scholarly journals The fractal dimension of the tree of life

2014 ◽  
Author(s):  
Xiaofei Lv ◽  
Yuping Wu ◽  
Bin Ma
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Fractal Dimensions ◽  
Predictive Modelling ◽  
Tree Of Life ◽  
Fundamental Question ◽  
Theoretical Studies ◽  
Structure Pattern ◽  
Intermediate Size ◽  
Taxonomic Groups

The structure pattern of the tree of life clues on the key ecological issues; hence knowing the fractal dimension is the fundamental question in understanding the tree of life. Yet the fractal dimension of the tree of life remains unclear since the scale of the tree of life has hypergrown in recent years. Here we show that the tree of life display a consistent power-law rules for inter- and intra-taxonomic levels, but the fractal dimensions were different among different kingdoms. The fractal dimension of hierarchical structure (Dr) is 0.873 for the entire tree of life, which smaller than the values of Dr for Animalia and Plantae but greater than the values of Dr for Fungi, Chromista, and Protozoa. The hierarchical fractal dimensions values for prokaryotic kingdoms are lower than for other kingdoms. The Dr value for Viruses was lower than most eukaryotic kingdoms, but greater than prokaryotes. The distribution of taxa size is governed by fractal diversity but skewed by overdominating taxa with large subtaxa size. The proportion of subtaxa in taxa with small and large sizes was greater than in taxa with intermediate size. Our results suggest that the distribution of subtaxa in taxa can be predicted with fractal dimension for the accumulating taxa abundance rather than the taxa abundance. Our study determined the fractal dimensions for inter- and intra-taxonomic levels of the present tree of life. These results emphases the need for further theoretical studies, as well as predictive modelling, to interpret the different fractal dimension for different taxonomic groups and skewness of taxa with large subtaxa size.

scholarly journals The fractal dimension of the tree of life

2014 ◽  
Author(s):  
Bin Ma ◽  
Xiaofei Lv ◽  
Jun Gong
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Fractal Dimensions ◽  
Predictive Modelling ◽  
Tree Of Life ◽  
Fundamental Question ◽  
Theoretical Studies ◽  
Structure Pattern ◽  
Intermediate Size ◽  
Taxonomic Groups

The structure pattern of the tree of life clues on the key ecological issues; hence knowing the fractal dimension is the fundamental question in understanding the tree of life. Yet the fractal dimension of the tree of life remains unclear since the scale of the tree of life has hypergrown in recent years. Here we show that the tree of life display a consistent power-law rules for inter- and intra-taxonomic levels, but the fractal dimensions were different among different kingdoms. The fractal dimension of hierarchical structure (Dr) is 0.873 for the entire tree of life, which smaller than the values of Dr for Animalia and Plantae but greater than the values of Dr for Fungi, Chromista, and Protozoa. The hierarchical fractal dimensions values for prokaryotic kingdoms are lower than for other kingdoms. The Dr value for Viruses was lower than most eukaryotic kingdoms, but greater than prokaryotes. The distribution of taxa size is governed by fractal diversity but skewed by overdominating taxa with large subtaxa size. The proportion of subtaxa in taxa with small and large sizes was greater than in taxa with intermediate size. Our results suggest that the distribution of subtaxa in taxa can be predicted with fractal dimension for the accumulating taxa abundance rather than the taxa abundance. Our study determined the fractal dimensions for inter- and intra-taxonomic levels of the present tree of life. These results emphases the need for further theoretical studies, as well as predictive modelling, to interpret the different fractal dimension for different taxonomic groups and skewness of taxa with large subtaxa size.

scholarly journals The fractal dimension of the tree of life

2014 ◽  
Author(s):  
Xiaofei Lv ◽  
Yuping Wu ◽  
Bin Ma
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Fractal Dimensions ◽  
Predictive Modelling ◽  
Tree Of Life ◽  
Fundamental Question ◽  
Theoretical Studies ◽  
Structure Pattern ◽  
Intermediate Size ◽  
Taxonomic Groups

The structure pattern of the tree of life clues on the key ecological issues; hence knowing the fractal dimension is the fundamental question in understanding the tree of life. Yet the fractal dimension of the tree of life remains unclear since the scale of the tree of life has hypergrown in recent years. Here we show that the tree of life display a consistent power-law rules for inter- and intra-taxonomic levels, but the fractal dimensions were different among different kingdoms. The fractal dimension of hierarchical structure (Dr) is 0.873 for the entire tree of life, which smaller than the values of Dr for Animalia and Plantae but greater than the values of Dr for Fungi, Chromista, and Protozoa. The hierarchical fractal dimensions values for prokaryotic kingdoms are lower than for other kingdoms. The Dr value for Viruses was lower than most eukaryotic kingdoms, but greater than prokaryotes. The distribution of taxa size is governed by fractal diversity but skewed by overdominating taxa with large subtaxa size. The proportion of subtaxa in taxa with small and large sizes was greater than in taxa with intermediate size. Our results suggest that the distribution of subtaxa in taxa can be predicted with fractal dimension for the accumulating taxa abundance rather than the taxa abundance. Our study determined the fractal dimensions for inter- and intra-taxonomic levels of the present tree of life. These results emphases the need for further theoretical studies, as well as predictive modelling, to interpret the different fractal dimension for different taxonomic groups and skewness of taxa with large subtaxa size.

