Fractal Reptiles of the Plane with Holes using Reflections and Rotations

A reptile in the plane is a compact set with non-empty interior that is made up of finitely many congruent tiles. In this chapter, we present a concise summary of existing general methods to obtain connected fractal reptiles with holes for any even integer n ≥ 4 and construct new examples of n − rep...

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1. Verfasser: Husain, Akhlaq
Format: Buchkapitel
Sprache:eng
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Zusammenfassung:A reptile in the plane is a compact set with non-empty interior that is made up of finitely many congruent tiles. In this chapter, we present a concise summary of existing general methods to obtain connected fractal reptiles with holes for any even integer n ≥ 4 and construct new examples of n − reptile with holes involving reflections and rotations. We also consider some variations of these reptiles and self-similar fractals as special cases and examples of reptiles with holes using integer matrices. This chapter presents a concise summary of existing general methods to obtain connected fractal reptiles with holes for any even integer n = 4 and construct new examples of n - reptile with holes involving reflections and rotations. The simplest examples of reptiles are convex polygons such as parallelogram and the right-angled isosceles triangles with side lengths in the ratio 1:2, and these are the only examples of 2-reptiles with this ratio. Therefore, the reptiles with holes are the reptiles having disconnected interiors. Jordan and Ngai proved a general result and described a geometric method to construct 2m-reptiles with holes for every positive integer m=2, which is given in the following theorem.
DOI:10.1201/9781003222255-9