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# Talk:Tetration

## Issues with tetra-roots

### A failed attempt at an interpretation of a rational [power tower] height

From the article:

Now, if we have

${\displaystyle y{\uparrow \uparrow }h=x{\uparrow \uparrow }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$

then

${\displaystyle y=x{\uparrow \uparrow }\left({\tfrac {g}{h}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+}\,}$

gives us an interpretation for a rational height.

Now let x = 6.121278550700... so that

${\displaystyle x^{x}=65536=2\uparrow \uparrow 4}$

and thus

${\displaystyle x=2\uparrow \uparrow 2=2^{2}=4}$

But this is a contradiction, so the rule proposed above is invalid. I'm going to remove the section.

Charles R Greathouse IV 21:27, 14 May 2013 (UTC)

Over the next few days, I will meticulously go through the whole tetration topic and the related tetra-powers (and the inverse tetra-roots) and tetra-exponentials (and the inverse tetra-logarithms) to find out whether those can be well-defined on some domain. I'm aware that are many potential issues. Since L. R. Goodstein and other people investigated the higher operations, I've been assuming that they resolved those issues. So I will investigate more. The tetration article starts (I will look for the source...)
The term "tetration" (4th iteration) was coined by L. R. Goodstein in his consideration of the hierarchy of binary operations.
Some links (I will look for more):
Henryk Trappman, Andrew Robbins, Tetration FAQ, January 12, 2008.
The above has in its TOC:
2.2.1 Hyper-exponentials . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Hyper-logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Hyper-powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.4 Hyper-roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
http://de.cyclopaedia.net/wiki/Tetration (in German)
http://www.tetration.org/
Daniel Forgues 02:42, 15 May 2013 (UTC) — Daniel Forgues 04:58, 15 May 2013 (UTC)
After I logged off yesterday, I had a hint that I did not have an interpretation for a rational height [of a power tower]. What works with multiplication (since addition over the complex numbers is commutative and associative) and exponentiation (since multiplication over the complex numbers is commutative and associative) doesn't work with tetration (since exponentiation over the complex numbers is neither commutative nor associative).
With multiplication (defined as repeated addition), if we have
${\displaystyle y{\cdot }h=x{\cdot }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
then
${\displaystyle y=x{\cdot }\left({\tfrac {g}{h}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+}\,}$
gives us an "interpretation" for a rational multiplier. (Hey, this is how the rational numbers were defined in the first place!)
This is true because
${\displaystyle y{\cdot }(kh)=x{\cdot }(kg),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to (because addition over the complex numbers is commutative and associative)
${\displaystyle (y{\cdot }h){\cdot }k=(x{\cdot }g){\cdot }k,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to
${\displaystyle y{\cdot }h=x{\cdot }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
and thus
${\displaystyle y=x{\cdot }\left({\tfrac {kg}{kh}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to
${\displaystyle y=x{\cdot }\left({\tfrac {g}{h}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
and the rational multiplier interpretation makes sense because the two forms ${\displaystyle {\tfrac {kg}{kh}},\,k\neq 0,}$ and ${\displaystyle {\tfrac {g}{h}}}$, which represent the same rational number, also represent the same multiplier. (Now, we can swap the multiplier with the multiplicand and repeat the process a second time, to get an interpretation of the product of two rational numbers.)
