Abstract
Numerous irregular and fragmented complex forms in nature can be effectively generated and described by fractal geometry through self-similar iteration. However, fractal self-similarity in nature is limited to a specific scale range, with a clear scale boundary. An imaginary dimensional cohaesal iteration was proposed to overcome this problem. Through imaginary dimensional cohaesal iteration, it generates complete systems, where fractals can be serve as generating subsystems. Through variations in iteration, it achieves pattern evolution, breaks through scale limitations, and better describes complex object systems in nature (such as from branches to leaves). While cohaesal has expanded the specific scale range of fractals, it still has certain size constraints and overlooks details. Studies have demonstrated that under imaginary iteration, patterns derived from different original patterns exhibit mutual similarity. It can simulate various natural and physical processes (e.g., leaf veins developing mesophyll), facilitate the modeling of processes such as egg hatching, and realize the iterative generation of images from branches to leaves.
Full Text
Preamble
Fractal imension, ohaesal maginary imension Zehua School Information Communication Engineering, Beijing University Posts Telecommunications, Beijing 100876, China Zehua responsible
Abstract
Objective: Numerous irregular fragmented complex forms nature effectively generated described fractal geometry through self-similar iteration. owever, fractal self-similarity nature limited specific scale range, clear scale boundary. imaginary dimensional cohaesal iteration proposed overcome problem Methods:
Through imaginary dimensional cohaesal iteration, generates complete systems where fractals serve generating subsystems Results:
Through variations iteration, achieves pattern evolution, breaks through scale limitations, better describes complex object systems nature (such branches leaves).
Limitations: While cohaesal expanded specific scale range fractals, still certain constraints overlooks details.
Conclusions: Studies demonstrated under imaginary iteration, patterns derived different original patterns exhibit mutual similarity. simulate various natural physical processes (e.g., veins developing mesophyll), facilitate modeling processes hatching, realize iterative generation images branches leaves.
Keywords
Fractal Imaginary Dimension Cohaesal Mutual Similarity Graphic Evolution supported National Program China No.2023YFC3010700 (E-mail:
1. I
ntroduction major scientific achievements century include relativity, quantum mechanics double helix structure fractals-chaos, Among these, fractal geometry [1-7] broke through traditional framework Euclidean geometry, while chaos theory [8-12] explained characteristics "sensitivity initial conditions" "topological transitivity" through mathematical definitions.
Since century, fractal chaos theories further deeply integrated complex systems science [13-22], become common research framework multiple disciplines physics biology [23-29].
Fractals originate geometric
Abstract
"irregular complex systems" represented geometric forms coastlines, branches, lightning.
Although systems irregular shapes, possess characteristic "infinite nested self-similarity".
Based theory, calculate fractal dimension certain section coastline between [30]; fractals that, first time, expanded geometric dimensions integer dimension non-integer dimension.
Before emergence fractal theory, traditional Euclidean geometry described world using regular shapes straight lines, circles, cones. struggled effectively handle irregular, fragmented, complex forms ubiquitous nature.
Fractals provided mathematical language describing complex natural forms introduced powerful "fractal dimension," which quantify "roughness" "complexity" these complex forms.
Fractals offer brand-new "grammar" "vocabulary" interpret, quantify, simulate natural universe inherently irregular, complex, rough.
Fractals proven simple iterative rules behind "complexity," marking major human cognition.
Fractal cross-disciplinary universal mathematical tool, providing researchers different fields common language perspective. field medicine, analyze vascular networks, tumor vascular distribution, bronchial structures, neuron morphologies, electrocardiograms [32-37], providing quantitative indicators disease diagnosis. field materials science, study fracture surfaces materials structures porous media (such rocks soil), while establishing connections between these structures their physical properties (such strength permeability) [38-40]. field geology, fractals applied analyze structures [41-42]. universality facilitated interdisciplinary collaboration innovation.
Currently, application fractal theory still certain limitations [43], "multi- physical field coupling" [44-46].
