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.

The image by Andrew Pike was found at http://www.bridgesmathart.org/art-exhibits/jmm08/pike.html

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.

Wood work by Hans de Koning

Today I received a postage with wood work by Hans de Koning (see above). The shapes of the most of his works are based on impossible figures. His works are flat, but he uses different kinds of wood to make three-dimensional effect. Wood planks with different hues imitate sides slope and opacity of the impossible figure.

You can see more his work in his Picasa album and at the site Impossible World.

Fractal trees

Some time ago we have talked about Pythagoras tree, which represents simple fractal structure consisting of squares. Also, there are many variations of fractal trees, which are consist of lines and curves.

The three-dimensional fractal tree above is constructed from lines. It belongs to L-system class of fractals. Associated as trunk and branches brown lines of the tree are elements of low generations of the fractal. Green lines are elements of higher generations of the fractal. They remind us leafs. So, the whole fractal structure resembles real tree.

The rainbow fractal Julius tree below was crated with help of the computer program Fractal Imaginator. The tree reminds rounded Pythagoras tree, where squares were replaced to thin rectangles. The tree fractal can be created not only with help of straight lines or rectangles, but also with help of curves and spirals. Below, you can see a title for the High School Course "Gödel, Escher, Bach: A Mental Space Odyssey" by Justin Curry and Curran Kelleher, where curved fractal tree is used. The spiral was chosen as base element for this fractal, which gives many elegant curls.

Hilbert curve

Hilbert curve is a continuous fractal space filling curve, which was first described by German mathematician David Hilbert in 1891. Below you can see 5 first steps of the plane Hilbert curve.

But the Hilbert curve looks more interesting if it represent in three dimensions. Carlo H. Séquin, a professor of Berkley, created a small 5" metal sculpture of Hilbert curve, which he called "Hilbert 512". You can see it below.

It was also be modeled by Torolf Sauermann in the program Maxwell renderer. Below, you can see the second step of the three-dimensional Hilbert curve and two versions of cubic Hilbert curve.