With exponentiation (defined as repeated multiplication), if we have
${\displaystyle y{\uparrow }h=x{\uparrow }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
then
${\displaystyle y=x{\uparrow }\left({\tfrac {g}{h}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+}\,}$
gives us an interpretation for a rational exponent.
This is true because
${\displaystyle y{\uparrow }(kh)=x{\uparrow }(kg),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to (because multiplication over the complex numbers is commutative and associative)
${\displaystyle (y{\uparrow }h){\uparrow }k=(x{\uparrow }g){\uparrow }k,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to
${\displaystyle y{\uparrow }h=x{\uparrow }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
and thus
${\displaystyle y=x{\uparrow }\left({\tfrac {kg}{kh}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is equivalent to
${\displaystyle y=x{\uparrow }\left({\tfrac {g}{h}}\right),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
and the rational exponent interpretation makes sense because the two forms ${\displaystyle {\tfrac {kg}{kh}},\,k\neq 0,}$ and ${\displaystyle {\tfrac {g}{h}}}$, which represent the same rational number, also represent the same exponent.
Not so for tetration (defined as repeated exponentiation), because
${\displaystyle y{\uparrow \uparrow }(kh)\neq (y{\uparrow \uparrow }h){\uparrow \uparrow }k,}$
i.e.
${\displaystyle \underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{kh}\neq \underbrace {{\left(\underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{h}\right)}^{{.\;}^{{.\;}^{{.\;}^{{\left(\underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{h}\right)}^{{\left(\underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{h}\right)}^{{\left(\underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{h}\right)}^{\left(\underbrace {{y}^{{.\,}^{{.\,}^{{.\,}^{{y}^{{y}^{{y}^{y}}}}}}}} _{h}\right)}}}}}}}} _{k},}$
since exponentiation over the complex numbers is neither commutative nor associative.
It means that the form
${\displaystyle y{\uparrow \uparrow }(kh)=x{\uparrow \uparrow }(kg),\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
is not equivalent to (since exponentiation over the complex numbers is neither commutative nor associative)
${\displaystyle (y{\uparrow \uparrow }h){\uparrow \uparrow }k=(x{\uparrow \uparrow }g){\uparrow \uparrow }k,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,k\neq 0,\,}$
so we don't have
${\displaystyle y{\uparrow \uparrow }h=x{\uparrow \uparrow }g,\quad g\in \mathbb {Z} ,\,h\in \mathbb {N} ^{+},\,}$
which implies that for each ${\displaystyle k,\,k\neq 0,}$ we have a distinct form. This means that we do not have an interpretation of a rational height ${\displaystyle {\tfrac {g}{h}}}$. It's more like we have a tuple ${\displaystyle (g,h)}$, e.g.
${\displaystyle y{\uparrow \uparrow }1=x{\uparrow \uparrow }2}$
is not equivalent to
${\displaystyle y{\uparrow \uparrow }2=x{\uparrow \uparrow }4}$
is not equivalent to
${\displaystyle y{\uparrow \uparrow }3=x{\uparrow \uparrow }6}$
and so on...
So in your example, the form with ${\displaystyle (g,h)=(4,2)}$
${\displaystyle x{\uparrow \uparrow }2=2{\uparrow \uparrow }4}$
has solution
${\displaystyle x}$ = 6.121278550700...
while the form with ${\displaystyle (g,h)=(2,1)}$
${\displaystyle x{\uparrow \uparrow }1=2{\uparrow \uparrow }2}$
has solution
${\displaystyle x}$ = 4.
Indeed, we do not have a rational interpretation for the height. (Did someone find a way...? I'll read more about it...) — Daniel Forgues 04:58, 15 May 2013 (UTC) — Daniel Forgues 04:14, 17 May 2013 (UTC)
Should I add the above explanation (A failed attempt at an interpretation of a rational [power tower] height) as a subsection of tetra-roots, or would it clutter the article too much? — Daniel Forgues 04:14, 17 May 2013 (UTC)