Although fractals constructed through iterative functions, real-world fractals (e.g., mountain ranges, cloud formations) often shaped combined action multi-scale physical processes, mathematical models struggle fully capture their dynamic evolution laws. engineering systems (e.g., aero-engines, chips), phenomenon "mechanical- thermal-electromagnetic" multi-physical field coupling widespread.
However, existing fractal chaos models mostly limited single-physical field research (e.g., analyzing fractal characteristics vibration) [48-49], fully consider coupling effects between different physical fields. addition, fractals nature (such cloud formations, coastlines, mountain ranges) strictly self-similar; exhibit approximate self-similarity statistical sense.
Meanwhile, their self-similarity limited specific scale range, clear minimum maximum scale boundaries.
Self-similarity holds within limited scale range, beyond self-similarity ceases exist [50]. example, branching structure levels self-
similar characteristics trunk twigs, scale reduced level leaves cells, self-similarity longer exists.
Fractal dimension, while capable converting complexity complex forms quantifiable values, single statistical measure. extracting features, inevitably compresses information inherent shape itself characteristics branching patterns structures, texture details, local morphological differences weakened overlooked. objects strikingly different shape structure highly similar fractal dimensions after measurement. using fractal
analysis
methods, there counterintuitive phenomenon: structures different initial forms fractal dimensions after multiple iterations, converge overall macroscopic structural shape visually identical highly similar. characteristic stems inherent nonlinear dynamical systems. final presentation fractal related initial shape, iteration rules, parameters; differences initial dimensions "weakened" "neutralized" synergistic effects parameters during repeated iterations, ultimately leading similar structural characteristics [51]. example, certain iterated function systems chaotic systems strange attractors, regardless different initial points system evolution eventually attracted fractal structure. example, "bifurcating fractals" natural simulation: initial parameters fractal dimensions group binary trees, where length branch level times previous level bifurcation angle other group ternary trees, where length branch level times previous level bifurcation angle after iterations, overall forms simulation
results
present vein-like pattern "main veins extending lateral veins distributed crisscross pattern". there significant differences between them, differences existing microscopic level. aforementioned bifurcating trees consider length-wise branch proportion account lateral dimension branches.
results
bifurcating models binary trees ternary trees describe longitudinal iteration branches, veins, blood vessels, cannot describe iteration mesophyll animal tissue. introducing dimensional extension expanding dimensions imaginary dimensions performing iteration within imaginary dimensions (with variations occurring during iteration), paper breaks through scale limitations fractals dimensions. enables veins mesophyll, blood vessels tissue, achieves iterative image generation branches leaves, thereby better describing complex objects nature (e.g., branches leaves). found through iterative generation imaginary dimensions, different iteration parameters similarity between different patterns, mutual similarity achieved through different iterations.
Fractal iteration determines contour shape high-frequency details.
Through imaginary iteration, high-frequency information fractal iteration covered, resulting higher degree mutual similarity.
While fractal veins obtained removing imaginary leaves, bifurcation obtained removing imaginary crown. cohaesal inverse process fractals. fractal dimension obtained through iteration certain direction. positioning performed establishing coordinates dimensions, positions imaginary dimensions deviate real-dimensional coordinates.
These positions depend attach parasitic
real-dimensional coordinates. neither independent dimensions, cannot separated real-dimensional coordinates.
Imaginary coordinates established based imaginary dimensions. addition uniform expansion imaginary direction, exponential, trigonometric functions, probability distributions. particular, probability distributions provide coordinate system quantum mechanics: function propagation transformed description using imaginary dimensions, which greatly simplify quantum mechanical calculations. example, electronic motion probability distribution vertical direction motion, which attached direction dimension motion.
2. Bifurcating
Trees Imaginary Dimension
2.1 Bifurcating
Trees Fractal Iteration binary fractal iteration designed, parameters follows: angle between branch parent branch fixed ensure fractal structure exhibits regular symmetrical divergent morphology. recursive depth levels, fractal start initial branch, generate levels branches layer layer, ultimately forming tree-like structure hierarchical levels. initial length branch pixels, width branches pixel ensure clarity visual consistency branch lines. branch length decay coefficient i.e., length branch times parent branch. controlling decay rhythm branches through proportion, overall fractal neither sparse crowded, ultimately binary fractal graph coordinated morphology distinct details generated, shown fractal dimension binary fractal pattern perfectly exhibits expected structural characteristics under parameter setting: starting initial branch visual center, branches right sides symmetrically diverge strict angle forming highly regular symmetric morphology.
