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