Question

The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. Though the Sierpinski triangle looks complex, it can be generated with a short recursive function. Your main task is to write a recursive function Sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: Sierpinski() should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth.
API specification:
When writing your program, exercise modular design by organizing it into four functions, as specified in the following API:

public class Sierpinski {

// Height of an equilateral triangle whose sides are of the specified length.
public static double height(double length)

// Draws a filled equilateral triangle whose bottom vertex is (x, y)
// of the specified side length.
public static void filledTriangle(double x, double y, double length)

// Draws a Sierpinski triangle of order n, such that the largest filled
// triangle has bottom vertex (x, y) and sides of the specified length.
public static void Sierpinski(int n, double x, double y, double length)

// Takes an integer command-line argument n;
// draws the outline of an equilateral triangle (pointed upwards) of length 1;
// whose bottom-left vertex is (0, 0) and bottom-right vertex is (1, 0); and
// draws a Sierpinski triangle of order n that fits snugly inside the outline.
public static void main(String[] args)
}

Restrictions: You may not change either the scale or size of the drawing window.
Observe the following rules:

DO NOT use System.exit().
DO NOT add the project or package statements.
DO NOT change the class name.
DO NOT change the headers of ANY of the given methods.
DO NOT add any new class fields.
ONLY display the result as specified by the example for each problem.
DO NOT print other messages, follow the examples for each problem.
USE StdIn, StdOut, and StdDraw libraries.

Answer 1

Answer:

/ Sierpinski.java

public class Sierpinski {

     // method to find the height of an equilateral triangle with side length =

     // length

     public static double height(double length) {

           // formula= length*sqrt(3)/2

           double h = (length * Math.sqrt(3)) / 2.0;

           return h;

     }

     // method to draw a filled triangle (pointed downwards) with bottom vertex

     // x,y

     public static void filledTriangle(double x, double y, double length) {

           // finding height

           double h = height(length);

           // filling triangle as a polygon

           // passing x, x-length/2, x+length/2 as x coordinates

           // and y, y+h, y+h as y coordinates

           StdDraw.filledPolygon(new double[] { x, x - (length / 2.0),

                       x + (length / 2.0) }, new double[] { y, y + h, y + h });

     }

     // method to draw an n level sierpinski triangle

     public static void sierpinski(int n, double x, double y, double length) {

           // checking if n is above 0 (base condition)

           if (n > 0) {

                 // drawing a filled triangle with x, y as bottom coordinate, length

                 // as side length

                 filledTriangle(x, y, length);

                 // drawing the triangle(s) on the top

                 sierpinski(n - 1, x, y + height(length), length / 2);

                 // drawing the triangle(s) on the left side

                 sierpinski(n - 1, x - (length / 2.0), y, length / 2);

                 // drawing the triangle(s) on the right side

                 sierpinski(n - 1, x + (length / 2.0), y, length / 2);

           }

     }

     public static void main(String[] args) {

           // parsing first command line argument as integer, if you dont provide

           // the value while running the program, this program will cause

           // exception.

           int n = Integer.parseInt(args[0]);

           // length of the outline triangle

           double length = 1;

           // finding height

           double h = height(length);

           // drawing a triangle (pointed upwards) to represent the outline.

           StdDraw.polygon(new double[] { 0, length / 2, length }, new double[] {

                       0, h, 0 });

           // drawing n level sierpinski triangle(s) with length / 2, 0 as x,y

           // coordinates and length / 2 as initial side length

           sierpinski(n, length / 2, 0, length / 2);

     }

}