scholarly journals The fractal dimension of the tree of life

2014 ◽  
Author(s):  
Bin Ma ◽  
Xiaofei Lv ◽  
Jun Gong
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Low Frequency ◽  
Predictive Modelling ◽  
Tree Of Life ◽  
Fundamental Question ◽  
Theoretical Studies ◽  
Structure Pattern ◽  
Intermediate Size ◽  
Taxonomic Groups

The structure pattern of the tree of life clues on the key ecological issues; hence knowing the fractal dimension is the fundamental question in understanding the tree of life. Yet the fractal dimension of the tree of life remains unclear since the scale of the tree of life has hypergrown in recent years. Here we show that the tree of life displays a consistent power-law rules for inter- and intra-taxonomic levels. The values of fractal dimension for both inter- and intra-taxonomic levels were different among different kingdoms. The distribution of taxa size is governed by fractal diversity but skewed by overdominating taxa with low frequency; the proportion of subtaxa in taxa with small and large sizes was greater than in taxa with intermediate size. Our results suggest that the abundance of subtaxa in taxa with small and large sizes can be predicted with fractal dimension for the accumulating taxa abundance rather than the taxa abundance. These results emphasize the need for further theoretical studies, as well as predictive modelling, to interpret the different fractal dimension for different taxonomic groups.

ANALYSIS AND NUMERICAL COMPUTATION OF THE DIMENSION OF COLORED NOISE AND DETERMINISTIC TIME SERIES WITH POWER-LAW SPECTRA

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 53-64
Author(s):  
WEN ZHANG ◽  
BRUCE J. WEST
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Numerical Computation ◽  
Trigonometric Series ◽  
Power Law ◽  
Colored Noise ◽  
Fractal Dimensions ◽  
Box Dimension ◽  
Inverse Power Law ◽  
High Degree

We investigate the box dimension of a graph of time series generated by trigonometric series with an inverse power-law spectrum. Such time series can either be random, in which case it is called colored noise, or deterministic in nature. We show analytically that both series can have fractal dimensions depending on the value of the exponent in the power-law. However, the fractal dimension of colored noise is 0.5 higher than that of the corresponding deterministic series. We comment on calculating fractal dimensions and present a reliable numerical algorithm which yields a high degree of consistency between experimental and analytical results.

Light-scattering studies of aggregation

1989 ◽  
Vol 423 (1864) ◽  
pp. 89-102 ◽  
Keyword(s):  
Fractal Dimension ◽  
Light Scattering ◽  
Dynamic Light Scattering ◽  
Power Law ◽  
Fractal Geometry ◽  
Linear Growth ◽  
Fractal Dimensions ◽  
Power Law Model ◽  
Aggregate Distribution ◽  
Scattering Measurements

We discuss dynamic and static light-scattering measurements made during slow (reaction-limited) aggregation of model colloids and immune complex forming proteins. Analysis of the results leads to an understanding of the random aggregates formed in terms of a fractal geometry and measurement of the fractal dimension. Differences in the measured fractal dimensions of the model and protein systems are discussed. The aggregation appears to follow ‘Smoluchowski-like’ kinetics as measured by a near linear growth of the low-angle light scattering with time. However, the dynamic light-scattering results support a simple power-law model for the aggregate distribution and allow an estimate of this power law to be made.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using 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 Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Fractal dimension based features are useful descriptors of leaf complexity and shape

1999 ◽  
Vol 29 (9) ◽  
pp. 1301-1310 ◽  
Author(s):  
Wojciech Borkowski
Keyword(s):  
Fractal Dimension ◽  
Tree Species ◽  
Fractal Dimensions ◽  
Plant Identification ◽  
Simple Method ◽  
Mass Dimension ◽  
Scale Range ◽  
Computer Identification

An application of fractal dimensions as measures of leaf complexity to morphometric studies and automated plant identification is presented. Detailed algorithms for the calculation of compass dimension and averaged mass dimension together with a simple method of grasping the scale range related variability are given. An analysis of complexity of more than 300 leaves from 10 tree species is reported. Several classical biometric descriptors as well as 16 fractal dimension features were computed on digitized leaf silhouettes. It is demonstrated that properly defined fractal dimension based features may be used to discriminate between species with more than 90% accuracy, especially when used together with other measures. It seems, therefore, that they can be utilized in computer identification systems and for purely taxonomical purposes.

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