## Charles' observation

Well, as the initial "originator" of the first version of the article I've taken me out after I'd seen, that things, which I deliberately had left open or vague/in daily-speak was supplied with full-featured statements of existence/correctness, although the extension of the tetration to non-integer heights is still open in number theory... Also there is no real consensus about the best perspective in which the "fourth hyperoperation" should be seen. The problem of "hyperroots" and "hyperheights" should be seen different: (sorry, at the moment I do not see how I can invoke the "math-mode" in writing, I'll improve this later when I've found the needed conventions)

"hyper-root" asks for the basis with which must be exponentiated a given height to arrive from a given start-value at a given end-value, such that ${\displaystyle x_{h}=\exp _{b}^{[h]}(x_{0})}$ with given ${\displaystyle x_{0},x_{h}}$ and ${\displaystyle h}$.

"hyper-height" (or short: (iteration-)height-function "hgh") asks for the needed height ${\displaystyle h}$ to iterate from ${\displaystyle x_{0}}$ to ${\displaystyle x_{h}}$ using a known base ${\displaystyle b}$

Algebraic operations on the height-parameter besides additive ones are not yet known because of the missing associativity in the expression ${\displaystyle x_{h}=(b^{(b^{(b^{(b^{x_{0}})})})})}$ (The parentheses cannot be set differently). There are attempts to this, for instance of Ioannis Galidakis who is now no more active in this field, to solve for multiplicative operations on the height parameter using continued fractions, but this failed. Because the change of the basis-parameter would be involved, and such change has also still no known formula, we are here still in the situation of open research, not even armed with partial results.

So, much of this should be restated as "open problem", "unsolved" in the "tetration-article". And my intention has thus been, to give a short allusion for the reader what is there, what are the mental/algebraic concepts needed, and not to inject the impression, that this subject is already in a shape which would allow to put it in, say, formula-manuals like Abramowitz/Stegun or similar ones.

--Gottfried Helms 08:25, 15 May 2013 (UTC)

I completely agree, Gottfried. There are many flavors of tetration 'out there', and none have been widely accepted by the mathematical community.
It would be nice to have an overview of the types that have been tried, the properties they satisfy, the types of numbers they apply to, and how they can be computed. But this is not a small project, it would take at least as long as it took me to write Density.
Charles R Greathouse IV 18:29, 16 May 2013 (UTC)
Charles/Daniel: I've uploaded a first version/first part of a new version of an article. I'll use that version later on my own site, but have it a bit focused for the OEIS/Seqfan user. I've uploaded in an editable format (winword) and if you find this worth in style and approach then you might even add improvements directly in the text and send me your ideas (or just extract text for the OEIS/Tetration-article). See http://go.helms-net.de/math/tetdocs/TetrationForSeqFans.zip
--Gottfried Helms 11:04, 17 May 2013 (UTC)
I avoid opening downloaded Microsoft Word documents (they're not safe...). Even OEIS Wiki doesn't support .doc (nor any other MS Office formats for the same reason) uploads! Check https://oeis.org/wiki/Special:Upload
Permitted file types: bmp, c, cc, eps, gif, gz, java, jpeg, jpg, mid, mp, mp3, nb, pdf, png, ps, rtf, svg, tex, tiff, txt.
Daniel Forgues 22:24, 17 May 2013 (UTC)
Daniel, I've inserted a rtf-version in the zip. (It's nearly the same contents - I did not work much on the text today) Gottfried --Gottfried Helms 19:40, 18 May 2013 (UTC)
You are welcome to add ideas from your article to the tetration article. The article needs more contributions, it is far from complete: for example, it needs a well-defined definition for a rational height [of a power tower], if its ever possible! — Daniel Forgues 03:15, 19 May 2013 (UTC)

## A bit of Intuition for "complex heights"

In the section for "complex heights" we find a simple word: "how?".

I had dealt with that question sometimes explicitely; perhaps the two following links illustrate how such a thing can be understood.

In the first example I discuss (as an exercise) the iteration of the function f(x)=1/(1+x) and find explicite expression for the height parameter in the formula for the powerseries/rational expression as a simple argument similar like a monomial, where then the formal "h" can get a fractional and then also a complex value. See http://go.helms-net.de/math/tetdocs/FracIterAltGeom.htm mainly section 3) and 5) .

Next example: I discuss this in the consideration of tetration with a small base in the complex plane. The trajectory made a continuously increased real height h can describe a rotation/spiral in the complex plane towards the fixpoint (in the case if is not the infinity). Then a "complex height" of iteration simply takes another direction; and in case of a spiral for real height iteration one full winding of the curve might be short-cuttet by a little complex height iteration which is vertical to the spiralic trajectory made by real heights only. See par 1.5 and 1.7 in http://go.helms-net.de/math/tetdocs/Base004.htm

Gottfried --Gottfried Helms 22:49, 18 May 2013 (UTC)