ORIGINAL RESEARCH article
Front. Phys.
Sec. Interdisciplinary Physics
This article is part of the Research TopicGolden Fractal Jubilee: 50 Years of Bridging Art and ScienceView all 5 articles
Fractal Rep-tiles of the Plane via Reflections and Integer Matrices
Provisionally accepted
- 1Qassim University, Buraydah, Saudi Arabia
- 2Jamia Millia Islamia Faculty of Sciences, New Delhi, India
- 3University of Delhi, New Delhi, India
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Fractal rep-tiles of the Euclidean plane are considered in this article which are essentially exam-ples of rep-tiles (tilings) having fractal boundaries. Several new examples of fractal rep-tiles are constructed using reflection transformations and integer matrices. A new class of foldable fractal rep-tiles is introduced which is based on general reflection mappings and it is shown that these rep-tiles tile the plane using similitudes including reflections if the IFS satisfies lattice tiling condi-tions. We prove the existence of foldable fractal 2-rep and 4 rep-tiles that exhibit novel properties (chirality and aperiodicity) caused by reflection mappings. Fractal variations of foldable rep-tiles are also constructed. The fractal rep-tiles and the foldable rep-tiles presented here are in one-to-one correspondence with finite reflection groups and this novel class of foldable rep-tiles can be lifted to construct new classes of fractal rep-tiles having roots in classical reflections groups. The images of rep-tiles are rendered using the random iteration algorithm which is a popular iterative method to generate self-similar fractals and tilings.
Keywords: Algorithms, Foldablerep-tiles, Fractal rep-tiles, Integer matrices, Iterated function system, Iterative method, Reflections
Received: 05 Sep 2025; Accepted: 20 Jan 2026.
Copyright: © 2026 Sajid, Husain and Kumar. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Akhlaq Husain
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