Similar dissection of sets

Similar Dissection of Sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let D ⊂ R be a given set and let f1, . . . , fk be injective continu...

Finding Similar Sets

Computability of Self-Similar Sets

We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar is a recursive compact set. A further result on the case that the ...

Statistically Self - Similar Fractal Sets

In the present paper we define statistically self-similar sets, and, using a modification of method described in [2], [3], find a Hausdorff dimension of a statistically self-similar set.

On the classification of self-similar sets

We classify binary self-similar sets, which are compact sets determined by two contractions on the plane, into four classes from the viewpoint of functional equations. In this classification, we can not only show close relationships among functions with self-similarity but also give solutions to a few open problems in other field.