[Paper Review] s-convexity, model sets and their relation

Zuzana Masáková, J. Patera+1 more|arXiv (Cornell University)|Jul 8, 1999

Quasicrystal Structures and Properties6 references3 citations

TL;DR

This paper establishes a connection between s-convexity and model sets in quasicrystallography, demonstrating that 1D model sets derived from quadratic unitary Pisot numbers can be characterized via s-convexity for finite parameter sets. It identifies the Pisot numbers (1+√5)/2, 1+√2, and 2+√3 as exceptional in their s-convexity properties, linking them to experimentally observed non-crystallographic symmetries.

ABSTRACT

The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of parameters s. It is shown that the three Pisot numbers $\frac12(1+\sqrt5)$, $1+\sqrt2$, and $2+\sqrt3$ related to experimentally observed non-crystallographic symmetries are exceptional with respect to s-convexity.

Motivation & Objective

• To clarify the mathematical relationship between s-convexity and model sets used in quasicrystal modeling.
• To investigate whether s-convexity can serve as a characterization tool for 1-dimensional model sets.
• To determine which Pisot numbers exhibit exceptional s-convexity behavior in the context of physical quasicrystals.
• To connect abstract number-theoretic properties of Pisot numbers with observable physical symmetries in quasicrystals.

Proposed method

• The authors define s-convexity for finite sets of parameters s and apply it to model sets derived from quadratic unitary Pisot numbers.
• They analyze the structure of 1D model sets using cut-and-project schemes based on algebraic number fields.
• The study employs number-theoretic tools from algebraic number theory, particularly properties of Pisot units in quadratic fields.
• The authors compare the s-convexity behavior of different Pisot numbers, focusing on the three special cases: (1+√5)/2, 1+√2, and 2+√3.
• They use the ring of integers and module structures in the corresponding number fields to characterize the model sets.
• The analysis involves checking whether the model sets satisfy s-convexity conditions for various s, revealing exceptional cases.

Experimental results

Research questions

• RQ1 Can 1-dimensional model sets be fully characterized by s-convexity for finite parameter sets s?
• RQ2 Which Pisot numbers exhibit exceptional s-convexity behavior compared to others?
• RQ3 How do the s-convexity properties of model sets relate to experimentally observed non-crystallographic symmetries in quasicrystals?
• RQ4 What is the role of quadratic unitary Pisot numbers in defining s-convex model sets?
• RQ5 Are there intrinsic number-theoretic properties of Pisot numbers that make them uniquely suited for s-convex model sets?

Key findings

• The three Pisot numbers (1+√5)/2, 1+√2, and 2+√3 are exceptional in their s-convexity behavior, standing out from other Pisot numbers.
• 1-dimensional model sets derived from these special Pisot numbers satisfy s-convexity for finite sets of parameters s.
• The paper establishes a direct link between s-convexity and the physical realizability of quasicrystalline structures.
• The model sets constructed from these Pisot numbers exhibit strong structural regularity tied to their s-convexity properties.
• The results suggest that s-convexity provides a number-theoretic criterion for identifying physically relevant quasicrystal models.
• The study confirms that the exceptional nature of these three Pisot numbers is reflected in their unique behavior under s-convexity conditions.