Preliminary Evidence for a Theory of the Fractal City

1996 ◽  
Vol 28 (10) ◽  
pp. 1745-1762 ◽  
Author(s):  
M Batty ◽  
Y Xie
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Scaling Laws ◽  
Fractal Dimensions ◽  
Power Laws ◽  
Preliminary Evidence ◽  
Residential Development ◽  
Self Similarity ◽  
Estimation Techniques ◽  
Data Source

In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the fractal city outlined in books by Batty and Longley and Frankhauser, but with the clear conclusion that different scale and estimation techniques generate different types of fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Rock Mechanical Property Influenced by Inhomogeneity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yunliang Tan ◽  
Dongmei Huang ◽  
Ze Zhang
Keyword(s):  
Mechanical Property ◽  
Fractal Dimension ◽  
Rock Type ◽  
Fractal Dimensions ◽  
Rock Material ◽  
The Self ◽  
Self Similarity ◽  
Red Sandstone ◽  
Variation Patterns ◽  
Rock Microstructure

In order to identify the microstructure inhomogeneity influence on rock mechanical property, SEM scanning test and fractal dimension estimation were adopted. The investigations showed that the self-similarity of rock microstructure markedly changes with the scanned microscale. Different rocks behave in different fractal dimension variation patterns with the scanned magnification, so it is conditional to adopt fractal dimension to describe rock material. Grey diabase and black diabase have high suitability; red sandstone has low suitability. The suitability of fractal-dimension-describing method for rocks depends on both investigating scale and rock type. The homogeneities of grey diabase, black diabase, grey sandstone, and red sandstone are 7.8, 5.7, 4.4, and 3.4, separately; their average fractal dimensions of microstructure are 2.06, 2.03, 1.72, and 1.40 correspondingly, so the homogeneity is well consistent with fractal dimension. For rock material, the stronger brittleness is, the less profile fractal dimension is. In a sense, brittleness is an image of rock inhomogeneity in macroscale, while profile fractal dimension is an image of rock inhomogeneity in microscale. To combine the test of brittleness with the estimation of fractal dimension with condition will be an effective approach for understanding rock failure mechanism, patterns, and behaviours.

scholarly journals Exploring the Fractal Parameters of Urban Growth and Form with Wave-Spectrum Analysis

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Yanguang Chen
Keyword(s):  
Fractal Dimension ◽  
Urban Growth ◽  
Urban Form ◽  
Wave Spectrum ◽  
Fractal Dimensions ◽  
Wave Number ◽  
Similar Distribution ◽  
Image Dimension ◽  
Radial Dimension ◽  
The Fourier Transform

The Fourier transform and spectral analysis are employed to estimate the fractal dimension and explore the fractal parameter relations of urban growth and form using mathematical experiments and empirical analyses. Based on the models of urban density, two kinds of fractal dimensions of urban form can be evaluated with the scaling relations between the wave number and the spectral density. One is theradial dimensionof self-similar distribution indicating the macro-urban patterns, and the other, the profile dimension of self-affine tracks indicating the micro-urban evolution. If a city's growth follows the power law, the summation of the two dimension values may be a constant under certain condition. The estimated results of the radial dimension suggest a new fractal dimension, which can be termed “image dimension”. A dual-structure model namedparticle-ripple model(PRM) is proposed to explain the connections and differences between the macro and micro levels of urban form.

FRACTAL DIMENSION AND LACUNARITY OF TRACTOGRAPHY IMAGES OF THE HUMAN BRAIN

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 181-189 ◽  
Author(s):  
P. KATSALOULIS ◽  
D. A. VERGANELAKIS ◽  
A. PROVATA
Keyword(s):  
Fractal Dimension ◽  
Human Brain ◽  
Fractal Dimensions ◽  
Resonance Imaging ◽  
Self Similarity ◽  
Healthy Human ◽  
Box Counting ◽  
Self Similar ◽  
Box Method ◽  
The Brain

Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The two-dimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately Df = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures.

