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.

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

Rapid prototyping sculptures

George Hart is active using rapid prototyping (RP) technology for modeling sculptures.

Rapid Prototyping or Solid Freeform Fabrication refers to a range of new technologies which construct physical three-dimensional objects by assembling thin layers of material under computer control. Objects can be made which are extremely accurate, complex, and beautiful, and which no other technology can produce.

The image left shows he and his model of Sierpinski triangle.

He also interested in four-dimensional geometry. From a 4D object, one can calculate 3D "shadows" which are often beautiful but very complex objects.

The image below shows a 4D structure made of 120 regular dodecahedra. This "shadow" of it has the form of one large dodecahedron filled in with 119 smaller dodecahedra. In 4D all the dodecahedra are regular, but in this 3D shadow, angles are necessarily distorted, so only the innermost and outermost dodecahedra appear regular.

Even more beautiful and intricate is the truncated 120-cell, a 4D object made of 120 truncated dodecahedra and 600 tetrahedra.

Below is two variants of a woven assemblage of Salamanders, in homage to M.C. Escher. The left image is a rapid prototype and right image is a laser-cut wooden sculpture.