Some π-ography for Pi day | Hacker News Reader

Some π-ography for Pi day

julialang.org

39 pointsViralBShah21 comments

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.

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.

Both log() in C and Math.log() in JS are natural logarithms, so there's no need to define your own ln() functions.

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.

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?

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

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

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.

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

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.

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().

You got the sign wrong.

e^(iπ) = −1

Ha! Serves me right for trying to comment from my phone. You're right, but I can no longer edit the comment.

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.

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.

The author of this post is the creator of the Julia visualization package 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).

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/

Pi day works in that format...

2018-03-14

You might appreciate one of my favorite Pi Day related comics, then: http://www.qwantz.com/index.php?comic=955

The value on pi approximation day (22/7) is closer to pi than the value on pi day (3.14).

Also, my favorite pi related comic: https://www.smbc-comics.com/?id=1777

It's not specific to pi-day, but it still seems appropriate.

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