Geometric Properties of Aperiodic Monotile Supertiles Involving Fibonacci and Lucas Sequences

Core Concepts

The supervectors of the supertiles in the aperiodic hat family of monotiles can be expressed using Fibonacci and Lucas sequences, and a new integer sequence linearly related to the Lucas sequence governs the rotational angles of the supertiles.

Abstract

The paper discusses the geometric properties of the supertiles formed by the aperiodic hat and spectre monotiles. It first reviews the recursive rules for generating the supertiles, where each supertile is composed of multiple copies of the original tiles arranged in a specific pattern.
The author then derives a simple recurrence relation for the supervectors (𝑉𝑛) that describe the size and orientation of the supertiles. It is shown that 𝑉𝑛can be expressed in terms of the Fibonacci and Lucas sequences. Additionally, the author discovers a new integer sequence (𝐺𝑛) that is linearly related to the Lucas sequence and governs the rotational angles between successive generations of supertiles.
The results are first proven for the hat tiles and then generalized to any aperiodic tile in the hat family. The paper also discusses the fractal-like scaling and rotation properties of the supertiles in the limit of infinitely many generations.
Overall, the paper reveals deep connections between the geometric structure of aperiodic monotile supertiles and well-known integer sequences, suggesting rich combinatorial and mathematical insights into these fascinating tiling structures.

Stats

The short edge of the original hat tile is 1.
𝑉1 = (1, 3√3)
𝑉2 = (3, 7√3)
𝑉3 = (8, 18√3)
The first few terms of the 𝐺𝑛 sequence are:
3, 11, 67, 451, 3083, 21123, 144771, 992267, 6801091, 46615363, 319506443, 2189929731, 15010001667.

Quotes

"The supervectors of the supertiles take an elegant form involving Fibonacci and Lucas sequences, indicating yet another deep combinatorial meaning of aperiodic monotiles in the hat family."
"A new integer sequence linearly related to the Lucas sequence is in the rotational angles of the supertiles."

Key Insights Distilled From

Fibonacci and Lucas Sequences in Aperiodic Monotile Supertiles

by Shiying Dong at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19621.pdf

Deeper Inquiries

How can the insights from the geometric properties of aperiodic monotile supertiles be leveraged to develop new algorithms or techniques for tiling problems

The insights gained from the geometric properties of aperiodic monotile supertiles can be instrumental in developing new algorithms and techniques for tiling problems. By understanding the recursive nature of the supertiles and the relationship between their supervectors, one can create efficient algorithms for generating complex tilings. The recurrence relation discovered in the supervectors, such as 𝑉𝑛= 3𝑉𝑛−1 −𝑉𝑛−2, can be utilized to iteratively construct intricate tiling patterns. This recursive approach can be extended to develop algorithms for generating a wide variety of aperiodic tilings with different geometric properties. Additionally, the rotational angles and scaling factors observed in the supertiles can be leveraged to optimize tiling processes, ensuring seamless and aesthetically pleasing tile arrangements.

What other mathematical structures or sequences might be uncovered by further studying the combinatorial and fractal-like properties of these aperiodic monotile systems

Further studying the combinatorial and fractal-like properties of aperiodic monotile systems may uncover additional mathematical structures and sequences beyond the Fibonacci and Lucas sequences. Exploring the intricate relationships between the supervectors, rotational angles, and scaling factors of the supertiles could lead to the discovery of new mathematical patterns and sequences. For instance, investigating the behavior of the tangent of incremental rotational angles in the fractal limit could reveal novel mathematical properties that extend beyond the known sequences. By delving deeper into the geometric and combinatorial aspects of aperiodic monotile systems, mathematicians may uncover hidden mathematical structures that have applications in various fields of mathematics.

Could the connections between the supertile geometry and integer sequences like the Fibonacci and Lucas sequences lead to applications in fields beyond just tiling, such as in number theory or computer science

The connections between the supertile geometry and integer sequences like the Fibonacci and Lucas sequences have the potential to have applications beyond tiling, such as in number theory and computer science. The presence of these well-known integer sequences in the supervectors of the supertiles suggests a deeper connection between aperiodic monotiles and fundamental mathematical concepts. These connections could be leveraged to develop new algorithms for generating Fibonacci and Lucas sequences efficiently. Moreover, the discovery of a new integer sequence linearly related to the Lucas sequence in the rotational angles of the supertiles opens up possibilities for applications in number theory. Understanding how these integer sequences manifest in the geometric properties of aperiodic monotiles could lead to advancements in algorithm design, cryptography, and other computational fields that rely on number theory principles.