Tessellations of David Bailey

M.C. Escher was the first, who used figures of birds, fishes, lizards and other animals for artistic regular plane division. Many followers then created numerous tessellation images.

One of them is artist from England David Bailey. He creates his images in pen and watercolour.

The main motifs of his tessellations are birds.


The more complex constructions come in, when two distinct motifs are used in conjunction with each other.



Besides usual animals David Bailey uses imaginary creatures to create his wonderful artworks. Below you can see dinosaur-like creatures. The main distinguishing feature of this drawing is that only part of the image was used for animals, when other space shows only borders of elementary tiles. It helps to better understand, which kind of symmetry was used in every case.

Also he created several artworks, which reminds Escher's artworks with mutable tessellations like Metamorphoses. The image below shows the same process as in Escher's lithograph Development I. But David Bailey used birds as motif for his artwork instead of Escher's lizards.

The reverse process is shown in the image below.
More images by David Bailey you see at his personal site http://www.tess-elation.co.uk/. Besides this you you read some articles about tessellation art and Escher's artworks.

Fractal waves

On the image above we see artwork "The Great Wave off Kanagawa" by Japanese artist Hokusai, which was published in 1832 as the first in Hokusai's series 36 Views of Mount Fuji. It depicts an enormous wave threatening boats near the Japanese prefecture of Kanagawa; Mount Fuji can be seen in the background. The main reason of publishing this artwork here is highly detailed painted wave. As we know, some artworks, which are close to fractal images by detailed elaboration, were created long before the inventing fractals by Benoît Mandelbrot. Sea waves can be represented by many types of fractals, as you can see below.
Fractal artworks with waves are created by modern artists too. For example, Robert Fathauer depicted a wave fractal with Escher-like fish tiling in his artwork "Fractal Fish - Grouped Groupers" (2001).
Also, a wave with horses figures are depicted on the cover of English rock group Keane "Under The Iron Sea" (2006).

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.

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.

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