Tetration Forum

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I have the intuition that tetration requires the introduction of numbers with real dimension (non integer, like complex or quaternion sets).

A problem is that those numbers have to include hypercomplex as a subset, but, for example a set with the dimension of a serpinsky triangle, would be a subset of the complex, and it cannot be closed under addition/multiplication, without including all of the complex numbers, but then it would just be equal to the complex.

So, closure has to be abandoned, or the set has to be a set of sets with all the possible dimensions.

Also, sometimes tetration leads to step functions or dirac delta functions. A diract delta is similar to a segment of line with unit length. It is one dimensional, meanwhile an hypercomplex number is a point, which is zero-dimensional.

It is analogous to surreal numbers, where a segment of line would be like a number infinitely larger than a point.

So maybe the serpinsky triangle should be taken as something in between a point and a surface, instead of a subset of the complex.

Instead of thinking of numbers as points, we should think of then as fractal sets.

An exponentiation to a real exponent, in general, has infinite solutions (because a root has infinite solutions in the complex field). Those infinite solutions are a set.

Probably we should think of tetration as operations between sets instead of between numbers. Probably fractal sets.

Possibly singularities in functions are just numbers with a different dimension. A pole in a complex function may be a line number with length equal to the pole's residue.

Possibly Taylor series require numbers with a different dimension outside of his radius of convergence, to be able to calculate pass singularities.
For the " tommy octonion " we have the following Jacobi matrix.
By that I mean the Jacobi matrix for taking f(X) = X^2 in that number system.

see pictures.

The inverse of that matrix corresponds to taking the square root whenever the matrix is invertible (the determinant of the original Jacobi is not zero).
( see : https://en.wikipedia.org/wiki/Inverse_function_theorem )


Tom Marcel Raes
Off topic but I like that you use a day planner/agenda to write your math in, lol.
My friend posted the question at Mathoverflow :


So it is more formal now.

Some of you are probably on MO.


another idea are my " xyz numbers ".

They are also a commutative 4d type of number.

x*x = y*y = -1

x*y = y*x = 1.

x*z= z*x = y

y*z = z*y = - x

z*z = - 1 + x + y

notice many properties are not present ; not associative , nilpotent , no unique inverses etc

For instance 

(x+y)^2 = 0

but x+y is not 0.

-x*x = y*x = 1.

but -x is not y.

still investigating.

what do you think ?


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