Moreover, symmetry persists across iterative levels, maintaining consistent left-right mirror distribution throughout. self-similarity. characteristic, defined "structural similarity between local parts whole," endows pattern rigor mathematical models implicit growth rhythm natural trees, resembling simplified
abstract
natural tree. ternary fractal iteration further designed shown parameters design details follows: angle between branch parent branch Three branches diverge uniformly node, angle between adjacent branches, which prevents spatial overlap branches subsequent iterations. recursive depth levels, iterating according upper-level branch generating three lower-level branches" balance
detail density visualization range. initial length branch pixels, width branches pixel ensure clarity visual consistency branch lines. branch length ratio since ternary branches, small ratio helps balance quantity space, preventing stacking sparsity. fractal dimension
2.2 Bifurcating
Trees Cohaesal Iteration aforementioned fractal trees, branch width However, actual branches exhibit varying widths viewed side: width iteration ratio trunk branches levels generally different length ratio.
Considering width-lateral expansion nature presents multiple forms including uniform expansion, exponential expansion, trigonometric expansion, probabilistic expansion, different density distributions expansion entirely distinct longitudinal fractal fractional dimension.
Specifically, there exists fractional dimension longitudinal direction, while lateral expansion iteration corresponds imaginary dimension. fractal dimension defined follows: graph composed similar graphs, obtained reducing original graph factor fractal dimension imaginary-real iteration
experiment
conducted binary tree, parameters follows: angle between branch parent branch fixed recursive depth levels, branch length decay coefficient 0.63; branch width decay coefficient
According to the definition, the dimension is:
generated graph shown Through binary cohaesal nature, making graph rigorous mathematical model expressing growth rhythm natural trees.
Imaginary-real iteration performed ternary tree, parameters follows: angle between branch parent branch fixed recursive depth levels, branch length ratio coefficient branch width ratio coefficient dimension generated Ternary graph shown presents Ternary branch length ratio coefficient branch width ratio coefficient observed
Figures these ternary cohaesal graphs closely resemble trees nature enhance rigor mathematical model convey growth rhythm natural trees.
2.3 Bifurcation
Trees Cohaesal Variational-Iteration cohaesal variant iteration bifurcating trees analyzed. graphical perspective, variation iteration
results
evolution graphs branch-like leaf-like. inary iteration performed: first iterations, branch width ratio coefficient remains constant (displayed black).
Variation occurs after iteration, where branch width ratio coefficients subsequent iterations variant parts displayed green, shown local within magnified variation region. observed process resembles transition leaves nature sprouting spring becoming fully developed summer.
Fractals perform branch length iteration. However, through imaginary-dimensional cohaesal iteration, graphical evolution achieved variation, iteration, flower iteration, fruit iteration. iterated time, characteristics temporal changes analyzed.
Before variation, dimension branch along longitudinal direction three- dimensional, there perpendicular directions, which extended imaginary dimension other imaginary dimension After iterative variation leaves, perspective dimension, three-dimensional branch changes two-dimensional thickness direction neglected graphical perspective, making quasi-two-dimensional). perspective imaginary dimensions, imaginary dimensions disappears (i.e., thickness direction), while other enhanced
result
variation. variation increases contact between external environment, enhances biological efficiency. contact sunlight increased after variation branch leaves. facilitating photosynthesis exchange.
Similarly, biological processes development blood vessels capillaries tissue, bronchi lung, neurons brain expand contact scope, thereby maximizing biological efficiency.
3. Cohaesal
Similarity
Analysis
graphs figures graphs similar before after imaginary iteration. analyze relationship between imaginary iteration similarity, cross- correlation
analysis
adopted herein, covering scenarios: cross-correlation bifurcating trees before after imaginary iteration, cross-correlation different Bifurcating trees. cross-correlation function images
similarity degree between graphs determined calculating cross-correlation value. cross-correlation value corresponds autocorrelation, indicating graphs identical. higher cross-correlation value, greater similarity degree between graphs.
