Approximate numbers e and pi using combinatorics and Fibonacci.

e.php

<?php
/**
 * Find the number e (2.7182818284590452354 => M_E constant in PHP).
 *
 * To find the number e using permutations:
 *  1: Generate a set of size S of random values,
 *  2: Find the number of permutations of that set, let it be #P0,
 *  3: With all the permutations of this set, let #P1 the count of the permutations where
 *     any items are not at the same index as in the original input,
 *  4: Divide #P0 by #P1, that's your approximation,
 *  5: Restart at 1 and increase the size of the set.
 *
 * Requirements:
 *   - https://packagist.org/packages/drupol/phpermutations
 *
 * Install:
 *   1: Git clone phpermutations,
 *   2: Run composer install,
 *   3: Create this file at the root of the project,
 *   4: Run php e.php.
 */

include "./vendor/autoload.php";

for ($i = 2; $i < 15; $i++) {
  $input = range(1, $i);
  $size = $i;

  echo "**********************************************************" . "\n";
  echo "Size: " . $i . "\n";

  $Permutations = new \drupol\phpermutations\Generators\Permutations($input, $size);

  $j = 0;
  foreach ($Permutations->generator() as $permutation) {
    if (!($array3 = array_intersect_assoc($input, $permutation))) {
      $j++;
    }
  }

  echo "Number of permutations (#P0): " . $Permutations->count() . "\n";
  echo "Number of permutations without item at the same place (#P1): " . $j . "\n";
  echo "(#P0) / (#P1): " . $Permutations->count()/$j . "\n";
}

pi.php

<?php

/**
 * Find the number pi (3.1415926535898 => M_PI constant in PHP).
 *
 * π/4 = arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/21) + ...
 *
 * You'll have already noticed the Fibonacci numbers here.
 * However, not all the Fibonacci numbers appear on the left hand sides.
 * For instance, we have no expansion for arctan(1/5) nor for arctan(1/13).
 * Only the even numbered Fibonacci terms seem to be expanded:
 * (F2=1, F4=3, F6=8, F8=21, ...)
 *
 * To find the number pi using Fibonacci:
 *  1: Generate a set of size S of Fibonacci numbers,
 *  2: Apply on each item of the set a function that will compute the inverse
 *     of the item,
 *  3: Apply on each item of the set the atan() function and multiply by 4,
 *  4: Filter out values where the key value is odd,
 *  3: Sum all the terms of the array.
 *
 * Requirements:
 *   - https://packagist.org/packages/drupol/phpermutations
 *
 * Install:
 *   1: Git clone phpermutations,
 *   2: Run composer install,
 *   3: Create this file at the root of the project,
 *   4: Run php pi.php.
 */
include "./vendor/autoload.php";

$Fibonacci = new \drupol\phpermutations\Generators\Fibonacci();

for ($i = 2; $i <= 50; $i = $i+2) {
  $gen = $Fibonacci->generator();

  $input = array();
  for ($s = 0; $s <= $i; $s++) {
    $input[] = $gen->current();
    $gen->next();
  }

  $input = array_sum(array_filter(array_map(function($item) {
    return 4 * atan(1/$item);
  }, array_slice($input, 3)), function($key) {
    return !($key%2);
  }, ARRAY_FILTER_USE_KEY));

  echo "**********************************************************" . "\n";
  echo "Size: " . $i . "\n";
  echo "Pi estimation: " . $input . "\n";
}