By slight modification of the data of the Sierpiński gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus, the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.

The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.

We define G-symmetric polygonal systems of similarities and study the properties of symmetric dendrites, which appear as their attractors. This allows us to find the conditions under which the attractor of a zipper becomes a dendrite.

A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some problems of computer-assisted research in geometry.

Part. Part. Syst. Charact. 23 (2006) 254-259 * E. I. Palchikov. Professor, PhD, Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Lavrentyev pr., 15, Novosibirsk 630090 (Russia). AbstractThe possibility of applying a digital X-ray laminography method to the example of mapping the fluid distribution in porous media is studied. The study is performed on the basis of a system which models the process of oil displaced by water and gas as occurs in oil field development. Two methods of tomographic section recovery are discussed: geometric and algebraic, and their application for different specimen structures is made. It is shown that when scanning with steps of 25 lm, 100 lm, 250 lm, or 2 mm it is possible to determine the distribution of concentrations of heterogeneities for multiphase fluid inside a porous core sample layer, giving advantages compared to simple shadow pictures.