Step by Step solution about How to resolve the algorithm Sierpinski triangle step by step in the PHP programming language

The provided PHP code generates a Sierpinski triangle, which is a fractal pattern that resembles a triangle and is constructed using a recursive algorithm. Here's a step-by-step explanation of the code:

1. Function Definition: The code defines a function named sierpinskiTriangle that takes an integer order as an argument. The order parameter determines the size of the triangle to be generated.

2. Initialization: Inside the function, several variables are initialized:

   • $char: This is the character used to represent the filled triangles in the pattern. It's set to # in this code.
   • $n: This variable calculates the height and width of the triangle based on the given order. The formula 1 << $order is used to compute $n.
   • $line: This is an array that represents each row of the triangle. It's initially filled with spaces (' ') and is of length 2 * $n.
   • The middle element of $line is set to the character $char, which marks the top of the triangle.

3. Triangle Generation: The main part of the code generates the Sierpinski triangle row by row. It enters a for loop that iterates from 0 to $n - 1:

   • In each iteration, it prints the current row stored in the $line array, followed by a newline character.
   • It then updates the $line array to generate the next row:
     • A variable $u is initialized with the value $char.
     • It iterates through elements in $line from index $n - $i to $n + $i. For each element:
       • It computes the value of $t based on the values of adjacent elements in $line. If the adjacent elements are the same, $t is set to ' '; otherwise, it's set to $char.
       • It updates the left adjacent element in $line with the previous value of $u.
       • It updates $u with the value of $t.
     • Finally, it updates the elements at indices $n + $i and $n + $i + 1 in $line to $t and $char, respectively.

4. Function Call: The function is called with sierpinskiTriangle(4), which means it will generate a Sierpinski triangle with an order of 4. The resulting pattern is printed to the console.

As you increase the order, the Sierpinski triangle becomes more complex and visually appealing. It's an example of how fractals, which are self-similar patterns, can be generated using recursive algorithms.

<?php

function sierpinskiTriangle($order) {
    $char = '#';
    $n = 1 << $order;
    $line = array();
    for ($i = 0 ; $i <= 2 * $n ; $i++) {
        $line[$i] = ' ';
    }
    $line[$n] = $char;
    for ($i = 0 ; $i < $n ; $i++) {
        echo implode('', $line), PHP_EOL;
        $u = $char;
        for ($j = $n - $i ; $j < $n + $i + 1 ; $j++) {
            $t = ($line[$j - 1] == $line[$j + 1] ? ' ' : $char);
            $line[$j - 1] = $u;
            $u = $t;
        }
        $line[$n + $i] = $t;
        $line[$n + $i + 1] = $char;
    }
}

sierpinskiTriangle(4);