ON SELF-SIMILARITY OF QUADRANGLE

Abstract

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC.