Sierpinski carpet

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The Sierpinski carpet is a fractal that goes back to the Polish mathematician Wacław Sierpiński , who presented it in a first description in 1916. It is related to the Sierpinski triangle and the Menger sponge .

Construction sketch

One ninth of the area is removed from a square in the middle. One ninth of the area is removed from each of the eight square fields around the hole, and so on.

Sierpinski carpet:
Sierpinski carpet 0.svg Sierpinski carpet 1.svg Sierpinski carpet 2.svg Sierpinski carpet 3.svg Sierpinski carpet 4.svg Sierpinski carpet 5.svg
Level 0 step 1 Level 2 level 3 Level 4 Level 5

The fractal dimension of the Sierpinski carpet is  - in particular its area (in Lebesgue measure ) is zero.

The construction is very similar to the construction of the Cantor set , where the middle part is removed from a line, or the Sierpinski triangle , in which the middle part is removed from a triangle. In 3 dimensions, the construction of the Sierpinski carpet becomes the construction of the Menger sponge .

Area

The area of the (remaining) carpet can be as a result pose: If we assume that the side length of the original square is 1, then for the explicit representation and for the recursive representation , .

Computer program

The following Java applet draws a Sierpinski carpet using a recursive method:

import java.awt.*;
import java.applet.*;

public class SierpinskiCarpet extends Applet
{
    private Graphics g = null;
    private int d0 = 729; // 3^6

    public void init()
    {
        g = getGraphics();
        resize(d0, d0);
    }

    public void paint(Graphics g)
    {
        // Rekursion starten:
        drawSierpinskiCarpet (0, 0, getWidth(), getHeight() );
    }

    private void drawSierpinskiCarpet(int xOL, int yOL, int breite, int hoehe)
    {
        if (breite>2 && hoehe>2)
        {
            int b = breite/3;
            int h = hoehe/3;
            g.fillRect (xOL+b, yOL+h, b, h);
            for (int k=0; k<9; k++) if (k!=4)
            {
                int i=k/3;
                int j=k%3;
                drawSierpinskiCarpet (xOL+i*b, yOL+j*h, b, h); // Rekursion
            }
        }
    }
}

topology

In topology one considers the Sierpinski carpet as a subspace of the Euclidean metric provided . It represents a nowhere dense , locally coherent , metric continuum and is - together with the Sierpinski triangle - not least for this reason a particularly remarkable topological space .

literature

  • PS Alexandroff : Introduction to set theory and general topology (=  university books for mathematics . Volume 85 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1984.
  • Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer Spectrum, Berlin, Heidelberg 2015, ISBN 978-3-662-45460-2 , doi : 10.1007 / 978-3-662-45461-9 .

Individual evidence

  1. a b P. S. Alexandroff: Introduction to set theory and general topology. 1984, pp. 191-192.
  2. ^ Claudi Alsina, Roger B. Nelsen: Pearls of mathematics. 2015, pp. 225–226.

Web links

Commons : Sierpinski Carpet  - Album with pictures, videos and audio files