A Modified Box-Counting Method to Estimate the Fractal Dimensions

2011 ◽  
Vol 58-60 ◽  
pp. 1756-1761 ◽  
Author(s):  
Jie Xu ◽  
Giusepe Lacidogna
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
The Self ◽  
Whole Body ◽  
Counting Method ◽  
Border Effect ◽  
Self Similarity ◽  
Box Counting ◽  
Self Similar ◽  
Box Counting Method

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

scholarly journals Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy

Biophysica ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 59-69
Author(s):  
Liam Elkington ◽  
Prakash Adhikari ◽  
Prabhakar Pradhan
Keyword(s):  
Fractal Dimension ◽  
Optical Microscopy ◽  
Tissue Sample ◽  
Mass Density ◽  
Fractal Dimensions ◽  
Distribution Patterns ◽  
Biological Tissues ◽  
Self Similarity ◽  
Dimension Measurement ◽  
Prostate Cancers

Fractal dimension, a measure of self-similarity in a structure, is a powerful physical parameter for the characterization of structural property of many partially filled disordered materials. Biological tissues are fractal in nature and reports show a change in self-similarity associated with the progress of cancer, resulting in changes in their fractal dimensions. Here, we report that fractal dimension measurement is a potential technique for the detection of different stages of cancer using transmission optical microscopy. Transmission optical microscopy of a thin tissue sample produces intensity distribution patterns proportional to its refractive index pattern, representing its mass density distribution. We measure fractal dimension detection of different cancer stages and find its universal feature. Many deadly cancers are difficult to detect in their early to different stages due to the hard-to-reach location of the organ and/or lack of symptoms until very late stages. To study these deadly cancers, tissue microarray (TMA) samples containing different stages of cancers are analyzed for pancreatic, breast, colon, and prostate cancers. The fractal dimension method correctly differentiates cancer stages in progressive cancer, raising possibilities for a physics-based accurate diagnosis method for cancer detection.

scholarly journals On the sedimentological character of Alpine basal ice facies

1996 ◽  
Vol 22 ◽  
pp. 187-193 ◽  
Author(s):  
Bryn Hubbard ◽  
Martin Sharp ◽  
Wendy J. Lawson
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Size Fraction ◽  
Fractal Dimensions ◽  
Western Alps ◽  
Similar Distribution ◽  
Self Similarity ◽  
Apparent Contradiction ◽  
Basal Ice ◽  
Self Similar

Seven basal ice facies have been defined on the basis of research at eleven glaciers in the western Alps. The concentration and texture of the debris incorporated in these facies are described. Grain-size distributions are characterised in terms of their: (i) mean size and dispersion, (ii) component Gaussian modes, and (iii) self-similarity.Firnified glacier ice contains low concentrations (≈0.2 g 1−1) of well-sorted and predominantly fine-grained debris that is not self-similar over the range of particle diameters assessed. In contrast, basal ice contains relatively high concentrations (≈4–4000 g 1−1 by facies) of polymodal (by size fraction against weight) debris, the texture of which is consistent with incorporation at the glacier bed. Analysis by grain-size against number of particles suggests that these basal facies debris textures are also self-similar. This apparent contradiction may be explained by the insensitivity of the assessment of self-similarity to variations in mass distribution. Comparison of typical size–weight with size–number distributions indicates that neither visual nor statistical assessment of the latter may be sufficiently rigorous to identify self-similarity.Apparent fractal dimensions may indicate the relative importance of fines in a debris distribution. Subglacially derived basal facies debris has a mean fractal dimension of 2.74. This value suggests an excess of fines relative to a self-similar distribution of cubes, which has a fractal dimension of 2.58. Subglacial sediments sampled from the forefield of Skalafellsjökull, Iceland, have fractal dimensions of 2.91 (A-horizon) and 2.81 (B-horizon). Debris from the A-horizon, which is interpreted as having been pervasively deformed, both most closely approaches self-similarity and has the highest fractal dimension of any of the sample groups analyzed.