During calculation process, digital images consist discrete pixels. discrete functions two-dimensional images denoted f[m,n] (Size g[m,n] (Size graph pixel value coordinate (m,n) denoted f[m,n], where Here, represent horizontal pixel index (column) vertical pixel index (row), respectively, denotes graph graph pixel value coordinate (m,n) denoted g[m,n], where Here, represent horizontal pixel index (column) vertical pixel index (row), respectively, denotes graph cross-correlation function graph1 graph2 Normalized cross correlation (NCC) calculated eliminate influence brightness difference:
Where, local value image current window value
3.1 Cross
Correlation Between Binary Ternary Cross-correlation computation performed binary (angle Ternary (angle normalized cross-correlation value 0.229239, which certain similarity, degree similarity binary compared Ternary (angle normalized cross-correlation value 0.249564, which slightly higher, degree similarity
3.2 Cross
Correlation After Cohaesal Iteration cross-correlation after superimposing imaginary iteration analyzed. binary (length decay width Ternary (length decay width cross-correlation binary Ternary (length decay width cross- correlation normalized cross-correlation values trees superimposed imaginary iteration higher those without.
3.3 Cross
Correlation Between Internal Bifurcating Trees Variational-Iteration cross-correlation bifurcation cohaesal variant iteration analyzed here.
results
binary tree, Ternary Ternary consistent trend: greater deviation width coefficient benchmark value, larger imaginary dimension, smaller internal cross-correlation value, indicating lower similarity relationship curve between internal cross-correlation imaginary dimension Binary tree, Ternary Ternary shown larger imaginary lower internal cross- correlation binary Ternary Tree, lower similarity.
3.4 Cross-correlation
Between Variational-Iterative Binary Trees Ternary Trees Cross-correlation analyses between different types (binary ternary) variational iteration performed.
results
normalized cross-correlation values generally indicating significant structural differences.
However, iterative width coefficient increases, cross-correlation values change.
Notably, comparisons, increases, cross-correlation gradually increases. exceeds certain threshold (e.g., 4.7), cross- correlation values between different types become higher internal cross-correlation values original trees. indicates large enough imaginary dimension (driven becomes difficult distinguish benchmark which iterated graphs originated
3.5 Cross-correlation
Between Leaves Binary Trees Ternary Trees
analysis
focused "leaf" portions trees, which generated later stages variational iteration.
results
reinforce previous findings. branch width iteration coefficient increases, imaginary dimension values increase, cross-correlation values bifurcating leaves become closer.
Increasing binary ternary leaves significantly enhances cross-type similarity. cross-correlation between binary leaves ternary leaves exceeds their respective internal cross-correlation values. means patterns become highly similar "cohaesal converge," making impossible distinguish their origins
4. Discussion
Imaginary iteration applied numerous fields. field biological camouflage, instance, generation patterns butterfly wings achieved through imaginary iteration shapes combined color.
Additionally, shape evolution processes hatching, embryonic development, fossil organism restoration realized imaginary iteration.
Beyond biology, imaginary iteration applicable visualization processes other disciplines including pattern generation chemical reaction processes condensation processes physics.
Through inverse process imaginary iteration, known inverse imaginary iteration, veins obtained leaves branches obtained crown. supports practical applications precise recognition restoration clear images blurred ones. dynamic continuous transformation between patterns (e.g., expression transition human "smiling" "frowning"), changes achieved using Manifolds.
However, imaginary iteration achieves transformation through discrete stepwise changes, making suitable implementation computer technology. discuss special imaginary iteration: self-iteration patterns constant capacity.
Cross-correlation analyses performed characters varying stroke widths. that: internal-symbol cross-correlation (i.e., cross-correlation character, either varying stroke widths: stroke width increases, cross-correlation values decrease, accompanied reduction similarity. cross-correlation between (stroke width pixel) varying stroke widths pixels): characters stroke width pixel, cross-correlation value lower their respective internal-correlation values still greater indicating certain degree similarity. stroke width increases, cross-correlation value decreases. cross-correlation between (stroke width pixel) varying stroke widths pixels): cross-correlation value lower their respective internal- correlation values.
