The inverse of tetration is just the number of times you have to press the log (base B) button on your calculator until it reaches 1. If you get to a number below B, then you’ll end up with a fractional value as the final answer. You could just use this as the fractional part–so 10^10^10^56, which is really just 10^10^10^10^1.74, gives a tetration value of 4.24. To do the forward tetration you take 10^frac and then make a stack of tens with the integer part.
A little ugly, but C0 continuous, and I think C1 continuous as well. Consider e^e for simplicity. At the join, our two functions are:
ln(x)
ln(ln(x))*e
The respective derivatives are:
1/x
e/(x*ln(x))
See what happens at e^e:
1/(e^e)
e/((e^e)*ln(e^e)) = e/((e^e)*e) = 1/(e^e)
Ok, so it seems to be continuous there also. Second derivatives:
-1/x^2
e(ln(x)+1)/((x*ln(x))^2)
Well, those don’t match up anymore. Well, C1 continuity isn’t bad. I’m assuming here that the constants work out for bases other than e; I haven’t proved to myself that this is the case.
The best superscripts can keep going up่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่ี่่้้้้้้้้้้้้้้้้้้้้้้้and up้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้้, even into the previous poster’s post!
septimus, I’m not sure what you were trying to do in those two posts, but I don’t think it worked in Firefox/OSX. Your first post has a bunch of characters that look like dotted-line circles with various symbols above them, and your second has a bunch of similar dotted circles with symbols below them.
Asimov wrote a couple of articles about very large numbers: “Skewered” and “T-formation” and Douglas Hofstadter wrote “On Number Numbness” on the same subject. Neither seems to be on line, but this Who Can Name the Bigger Number? covers much of the same material.
Got stacks of weird stuff in Firefox and Vivaldi. Both Chromium based.
The interesting (?) part is that stuff “bled over” into the adjoining posts. Which some prankster could exploit to change/add to other posts.
One difference between Firefox and old Opera is that the former shrinks the exponents as they get higher up while the latter keeps the numbers the same size (much more readable).
2[sup]2[sup]2[sup]2[sup]2[sup]2[sup]2[sup]2[sup]2[sup]Some years ago some folks did a study and tested all the major browsers on all the major platforms and found all didn’t perform very well under all the requirements of what was then the well establish HTML 4.0 standard … I guess that’s still the case today … such a shame … Firefox on a Mac … what the web was designed to do[/sup][/sup][/sup][/sup][/sup][/sup][/sup][/sup][/sup]
Sorry for intruding (with champagne bubbles) into others’ posts. I was deep into the drunken 2017 countdown my time, and I was celebrating … even reaching out to my neighbors.
All our keyboards here are dual-language so I didn’t need to look up any special Unicodes. The champagne bubbles arose from Thai vowels and tone-marks; hence their peculiar shapes.
