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Related Concept Videos

Radicals01:27

Radicals

Roots, often written as radicals, identify the quantity that must be raised to a specific exponent to produce a given value. A radical expression consists of two main components: the radicand, which is the value placed inside the root symbol, and the index, which indicates the degree of the root being taken. The notation n√a indicates the principal nth root of a. If n equals 2, the operation is the square root, while n = 3 defines the cube root. When n is even, a negative radicand does not...
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Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific requirements are not imposed on the...
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Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c, where c is a constant.
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Fractal tiles associated with shift radix systems.

Valérie Berthé1, Anne Siegel, Wolfgang Steiner

  • 1LIRMM, CNRS UMR 5506, Université Montpellier II, 161 rue Ada, 34392 Montpellier Cedex 5, France ; LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France.

Advances in Mathematics
|September 27, 2013
PubMed
Summary
This summary is machine-generated.

Shift radix systems generalize numeration systems and are linked to fractal shapes. This study introduces fractal tiles for these systems, revealing complex tilings of space with potentially infinite tile shapes.

Keywords:
Beta expansionCanonical number systemShift radix systemTiling

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Area of Science:

  • Dynamical Systems
  • Number Theory
  • Fractal Geometry

Background:

  • Shift radix systems generalize established numeration systems like beta-expansions and canonical number systems.
  • These numeration systems are intrinsically linked to fractal structures, notably the Rauzy fractal and twin dragon.
  • Fractals play a crucial role in understanding the properties of number expansions.

Purpose of the Study:

  • To associate fractal tiles with shift radix systems.
  • To investigate the relationship between these new tiles and those from beta-expansions and canonical number systems.
  • To explore the tiling properties of these fractal tiles in d-dimensional spaces.

Main Methods:

  • Associating fractal tiles with shift radix systems based on parameter r.
  • Comparing these tiles with existing tiles from beta-expansions and canonical number systems.
  • Analyzing tiling properties under specific algebraic conditions on the parameter r.

Main Results:

  • Identified fractal tiles for shift radix systems, which align with existing tiles for specific parameter classes.
  • Demonstrated that these tiles can form multiple tilings of the d-dimensional real vector space.
  • Showcased that these tilings can be more complex, featuring infinitely many tile shapes and not necessarily being self-affine.

Conclusions:

  • Fractal tiles associated with shift radix systems offer a generalized framework for studying tiling phenomena.
  • The complexity of the resulting tilings highlights novel aspects compared to classical systems.
  • These findings extend the understanding of the interplay between number systems, fractals, and tiling theory.