However, stroke width pixels, cross- correlation value remains greater demonstrating relatively similarity. stroke width increases, cross-correlation value first increases decreases. cross-correlation between identical stroke widths pixels, characters sharing stroke width pair): stroke width increases, cross-correlation value consistently rises. stroke width exceeds pixels, cross- correlation value surpasses internal-correlation values Specifically, stroke width pixels, cross-correlation value reaches higher internal- correlation value between pixel) "E15" pixels), internal-correlation value between pixel) pixels).
Cross-correlation analyses conducted between observations follows:
cross-correlation between (stroke width pixel) varying stroke widths pixels): cross-correlation values remained consistently indicating degree similarity between characters. cross-correlation between identical stroke widths pixels, characters sharing stroke width pair): cross-correlation values increased consistently stroke width increased.
These
results
indicate exhibits higher similarity Specifically, mutual similarity cohaesal between achieved through imaginary iteration.
5. Conclusion
study introduces novel conceptual framework, "cohaesal iteration," extending fractal geometry imaginary dimensions. demonstrated approach successfully models evolution complex natural systems, breaking through scale limitations inherent traditional fractal self-similarity. introducing variations imaginary dimension during iteration, simulate development complete systems their generating subsystems, growth mesophyll veins formation crown branches. approach enables simulation various natural physical processes: allows veins develop mesophyll, blood vessels generate capillaries eventually muscle tissue, bronchi develop alveoli ultimately organ, neurons develop brain. supports modeling processes hatching, embryonic development, fossil organism restoration, chemical reactions, physical condensation.
Additionally,
method
realizes iterative generation images branches leaves, producing complete systems (fractals regarded generators subsystems). finding phenomenon "cohaesal," mutual similarity, where patterns originating structurally different initial forms converge towards degree similarity after undergoing imaginary iteration. quantitative
analysis
using normalized cross-correlation confirms imaginary dimension component increases, similarity between disparate structures (e.g., binary ternary trees) exceed their internal similarity their less-developed variants. suggests imaginary dimension underlying fractal structure, leading convergence macroscopic form. inverse process, which inverse imaginary iteration, offers powerful feature extraction, allowing deconstruction complex systems their fundamental fractal skeletons instance, extracting veins restoring clear image blurred Furthermore, concept imaginary dimension based probability distributions holds speculative significant potential simplifying calculations quantum mechanics, describing function propagation.
While cohaesal still certain constraints overlooks details. summary, cohaesal iteration provides holistic dynamic descriptive language complex, multi-scale systems found nature, offering bridge between rigid self-similarity fractals rich, evolving morphology world.
Future explore applications biological modeling, materials science, fundamental physics.
ACKNOWLEDGEMENTS supported National Program China No.2023YFC3010700
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SUPPORTING INFORMASTION 论文责任者（论文作者）研究身份识别材料 .pdf]
Figure Legends Figures Binary Angle Between Branches Parent Branch Length Decay Coefficient
Ternary Angle Between Branches Parent Branch Length Decay Coefficient
Binary Angle Between Branches Parent Branch, Branch Length Decay Coefficient 0.63, Branch Width Decay Coefficient
Ternary Angle Between Branches Parent Branch, Branch Length Decay Coefficient Branch Width Decay Coefficient
Ternary Branch Length Ratio Coefficient Branch Width Ratio Coefficient
Binary Branch Width Iteration Coefficients First Iteration Levels, After Iteration
Cross correlation between Binary Ternary
Cross correlation between Binary Ternary
Cross correlation between Binary Ternary after superimposing imaginary iteration
Cross correlation between Binary Ternary after superimposing imaginary iteration
Relationship curve between internal cross correlation Binary tree, Ternary Ternary
relationship curve between internal cross-correlation imaginary dimension Binary tree, Ternary Ternary
relationship curve between normalized cross-correlation
Cross correlation curve leaves
Relationship between stroke width cross correlation different stroke widths