regular tetration base sqrt(2) : an interesting(?) constant 2.76432104

[Gottfried]

Hi Sheldon -

just "in shortness"

Then we use the inverse Schröder function of \( -s_0 \) to get Gottfried's number.
\( z = S^{-1} (-s_0) + 2 \approx 2.7643210400001 \)

Well, ;-) I think such a naming deserves at least one index, so let's index it with the related fixpoint...

Unfortunately, that constant has no obvious relation to the asum- zero-height from the other thread - if there were some elegant relation: that were really great. Also whether there might be a relation to the same effect /constant using the upper/repelling fixpoint were an interesting thing: if we perform a handful of integer-height iterations towards the upper fixpoint and find the fixpoint_4-dual and if we could make this somehow consistent, then we had a nice relation over/connecting the whole real line.

For the connection from the duals below 2, between 2 and 4 and above 4 I'd made a picture (and some mail here, around 07'2010) but where I did not yet understand fully the different fixpoint-implications. I used that dual for a "norming", setting one value between 2 and 4 as having height = 0, namely the dual of 1.
   

The last point (the only one which I cannot answer myself, perhaps you can look at it): Can we look at your Kneser-method what the dual of, say x_0 = 1 or x_0 = 0 or x_0 = -infty were? I think we need only the appropriate imaginary iteration height to compute the respectively duals. Levenstein- numbers? ;-)

Gottfried