I have learn more geometry from trying to solve the unsolvable "squaring the circle" problem than all the Geom. classes I had in my life. For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by myself doing the exercise of the unsolvable. Also, more recently early this year, I found that the diagonal/diameter of a Pentagon was the Cubic Root of 3, or this was the hexagon, and the heptagon was the S.q.rt.o'2 multiplied by C.rt.o'3 or something like this, the thing is that they follow a sequence, Its always something with the square and cubic roots of 2 and 3 and I guess 5 or 7 will appear later in the sequence for polygons with more sides, noting that for a "infinite sides 'polygon' " , which would be a circle, the number that relates the diagonal with the "'sides'", or in this analogy, the Perimeter , is Pi...Anyway, good exercises. p.s: I have remembered that back then I thought maybe Pi was Square Root of Infinity, and now just came to my mind that maybe would be the Infinite Root of something...But off course just joking thinking, but nice exercise.
MU puzzle
31–40 of 46 posts
Re: MU puzzle
#32Earlier quoted context omitted.
I'm curious, why do people on HN warn about PDFs? The worst things I can imagine happening with PDFs are (1) you don't have any software that can read it (probably like .2% of HNers?) and (2) it could potentially have some malware (but the same could be said of a website)
The mobile experience for PDFs is awful.
The mobile experience for HTML/JS/CSS is awful too.
Re: MU puzzle
#33I have learn more geometry from trying to solve the unsolvable "squaring the circle" problem than all the Geom. classes I had in my life. For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by…
> For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by myself doing the exercise of the unsolvable.
That's actually the Pythagorean theorem. How did you arrive at that by doing the "unsolvable"?
Re: MU puzzle
#34Is the Gödel Escher Bach worth the reading? I started, but I feel it's fluff. People say he is good, though. Are there some practical insights?
I would say the first half is great, second half is terrible.
It is a lot of fluff and meandering, but the dialogue are just too good, and I still enjoy them.
Some people have recommended reading "I am a strange loop" as a basically more mature and better written version of GEB.
Re: MU puzzle
#35I have learn more geometry from trying to solve the unsolvable "squaring the circle" problem than all the Geom. classes I had in my life. For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by…
> For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by myself doing the exercise of the unsolvable. That's actually the Pythagorean theorem. How did you arrive at that by doing the "unsolvabl…
The “unsolvable problem” the parent refers to is the problem of geometrically constructing a square with the same area as a circle. It has been proven to be impossible.
Re: MU puzzle
#36I have learn more geometry from trying to solve the unsolvable "squaring the circle" problem than all the Geom. classes I had in my life. For example: No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's, so for a square where the side is 1 meter, the diagonal will be Sq.Rt.o'2 meters (around 1,4 m), And I came to find that by…
> No one ever has told me in school that for all squares, the length of the diagonal equals The Square Root Of Two times the side of the square's
This is basic knowledge (and as someone said, comes from the Pythagorean theorem) and taught on school in a lot of places.
Re: MU puzzle
#37Is the Gödel Escher Bach worth the reading? I started, but I feel it's fluff. People say he is good, though. Are there some practical insights?
I would say the first half is great, second half is terrible. It is a lot of fluff and meandering, but the dialogue are just too good, and I still enjoy them. Some people have recommended reading "I am a strange loop" as a basically more mature and better written version of GEB.
There is a section in the book where he laments that certain mediums (like books, and movies) leak information about where the story ends (because in a book as you are reading it you will know you are coming towards the end). He explained one way to avoid this and retain some mystery in story-telling would be to finish the story some distance before the end and pad the remainder of the book with blank pages. However blank pages are too easy to spot if someone happens to flick through the book before hand, so instead it would be better to pad the book with text which appears to be related to the primary story and therefore is undetectable unless someone reads through the whole book in order.
Re: MU puzzle
#38People who like this might also enjoy the similarly simple (yet unsolved) Collatz Conjecture: https://en.wikipedia.org/wiki/Collatz_conjecture
On a slight tangent, I recently read a paper on an interesting relationship between Collatz path length and Mersenne primes. "Our main finding to report is the fact that a path length of a Mersenne prime is approximately proportional to its index for large n, namely, D(Mn) ≈ 13.45n." Paper is at https://arxiv.org/pdf/1104.2804.pdf . WARNING: PDF.
You can de-PDF anything on the arXiv by chopping off the .pdf suffix and changing the /pdf/ to /abs/.
For example, https://arxiv.org/abs/1104.2804
Re: MU puzzle
#39Earlier quoted context omitted.
On a slight tangent, I recently read a paper on an interesting relationship between Collatz path length and Mersenne primes. "Our main finding to report is the fact that a path length of a Mersenne prime is approximately proportional to its index for large n, namely, D(Mn) ≈ 13.45n." Paper is at https://arxiv.org/pdf/1104.2804.pdf . WARNING: PDF.
You can de-PDF anything on the arXiv by chopping off the .pdf suffix and changing the /pdf/ to /abs/. For example, https://arxiv.org/abs/1104.2804
Thanks! I love learning little tips like that.
Re: MU puzzle
#40Is the Gödel Escher Bach worth the reading? I started, but I feel it's fluff. People say he is good, though. Are there some practical insights?
The explanation of how Godel's Theorem works that I got from reading GEB is still the best one I have encountered.
I always find the Gödel representations using powers of primes too strange. It's technically correct, I can follow the proof, but there is a small corner of my mind that can't believe it.
The GEB book doesn't have all the details, but it uses the ASCII representations (actually base 20 but whatever). With the ASCII representation the theorem feels obvious.