Re: Bringing back an old tetration curiosity?

mike3 wrote:
[snip]

This would suggest then that the tetrational function is _close_ to
symmetrical but not exactly so. I'd be curious to know if the
difference
between the value of ^0.5 sqrt(2) obtained using this method and that
obtained by assuming it is the square tetraroot of sqrt(2) is small
enough that perhaps the _correct_ curve for sqrt(2)'s tetrational
function would be able to pass through it at 0.5...

The formal order-2 tetraroot of sqrt(2) is 1/2*ln(2)/W(1/2*ln(2))=1.304351179.

His method gives: sqrt(2)^^(1/2)=sqrt(2)^((sqrt(2)^sqrt(2)-1)=1.2451.

My method gives sqrt(2)^^(1/2)=sqrt(2)^(1/2)=1.189207115

Both estimates bound the order-2 tetraroot from below. The respective
differences are:

0.0592

0.1151,

so his method bounds the tetraroot better, but not "close enough". Of course the
question is, "how close is close enough", but still a difference of 0.06 is
large for success.

In general I don't expect ANY calculable tetration function to pass "through"
the tetraroots. Any tetration extension to the reals is like a mathematical
"abomination" if you will. The best "approximation" of this function so far (by
Andrew Robbins) doesn't pass through the tetraroots, so all the rest are
expected to fail in this regard.

Regardless, the symmetry is obviously _close_, but no _cigar_.
I'd be curious to hear some more opinions from others here on
this group. It seems though like the guy's original idea, at least
in that form, is likely another dead end...

It is. Searching for a "proper" extension of tetration to the reals is in
general a dead end as well. It is easy to see that any function that attempts to
define x^^y for real y, will be inconsistent (definition-wise) with the function
x^^(m/n) when y=m/n, _exactly because_ (x^^m)^^(1/n) =/= (x^^(1/n))^^m. Worst
yet, _it cannot be_ either one (unless one defines it that way), because either
definition is valid, because if it's either one, the other definition will
complain. That's very much like trying to give a definition of x^y for real
y=m/n, with

(x^m)^(1/n)=/=(x^(1/n))^m.

It cannot be done. For example, if one gives you the value for x^^0.66666... for
some x, then if this was the "correct" tetration extension to the reals, this
*must* be equal x^^(2/3), yet the latter can be defined using two different ways
as (x^^2)^^(1/3) or as (x^^(1/3))^^2.

The order-n tetraroots are the last bastion of tetration for non-integer
exponents. I don't think the people who work on tetration have realized this yet
;o)
--
I.N. Galidakis

.

Relevant Pages

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