4 ms·

Some π-ography for Pi day

39 points by ViralBShah 9y ago

3pt14159 9y ago
For obvious reasons, I'm going to love today. There are so many fun things with pi. 1. It's roughly equal to ln(6) ^ (ln(5) ^ (ln(4) ^ (ln(3) ^ ln(2)))) 2. It's not exactly hard to figure out whether pi ^ e is greater than e ^ pi (years ago I did it sans calculator on a test in a fast but brutish way), but there are some pretty terse, beautiful solutions that can be elusive. 3. Aside from the whole e^(pi * i) - 1 = 0 just looking too crazy to be true, I also love how it includes the basics tools of mathematics / philosophy. Subtraction (or addition if re-arranged), Multiplication (or division if re-arranged), exponentiation, true / 1, false / 0, equality, geometric ratio (pi) which can be thought of as representing the physical world, a growth ratio (e) which can be though of as representing life, and a number (i) literally name after an aspect of human thought / imagination.
- 52-6F-62 9y ago
  That's fun. I'm recently working my way back through my fundamentals prepping for some future hopeful coursework and had to play with number 1. #include "stdio.h" #include "math.h" long double ln ( long double n ) { return log(n) / log(M_E); } long double pi ( void ) { return pow(ln(6), pow(ln(5), pow(ln(4), pow(ln(3), ln(2))))); } int main( void ) { printf("%LE\n", pi()); return 0; } Or in JS function ln(n) { return Math.log(n) / Math.log(Math.E); } function pi() { return (ln(6) ** (ln(5) ** (ln(4) ** (ln(3) ** (ln(2)))))); } If anybody else wants to toy with it. Getting older and working outside of the fields that employ a lot of certain technicals doesn't leave as much time for exploration as one would like.
  - jwilk 9y ago
    Both log() in C and Math.log() in JS are natural logarithms, so there's no need to define your own ln() functions.
    - 3pt14159 9y ago
      If I ever make a programming language the log function will take two parameters, value and base. It's too confusing when engineers, mathematicians, and computer scientists all use different implicit bases.
      - 52-6F-62 9y ago
        I had to do a double take for that reason. They're 100% correct, but I assumed what I'd learned from math. Whenever I otherwise work with logarithms in programming I choose to work explicitly to at least save myself confusion. By default I assumed log()/Math.log() referred to common logarithms. I usually end up re-implementing the log function with 2 values, as with ln/logE above, but substituting the constant. long double logn( long double n, long double x ) { return log(x) / log (n); } But, yes, all my above code could be shorter. I haven't worked with any explicitly scientific languages like Mathlab— do any languages take a more traditionally mathematical approach? Or any libraries that have taken that torch up?
        jwilk 9y ago
        For common cases of n=2 and n=10, consider using dedicated functions log2() and log10(), because they are more precise. For example: log(1E42)/log(10) == 42 - 7.105427357601002e-15 log10(1E42) == 42.0
        52-6F-62 9y ago
        Noted. It’s more for other cases. While doing any algebra coursework I usually (loosely) confirm my thoughts by running them in a repl with said functions to at least keep me on track— so logn usually serves uncommon bases. My thinking that way might not make much sense, but I start to explain it in other comments. Anyway— thanks, I appreciate the insights. That’s the reason I like this forum
