Wednesday, April 30, 2008

Menger sponge

As the Sierpinski carpet is a generalization of the Cantor set from one dimension into two dimension, the Menger sponge is a generalization of the Sierpinski carpet into three dimensions. Sometimes this three-dimensional fractal called Menger-Sierpinski sponge or Sierpinski sponge. It was first described by Austrian mathematician Karl Menger in 1926.

Like the Sierpinski carpet begins from square, Menger sponge begins from cube. Every face of the cube is divided into 9 smaller squares. This operation divide the cube into 27 smaller cubes. Then center cubes from all faces and the inner center cube are removed, leaving 20. This is a level 1 Menger Sponge. The next levels forms by repeating these steps to all 20 cubes rest. Below you can see first four levels.

Below you can see the Menger sponge with cut off corner, which was designed by Seb Przd.

There's also similar three-dimensional fractal based on tetrahedron, which is a generalization of the Sierpinski triangle into three dimensions. Below, you can see two versions of the Sierpinski pyramid and the Menger sponge in a single image.

Saturday, April 26, 2008

Impossible triangle by Hans de Koning

Today I received a postage with new wooden work by Hans de Koning. It's a flat contruction of traditional Penrose tribar contructed from three kinds of wood.

Friday, April 18, 2008

Sierpinski carpet

The image above we see a portrait of Wacław Sierpiński, which was created by a student of Oberlin College Andrew Pike. It reminds us zoomed newspaper photos, when we can see particular dots of various size. But it's unusual image, because every element in it is not a simple dot, but one of several generations of the Sierpinski carpet fractal, which was first described by Wacław Sierpiński in 1916.

The forming of the Sierpinski carpet is like to forming of the Sierpinski triangle fractal, because the next generation of the fractal sets up by cutting removing elements from the source shape. Generation of the Sierpinski carpet begins from the square. Then it being divided into nine rectangles, and the center rectangle removed. This procedure continues for each of eight rest squares. You can see several first generations of the fractal on the image below.

Andrew Pike used two series of several generations of the fractal. The one series began from the black color, and another from white. He designed a computer program, which divided a photo of Wacław Sierpiński into squares of various values of grey color. To avoid strong color changing he used dithering technique.

So, the inventor of the fractal was pictured with his fractal.

The Sierpinski carpet is a two dimensional generalization of the one dimensional fractal Cantor dust. Also, there's generalization of the Sierpinski carpet into three dimensions, which named Menger sponge.

Friday, April 11, 2008

Escher's favorite building

A tower with very unusual shape in Beijing (China) will be completed for the Olympic games 2008. It's new China Central Television Tower (CCTV). It seems, that this building cannot exist in the our world, because it consists of two leaning towers, which are joined by a bridge with corner shape. The whole shape of the building seems like deformed square donut.

In 2002 two architects from Holland Rem Koolhaas and Ole Scheeren won an international competition for the CCTV tower and the project broke ground in September 2004. The project is even more complex because Beijing lies in an earthquake zone, and the tower is full of technical challenges.

Escher could like this paradoxical building of his fellow countrymen.