complex analysis). 
But let me congratulate you once more: you have found a treasure trove because you found the number $\alpha$! Don't know what the number $\alpha$ is?
Let's define:
$\alpha := \frac{1}{3}(1 + j + j^2)$ where $j^3 = 1 \in \mathbb{R}^3$ and it has the next basic properties:
$\alpha^2 = \alpha$ and $j \alpha = \alpha$
If you want boatloads and boatloads of math like this, I have a page for that:
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff.htm 

In short you have two number systems:
One with $j^3 = 1$ and
one with $j^3 = -1$.
These are strongly related but they both give good complex analysis:
Define $X \in \mathbb{R}^3$ via $X = x + yj + zj^2$ and all analytic functions on the real line $\mathbb{R}$ can be extended to 3 dimensions in $\mathbb{R}^3$.
Of course for any dimension you can craft such two systems of higher dimensional numbers. 

A further example of how stuff works in $\mathbb{R}^3$, from the complex plane most people know that if $z = x + iy$ the conjugate is given by $\overline{z} = x - iy$ and that $z \overline{z} = x^2 + y^2$.
How does this pan out in $\mathbb{R}^3$ with $j^3 = 1$?
For any $X \in \mathbb{R}^3$ and $X = x + jy + j^2z$ the conjugate is now given by $\overline{X} = x + jz + j^2y$ and we are going to calculate $X \cdot \overline{X}$.
Doing so we get:
$$
\begin{array}{rl}
X \overline{X} =  & (x + yj + zj^2)(x + zj + yj^2) = \\
=     & x^2 + y^2 + z^2 + j(yz + xz + xy) + j^2(yz + xz + xy)
\end{array}
$$ 
In the complex plane if you put $z \overline{z} = 1$ you have the equation of a circle, how does this compare here?
Let up put $X \overline{X} = 1$ we now get a system of 2 equations:
$$
\left\{
\begin{array}{l}
x^2 + y^2 + z^2 = 1 \\
yz + xz + xy = 0
\end{array}
\right.
$$
This is the sphere-cone equation, the sphere is of course $x^2 + y^2 + z^2 = 1$ while the cone is given by $yz + xz + xy = 0$.
This cone equations is a special case of the general cone equation, in our case it is the cone that contains the 3 coordinate axis.
Stuff like this is basic stuff for the higher dimensional complex number system (remark that using $j^3 = 1$ we have a system of circular numbers and their matrix representations are the circulant matrices).
Let's leave it with that.
