ON SELF-SIMILARITY OF QUADRANGLE

Jiajun Xu (), Jinghua Xi () and Kan Jiang
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Jiajun Xu: School of Mathematics and Statistics, Ningbo University, Ningbo, P. R. China
Jinghua Xi: Department of Mathematics, Southern University of Science and Technology, Shenzhen, P. R. China
Kan Jiang: School of Mathematics and Statistics, Ningbo University, Ningbo, P. R. China

FRACTALS (fractals), 2024, vol. 32, issue 05, 1-7

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.

Keywords: Self-Similar Set; Open Set Condition; Trapezoid; Concave Quadrangle (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X24500968

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