09/12/2007, 06:23 AM

Perhaps I'm missing the point of the tau function (other than to move the fixed point to 0 for building a power series). Granted, in order to take advantage of the ability (in the immediate vicinity off the fixed point) to continuously iterate exponentiations by using continuously iterated multiplication (vanilla exponentiation), we need some transformation that moves large numbers (e.g., 1) down into the "well".

It seems like you are using tau to move points to the vicinity of the fixed point, then using the inverse of tau to move the fixed point to 0, iterating, moving the fixed point back to its proper location, then moving the transformed point back to its new location. If I'm reading this correctly, I don't think your tau function is the right way to move points towards the fixed point. Preferably, it should be possible with a single function to drive all points towards the fixed point. For base eta, this is done by taking logarithms (above the asymptote) or exponentials (below the asymptote) to move points near e, then performing continuous parabolic iteration, then taking the inverse of the iterated operation to get back to the proper point.

My first try with base e was to use iterated logarithms. Unfortunately, logarithms introduce an undesirable fractal pattern that distorts the spacing between points, because integer tetrates of the base (0, 1, b, b^2, etc.) will be connected out at infinity.

My current hypothesis, based on computing Andrew's slog for complex numbers within the radius of convergence, is that the "proper" way to transform large numbers into infinitesimal numbers (relative to the fixed point) is to use imaginary iterations of exponentiation. This allows us to move zero towards the fixed point, bypassing the singularity of the logarithm. In other words, if +1 iteration is exponentiation, and -1 iteration is a logarithm, then we need a formula (and a name!) for i iteration. Of course, currently the only way I have of generating such a formula is by using the first derivative of a tetration solution. But deriving a formula for performing imaginary iterations from tetration and then using it to derive a tetration solution is bad circular reasoning. At best it proves internal consistency.

I'm wondering, therefore, whether it's possible to derive the imaginary equivalent of something between an exponentiation and a logarithm, not quite either but related to both by being halfway between them and out to the side, so to speak.

If we could independently derive such a formula, and if we could be sure of its uniqueness, then we could use it to derive a solution to tetration. The fixed parabolic point for base eta was easy. The fixed hyperbolic points of bases between 1 and eta were harder, but still relatively easy.

Bases between e^-e and 1 are harder still, but on the whole still easy. Bases between 0 and e^-e are harder yet, and I haven't taken the time to prove numerically what I'm already sure of conceptually.

But for bases greater than eta, where the fixed point is complex... These really are hard. After all the time and thought I've poured into trying to understand continuous exponentiation of base e, and how the fixed point fits in, looking back at all the bases between 0 and eta is like studying calculus and then looking back at algebra.

And so far I'm only even concerning myself with real bases. The next challenge, if and after we solve bases above eta, is to solve the general case for complex bases, including making a determination whether certain bases are strictly unsolvable (no continuously differentiable solution).

It seems like you are using tau to move points to the vicinity of the fixed point, then using the inverse of tau to move the fixed point to 0, iterating, moving the fixed point back to its proper location, then moving the transformed point back to its new location. If I'm reading this correctly, I don't think your tau function is the right way to move points towards the fixed point. Preferably, it should be possible with a single function to drive all points towards the fixed point. For base eta, this is done by taking logarithms (above the asymptote) or exponentials (below the asymptote) to move points near e, then performing continuous parabolic iteration, then taking the inverse of the iterated operation to get back to the proper point.

My first try with base e was to use iterated logarithms. Unfortunately, logarithms introduce an undesirable fractal pattern that distorts the spacing between points, because integer tetrates of the base (0, 1, b, b^2, etc.) will be connected out at infinity.

My current hypothesis, based on computing Andrew's slog for complex numbers within the radius of convergence, is that the "proper" way to transform large numbers into infinitesimal numbers (relative to the fixed point) is to use imaginary iterations of exponentiation. This allows us to move zero towards the fixed point, bypassing the singularity of the logarithm. In other words, if +1 iteration is exponentiation, and -1 iteration is a logarithm, then we need a formula (and a name!) for i iteration. Of course, currently the only way I have of generating such a formula is by using the first derivative of a tetration solution. But deriving a formula for performing imaginary iterations from tetration and then using it to derive a tetration solution is bad circular reasoning. At best it proves internal consistency.

I'm wondering, therefore, whether it's possible to derive the imaginary equivalent of something between an exponentiation and a logarithm, not quite either but related to both by being halfway between them and out to the side, so to speak.

If we could independently derive such a formula, and if we could be sure of its uniqueness, then we could use it to derive a solution to tetration. The fixed parabolic point for base eta was easy. The fixed hyperbolic points of bases between 1 and eta were harder, but still relatively easy.

Bases between e^-e and 1 are harder still, but on the whole still easy. Bases between 0 and e^-e are harder yet, and I haven't taken the time to prove numerically what I'm already sure of conceptually.

But for bases greater than eta, where the fixed point is complex... These really are hard. After all the time and thought I've poured into trying to understand continuous exponentiation of base e, and how the fixed point fits in, looking back at all the bases between 0 and eta is like studying calculus and then looking back at algebra.

And so far I'm only even concerning myself with real bases. The next challenge, if and after we solve bases above eta, is to solve the general case for complex bases, including making a determination whether certain bases are strictly unsolvable (no continuously differentiable solution).

~ Jay Daniel Fox