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The strange properties of the infinite power tower (arxiv.org)
150 points by amelius 37 days ago | hide | past | web | favorite | 9 comments



This is only loosely related, but reading this brings back a wonderful memory.

At Hampshire College in the early 90's, I took a "Math Concentrators' Seminar" which was really a free-form math club with free pizza one evening a week run by an offbeat, charismatic professor whose name I forget [edit: David Kelly[1]]. I wasn't focused on math at the time and didn't have anywhere near the foundation I should have for the class, but the professor was happy to have me.

We spent hours just informally batting around diverse, fun math concepts, and one of the topics I remember best was all the fun and counter-intuitive properties of infinite series like this. In any case, this paper is really a treat. Thank you OP.

[1] https://en.wikipedia.org/wiki/David_Kelly_(mathematician)


FWIW, thanks to DDG and a spammy PDF doc site ("yumpu"?), I found the old course description:

NS 322 MATH CONCENTRATORS' SEMINAR David Kelly This weekly gathering of students interested in mathematics and its applications will include lectures by Hampshire faculty and guests. presentations by Division III students. films, workshops. problem-solving sessions, puzzles, games. paradoxes. history, and philosophy. The seminar provides an opportunity for students to get to know each other and gain exposure to many active areas of mathematics. This class will meet once a week for two hours.


What an amazing professor. THIS is how you instill fascination with a subject, by making it quite approachable.

Thanks for sharing your story! :)


Kelly also started an annual summer program, along similar lines, for high school students.

https://hcssim.org/

It's still running after almost half a century (with Kelly still in charge!). I got to attend almost 30 years ago (wait, how has it been that long already?).


This analysis can also be expanded to the complex numbers yielding "power tower fractals" aka "tetration fractals".

http://paulbourke.net/fractals/tetration/

https://thatsmaths.com/2016/04/14/the-power-tower-fractal/


This is an excellent teaching resource. I really like these sorts of well-written, short-but-not-too-short deep dives on accessible topics. It's a great way to help students learn how to read math and build up confidence in their ability to learn on their own.


It's key to include false starts and promising paths that dead end. Learning to explore unknown solution spaces and refine the domain of the problem itself are critical skills in any open ended problem.

Spending an entire school lifetime being taught to "solve" "problems" and then being confronted with a world where problems aren't defined and solutions are ad-hoc and piecemeal is a rude awakening.

The strategies that made us a good students and made us feel good and smart in school aren't the same strategies that make for a good employee and those strategies set new devs up to fail when they can't "see the answer" to the current jira ticket they are tasked with.


It clearly did me in. It made me very sad that I could not write code to problems I have not seen before. I still haven't learned the mental skills necessary to face open problems with a curious mind. My first reaction to new problems is fear and anxiety to the extent I had to leave development entirely. I now do production support which is not what I intended to do. I wanted be a great developer.


That technique on page 3 of splitting off one of the powers and taking the result to be equal to the original reminds me of the proof that 0.999...=1

https://i.imgur.com/ECYa380.png




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