Computer Geometry: Rep-Tiles with a Hole

Abstract

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.

Fractals: An Eclectic Survey, Part-I

“…In [45] an example with m = 24 was given and in [46] a more sophisticated and interesting example with very large m is discussed. An interesting example of a new 8−rep tile with a hole was found very recently by Bandt and Mekhontsev in [33] (see Figure 13). A tile T is called a self-affine lattice tile if there is an affine expanding mapping g and a lattice L such that g preserves L and maps T to a union of tiles T + k i with k i ∈ L. With respect to the standard basis vectors e i , the map g has a matrix representation g(x) = Mx where M is an integer matrix.…”

“…We also refer to the recent works on fractal tilings by Bandt and Mekhontsev [32][33][34][35] for a comprehensive summary and current state of the art in the subject. The use of the inverses of integer matrices in the study of tilings is well-established, and can be found in particular in many of the references cited above.…”

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

“…With two or three pieces, we have few degrees of freedom, and only a small number of IFS with OSC, most of which are well known [35,36]. With eight or nine pieces, we already have a huge choice of parameters s k and v k which we cannot control even with a computer (see [6] for discussion of a similar case). A maximal number of five pieces is better to handle.…”

“…Computer studies with our package IFStile [3] revealed a huge number of new examples of this sort, with quite unexpected symmetries and properties. Starting with [4], see also [5,6], we have tried to present new classes of fractal sets in a systematic way. Here we focus on examples from quadratic number fields.…”

Tiling of polyhedral sets

“…It is worth mentioning that the the self-affine partitions of polyhedra have also been studied under less restrictive conditions, when not only parallel translations but also rotations are allowed [1,3,16,18].…”