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)

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) }