There is a question that bugs me for weeks now : **what type of mathematical object is it ?**

I have a quite decent knowledge of the usual types : integers, reals, complex numbers, and even some Galois fields over Z², for example. Quaternions, vectors and matrices are other types that are well defined. But the object under consideration puzzles me. Is it a ring, a field, a group, or something else ?

Let's start with the obligatory Wikipedia quote:

In mathematics, Formally, a |

So here is the new riddle : **does addition even exist in this object ?** The answer is easy for the previously cited types yet here, the elements are not numbers, but coordinates in a 2.5D closed space. Whether the carry counts as one individual dimension is still being debated. The description does not fit directly with a vector system. The behaviour is a bit reminiscent with a very weird Galois field but here, the "generator" generates only one half of a symmetrical set of coordinates (which I call orbits here). Really I know nothing of this kind.

**So what can we say so far about this family of objects ?**

- It is a family of sets defined by a single parameter (called here w, or width)
- Each set is organised as a vector space with coordinates X=[0..2
^{w}-1], Y=[0..2^{w}-1] and C=[0..1], hence cardinality of 2^{w+1}elements. - Each element has a predecessor and successor, defined by a reversible formula (with forward and backwards directions, already defined in other logs).
- Elements (0,0,0) and (2
^{w}-1,2^{w}-1,1) are their own precursor and successor. - Except for w=1, the space shows a symmetry and duality (invertibility?), where all elements (x,y,c) have an inverse value (2
^{w}-(x+1), .2^{w}-(y+1), 1-c), with the same behaviour. - Since each point has a successor and a precursor, and the number of points is finite, the series of successors must be finite too, leading the series back to its first element, and this creates an "orbit".
- Some values of w create a system with only two orbits (as calculated in this log) and these are the ones I'm seeking:
2, 3, 4, 10, 16

Other numbers may exist above 25 but have not been discovered so far.

.

Another interesting detail is the way the successor is calculated: Y gets the value of X. So in a way, when we map the coordinates on a 2D plane, each step represents a sort of **rotation** on this plane. This system makes me think of the logistic map **r×x×(1-x)**, such as:

Of course it is not directly related, if only because here the values are integer, modulo 2^{w} and every overflow causes an increment of x. The other detail is that there is no parabola, no multiply/squaring and the guiding function is Fibonacci's series. Is there a link I'm missing yet?

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