Infinite tetration and superroot of infinitesimal

[andydude]

Where can I read about existing geometrical interpretations of tetration,

You can't, because I don't think anyone is seriously researching this aspect. The only two places that come close are a link that Gottfried posted, and Tony Smith's "work" (author of Banned by Cornell) which gives an interesting discussion of dimensions.

both finite and infinite, and its opposite log(log(log(log(log(.....) both finite and infinite?

What?

If You involve all infinite values of log, this construction expands pretty fast into infinity^infinity trees of values.

What?

You mentioned rotation via new dimensions- how does it happen?

Well, mathematically we are not limited to 3 dimensions. There are about as many dimensions as you have "commas", meaning, if you can write a "comma", then you can make a new dimension... So for example, (1, 2) is a point 1 unit along the first dimension, then two units along the second dimension. There are many ways of looking at lists of numbers, most are part of what most mathematicians call "vector spaces" all vector spaces have a dimensionality to them, so for example, although (1, 2) and (1, 2, 0) can represent the same polynomial/series (infinite dimensional space), they are two completely different vectors because they come from different vector spaces. When you are interpreting a vector as an hypercube as opposed to a point, then you can consider the hyper-volume (length, area, volume, etc...) of a vector (also called a norm or a measure) as the volume enclosed by the vector's scaled basis vectors in each dimension. This is the sense in which I was talking about "rotating through dimensions" a rotation in this sense is a map from (..., 1, 0, ...) to (..., 0, 1, ...) all else being equal. The hypervolume of two different lengths in two different dimensions (a, b) is a*b, or in other words: Multiplication. The hypervolume of one length rotated through n dimensions (a, a, a, ..., a) with n a's is a^n, or in other words: Exponentiation. So if this is the pattern for exponentiation, and tetration's definition in terms of exponentiation is: \( {}^{n}{x} = x^{\left({}^{(n-1)}x\right)} \) then all we have to do is substitute this formula into our interpretation of exponentiation. This lexical substitution gives: \( {}^{n}x \) is the hypervolume of a length x rotated through \( {}^{(n-1)}x \) dimensions. Although this is not directly interpretable, it is recursive. This is as far as I've gotten in a semi-physical interpretation of tetration. I think trying to understand that statement intuitively is doomed, as recursive interpretation may lead to madness...

Andrew Robbins