First, how many is ?
arrows of integer numbers
You know these:
Look at the previous formulas again. You can see a pattern:
So, tetration is generalized like this:
Now I can answer the question.
This is proper because it is greater than and smaller than .
Next, I will make for n arrows.
It's almost same pattern:
I think this is not good, but it must be like this by the definition.
arrows of non-integer numbers
Next, I have to think about a half arrow.
Look at these more...
And one more...
Now I have done (but too assertive).
I will summarize all definitions:
There is a problem on the old version. It is that is not 4 when n is not an integer. Is can be 3.7683... when n is around 0.53. This is a gap of 8% and it is not ignoble.
I have thought how to fix it for a week, and I have an answer: linear interpolation.
In math language,
If I want to make always four, it must be like this.
comparizon between old one and new one
(not be written yet)