Related Experiment Video
Updated: Jul 13, 2026
16:24
Analyzing and Building Nucleic Acid Structures with 3DNA
Published on: April 26, 2013
Angular structure of lacunarity, and the renormalization group
1Department of Physics, University of Warwick, CV4 7AL Coventry, United Kingdom.
Physical Review Letters
|December 2, 2000
Summary
We introduce angular structure of lacunarity in fractals, providing new evidence for the equivalence between self-avoiding walks and percolation perimeters. This fractal analysis reveals renormalization group behavior in real space.
Area of Science:
- Fractal Geometry
- Statistical Physics
- Universality
Background:
- Lacunarity is a measure of the heterogeneity or 'gappiness' of fractal patterns.
- Understanding the angular structure of lacunarity can offer deeper insights into fractal properties.
Purpose of the Study:
- To formulate the angular structure of lacunarity in fractals.
- To investigate the relationship between lacunarity, universality, and renormalization group concepts.
- To provide new evidence for the equivalence between self-avoiding walks and percolation perimeters.
Main Methods:
- Symmetry reduction of the three-point correlation function.
- Exact calculations for random walks.
- Analysis of measured data for percolation clusters and self-avoiding walks.
Main Results:
- The angular structure of lacunarity serves as a probe of universality.
- New evidence supports the equivalence between self-avoiding walks (SAWs) and percolation perimeters in 2D.
- Lacunarity reveals aspects of the renormalization group in real space.
Conclusions:
- The formulation of angular lacunarity provides a powerful tool for fractal analysis.
- The study confirms the connection between different fractal models and renormalization group theory.
- The findings offer a general test for long-range interaction contributions in fractal systems.
Related Concept Videos
02:54Trends in Lattice Energy: Ion Size and Charge
An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:
02:22Structures of Solids
Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
02:33Lattice Centering and Coordination Number
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
Types of Unit Cells
Imagine taking a large number of identical...
02:06Newman Projections
Different notations are used to represent the three-dimensional structure of molecules on two-dimensional surfaces. One of the most commonly used representations is the dash-wedge formula. The dashed wedges, solid wedges, and the plane lines indicate the groups situated behind the plane, coming out of the plane, and in the plane, respectively.
The organic molecules rotate across the single bonds leading to numerous temporary three-dimensional structures of varying energy known as conformers.
The organic molecules rotate across the single bonds leading to numerous temporary three-dimensional structures of varying energy known as conformers.
01:18Deformations in a Symmetric Member in Bending
When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress...
When the member is segmented into tiny cubic elements, it is observed that the primary stress...
01:29Law of Rational Indices
The Law of rational indices is a fundamental principle in the field of crystallography. According to this law, the intercepts of a crystal face along the crystallographic axes (the three-dimensional axes along which a crystal is measured) can be expressed as either equivalent to the unit intercepts (a, b, c) or simple whole number multiples of them. These multiples are typically denoted as na, n'b, and n''c, where n, n', and n'' are simple whole numbers.To illustrate, consider a crystal with...