Fibonacci and Lucas Sequences in Aperiodic Monotile Supertiles
Shiying Dong
TL;DR
This paper explores the role of Fibonacci, Lucas, and related sequences in the structure of aperiodic monotile supertiles, generalizing findings across the hat tile family.
Contribution
It introduces a generalized framework linking Fibonacci and Lucas sequences to aperiodic monotile supertiles, extending previous specific cases.
Findings
Fibonacci and Lucas sequences are integral to the structure of hat supertiles.
The results are generalized to all aperiodic tiles in the hat family.
The paper provides a mathematical foundation for understanding aperiodic tilings with monotiles.
Abstract
This paper first discusses the size and orientation of hat supertiles. Fibonacci and Lucas sequences, as well as a third integer sequence linearly related to the Lucas sequence are involved. The result is then generalized to any aperiodic tile in the hat family.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Quasicrystal Structures and Properties · semigroups and automata theory