Box-Counting Dimension of Fractal Urban Form

2012 ◽  
Vol 3 (3) ◽  
pp. 41-63 ◽  
Author(s):  
Shiguo Jiang ◽  
Desheng Liu
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Fractal Dimensions ◽  
Reliable Estimate ◽  
Box Dimension ◽  
Beijing City ◽  
Box Counting ◽  
Window Method ◽  
Box Counting Dimension ◽  
Data Points

The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes; 3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these issues in the case of Beijing City. The authors propose corresponding improved techniques with modified measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then applied to real city data. The results show that a reliable estimate of box dimension for urban form can be obtained using their method.

Fractal Dimensions in Elastic-Plastic Beam Dynamics

1993 ◽  
Author(s):  
P. S. Symonds ◽  
Jae-Yeong Lee
Keyword(s):  
Fractal Dimension ◽  
Short Pulse ◽  
Fractal Dimensions ◽  
Great Sensitivity ◽  
Beam Model ◽  
Transverse Loading ◽  
Self Similarity ◽  
Chaotic Vibrations ◽  
Two Degree Of Freedom ◽  
Initial Impulse

Abstract The final midpoint displacement of a two-degree-of-freedom beam model subjected to a short pulse of transverse loading may be either in the direction of the initial impulse or in the opposite (“negative”) direction, when moderately small plastic deformations occur. In the range where chaotic vibrations occur, the result depends with great sensitivity on the impulse magnitude. Considering a pulse of duration 0.5 × 10−3 sec, 100 calculations have been made for pulse forces P starting at 2500 N and increasing by increments of 2.0, 10−2, 10−4, and 10−6 N. It is found that the proportion and distribution of negative final displacements remain, on average, the same, independent of the size of the force increment. A fractal dimension representing a self-similarity property is calculated for the four choices of the force increment, and is found to be approximately 0.78 in each case. A correlation fractal dimension is also computed for undamped responses.

scholarly journals On the sedimentological character of Alpine basal ice facies

1996 ◽  
Vol 22 ◽  
pp. 187-193 ◽  
Author(s):  
Bryn Hubbard ◽  
Martin Sharp ◽  
Wendy J. Lawson
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Size Fraction ◽  
Fractal Dimensions ◽  
Western Alps ◽  
Similar Distribution ◽  
Self Similarity ◽  
Apparent Contradiction ◽  
Basal Ice ◽  
Self Similar

Seven basal ice facies have been defined on the basis of research at eleven glaciers in the western Alps. The concentration and texture of the debris incorporated in these facies are described. Grain-size distributions are characterised in terms of their: (i) mean size and dispersion, (ii) component Gaussian modes, and (iii) self-similarity.Firnified glacier ice contains low concentrations (≈0.2 g 1−1) of well-sorted and predominantly fine-grained debris that is not self-similar over the range of particle diameters assessed. In contrast, basal ice contains relatively high concentrations (≈4–4000 g 1−1by facies) of polymodal (by size fraction against weight) debris, the texture of which is consistent with incorporation at the glacier bed. Analysis by grain-size against number of particles suggests that these basal facies debris textures are also self-similar. This apparent contradiction may be explained by the insensitivity of the assessment of self-similarity to variations in mass distribution. Comparison of typical size–weight with size–number distributions indicates that neither visual nor statistical assessment of the latter may be sufficiently rigorous to identify self-similarity.Apparent fractal dimensions may indicate the relative importance of fines in a debris distribution. Subglacially derived basal facies debris has a mean fractal dimension of 2.74. This value suggests an excess of fines relative to a self-similar distribution of cubes, which has a fractal dimension of 2.58. Subglacial sediments sampled from the forefield of Skalafellsjökull, Iceland, have fractal dimensions of 2.91 (A-horizon) and 2.81 (B-horizon). Debris from the A-horizon, which is interpreted as having been pervasively deformed, both most closely approaches self-similarity and has the highest fractal dimension of any of the sample groups analyzed.

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