    - 52-6F-62 9y ago
      TIL! I completely and foolishly assumed that when I saw log on its own that it referred to a common logarithm. About 2 seconds on MDN and a math.h reference and I could have known better.
  - AtomicOrbital 9y ago
    initially I tried pasting ln(6) ^ (ln(5) ^ (ln(4) ^ (ln(3) ^ ln(2)))) into bc to no avail ... magically paste it into google - Bam
    - jwilk 9y ago
      bc does have natural logarithm, but it's called l(), and you have to enable the math library with the -l option. There's another problem, though: the ^ operator requires that the second expression is integer.
  - jwilk 9y ago
    Another weird thing about your C code is that is uses "long double" as argument and return types; but pow() and log() functions use doubles, so there's some unexpected rounding going on. Possible ways to fix that: - Use the "double" type everywhere. - Use powl() and logl() functions instead. - Use tgmath.h instead of math.h. (But when I tried this with GCC 7 + glibc 2.27, the compiler ran out of memory. Go figure…) - Use C++, which has overloads for pow() and log().
- jwilk 9y ago
  You got the sign wrong. e^(iπ) = −1
  - 3pt14159 9y ago
    Ha! Serves me right for trying to comment from my phone. You're right, but I can no longer edit the comment.
- zamalek 9y ago
  The thing I love most about π is that its nature comparable to that of circles: circles have infinitely many corners (technically extreme points), just like π has infinitely many digits.
- madcaptenor 9y ago
  What do you have in mind as a terse solution? I'm thinking as folllows: Let f(x) = x/log(x). Then f has a minimum at e - for example f'(x) = (log(x) - 1)/(log(x)^2), so f is decreasing for x < e and increasing for x > e. In particular f(e) < f(pi)-- that is, e / log(e) < pi / log(pi). Multiply through by log(e) log(pi) to get e log(pi) < pi log(e) and exponentiate both sides to get pi^e < e^pi.
tim_hutton 9y ago
Is anyone else a bit sick of these 'pi days'? Every year we get the same old shtick. Pleasingly, under the one true date format [0] there are no pi days. (Well there are, but long past the heat death of the Universe.) [0] https://xkcd.com/1179/ https://xkcd.com/1179/
- shawabawa3 9y ago
  Pi day works in that format... 2018-03-14
- kej 9y ago
  You might appreciate one of my favorite Pi Day related comics, then: http://www.qwantz.com/index.php?comic=955 http://www.qwantz.com/index.php?comic=955
  - tim_hutton 9y ago
    The value on pi approximation day (22/7) is closer to pi than the value on pi day (3.14).
  - scrooched_moose 9y ago
    Also, my favorite pi related comic: https://www.smbc-comics.com/?id=1777 https://www.smbc-comics.com/?id=1777 It's not specific to pi-day, but it still seems appropriate.
simonbyrne 9y ago
The author of this post is the creator of the Julia visualization package Luxor.jl (https://github.com/JuliaGraphics/Luxor.jl https://github.com/JuliaGraphics/Luxor.jl). One thing I like about it is the nice turtle graphics feature (http://juliagraphics.github.io/Luxor.jl/stable/turtle.html http://juliagraphics.github.io/Luxor.jl/stable/turtle.html).

39 points by ViralBShah 9y ago

For obvious reasons, I'm going to love today. There are so many fun things with pi. 1. It's roughly equal to ln(6) ^ (ln(5) ^ (ln(4) ^ (ln(3) ^ ln(2)))) 2. It's not exactly hard to figure out whether pi ^ e is greater than e ^ pi (years ago I did it sans calculator on a test in a fast but brutish way), but there are some pretty terse, beautiful solutions that can be elusive. 3. Aside from the whole e^(pi * i) - 1 = 0 just looking too crazy to be true, I also love how it includes the basics tools of mathematics / philosophy. Subtraction (or addition if re-arranged), Multiplication (or division if re-arranged), exponentiation, true / 1, false / 0, equality, geometric ratio (pi) which can be thought of as representing the physical world, a growth ratio (e) which can be though of as representing life, and a number (i) literally name after an aspect of human thought / imagination.

52-6F-62 9y ago
That's fun. I'm recently working my way back through my fundamentals prepping for some future hopeful coursework and had to play with number 1. #include "stdio.h" #include "math.h" long double ln ( long double n ) { return log(n) / log(M_E); } long double pi ( void ) { return pow(ln(6), pow(ln(5), pow(ln(4), pow(ln(3), ln(2))))); } int main( void ) { printf("%LE\n", pi()); return 0; } Or in JS function ln(n) { return Math.log(n) / Math.log(Math.E); } function pi() { return (ln(6) ** (ln(5) ** (ln(4) ** (ln(3) ** (ln(2)))))); } If anybody else wants to toy with it. Getting older and working outside of the fields that employ a lot of certain technicals doesn't leave as much time for exploration as one would like.
- jwilk 9y ago
  Both log() in C and Math.log() in JS are natural logarithms, so there's no need to define your own ln() functions.
  - 3pt14159 9y ago
    If I ever make a programming language the log function will take two parameters, value and base. It's too confusing when engineers, mathematicians, and computer scientists all use different implicit bases.
    - 52-6F-62 9y ago
      I had to do a double take for that reason. They're 100% correct, but I assumed what I'd learned from math. Whenever I otherwise work with logarithms in programming I choose to work explicitly to at least save myself confusion. By default I assumed log()/Math.log() referred to common logarithms. I usually end up re-implementing the log function with 2 values, as with ln/logE above, but substituting the constant. long double logn( long double n, long double x ) { return log(x) / log (n); } But, yes, all my above code could be shorter. I haven't worked with any explicitly scientific languages like Mathlab— do any languages take a more traditionally mathematical approach? Or any libraries that have taken that torch up?
      - jwilk 9y ago
        For common cases of n=2 and n=10, consider using dedicated functions log2() and log10(), because they are more precise. For example: log(1E42)/log(10) == 42 - 7.105427357601002e-15 log10(1E42) == 42.0
        52-6F-62 9y ago
        Noted. It’s more for other cases. While doing any algebra coursework I usually (loosely) confirm my thoughts by running them in a repl with said functions to at least keep me on track— so logn usually serves uncommon bases. My thinking that way might not make much sense, but I start to explain it in other comments. Anyway— thanks, I appreciate the insights. That’s the reason I like this forum
  - 52-6F-62 9y ago
    TIL! I completely and foolishly assumed that when I saw log on its own that it referred to a common logarithm. About 2 seconds on MDN and a math.h reference and I could have known better.
- AtomicOrbital 9y ago
  initially I tried pasting ln(6) ^ (ln(5) ^ (ln(4) ^ (ln(3) ^ ln(2)))) into bc to no avail ... magically paste it into google - Bam
  - jwilk 9y ago
    bc does have natural logarithm, but it's called l(), and you have to enable the math library with the -l option. There's another problem, though: the ^ operator requires that the second expression is integer.
- jwilk 9y ago
  Another weird thing about your C code is that is uses "long double" as argument and return types; but pow() and log() functions use doubles, so there's some unexpected rounding going on. Possible ways to fix that: - Use the "double" type everywhere. - Use powl() and logl() functions instead. - Use tgmath.h instead of math.h. (But when I tried this with GCC 7 + glibc 2.27, the compiler ran out of memory. Go figure…) - Use C++, which has overloads for pow() and log().
jwilk 9y ago
You got the sign wrong. e^(iπ) = −1
- 3pt14159 9y ago
  Ha! Serves me right for trying to comment from my phone. You're right, but I can no longer edit the comment.
zamalek 9y ago
The thing I love most about π is that its nature comparable to that of circles: circles have infinitely many corners (technically extreme points), just like π has infinitely many digits.
madcaptenor 9y ago
What do you have in mind as a terse solution? I'm thinking as folllows: Let f(x) = x/log(x). Then f has a minimum at e - for example f'(x) = (log(x) - 1)/(log(x)^2), so f is decreasing for x < e and increasing for x > e. In particular f(e) < f(pi)-- that is, e / log(e) < pi / log(pi). Multiply through by log(e) log(pi) to get e log(pi) < pi log(e) and exponentiate both sides to get pi^e < e^pi.

tim_hutton 9y ago

Is anyone else a bit sick of these 'pi days'? Every year we get the same old shtick. Pleasingly, under the one true date format [0] there are no pi days. (Well there are, but long past the heat death of the Universe.) [0] https://xkcd.com/1179/ https://xkcd.com/1179/

shawabawa3 9y ago
Pi day works in that format... 2018-03-14
kej 9y ago
You might appreciate one of my favorite Pi Day related comics, then: http://www.qwantz.com/index.php?comic=955 http://www.qwantz.com/index.php?comic=955
- tim_hutton 9y ago
  The value on pi approximation day (22/7) is closer to pi than the value on pi day (3.14).
- scrooched_moose 9y ago
  Also, my favorite pi related comic: https://www.smbc-comics.com/?id=1777 https://www.smbc-comics.com/?id=1777 It's not specific to pi-day, but it still seems appropriate.

simonbyrne 9y ago

The author of this post is the creator of the Julia visualization package Luxor.jl (https://github.com/JuliaGraphics/Luxor.jl https://github.com/JuliaGraphics/Luxor.jl). One thing I like about it is the nice turtle graphics feature (http://juliagraphics.github.io/Luxor.jl/stable/turtle.html http://juliagraphics.github.io/Luxor.jl/stable/turtle.html).