Search Results (344)

In recent decades, multiple approaches have emerged to assess architectural visual character, including fractal dimension analysis, visual equilibrium calculations, and visual preference surveys. However, the relationships among these methods and their alignment with subjective perception remain unclear. This study applies all three techniques to sample mosques in Riyadh, Saudi Arabia, to evaluate their validity and interconnections. Findings reveal a within-sample tendency toward low visual complexity, with fractal dimensions ranging from 1.2 to 1.547. Within this small, exploratory sample of five large main-road mosques in Riyadh, correlations between computed visual equilibrium and survey results provide preliminary, sample-specific convergent-validity evidence for Larrosa’s visual-forces method, rather than general validation. Within this sample, traditional façades with separate minarets tended to score as more visually balanced than more contemporary compositions. This triangulated approach offers an exploratory framework for architectural visual assessment that integrates objective metrics with human perception. Full article

We present a systematic framework for the reverse fractal design of Lindenmayer (L-) systems from target box-counting dimensions. By inverting the self-similarity formula $D={\displaystyle \frac{lnn}{ln(1/r)}}$ to $r=n^{-1/D}$, we analytically determine L-system parameters (copy count n and scaling factor r) that generate fractals with prescribed dimensions. Our key innovation is a topology template library encoding eight spatial arrangements that separate dimensional constraints from geometric structure, enabling systematic design space exploration. For a single target dimension $D=1.5$, the framework generates 21 topologically distinct L-systems (spanning binary, ternary, quaternary, and pentagonal patterns), all achieving machine-precision accuracy (error $<{10}^{-15}$) with sub-millisecond efficiency. Comprehensive validation through 25 unit tests and standard fractals (Koch snowflake and Sierpinski triangle) confirms mathematical correctness. This template-based approach provides unprecedented flexibility: users control complexity through n selection, visual density through r, and esthetics through topology choice. Applications span fractal art generation, natural structure simulation, procedural content generation, and texture synthesis. Full article

Background: Glioblastoma (GBM) is among the most aggressive and morphologically heterogeneous brain tumors. Beyond static imaging biomarkers, its structural organization can be viewed as a nonlinear dynamical system. Characterizing morphodynamic attractors within such a system may reveal latent stability patterns of morphological change and potential indicators of morphodynamic organization. Methods: We analyzed 494 subjects from the multi-institutional BraTS 2020 dataset using a fully automated computational pipeline. Each multimodal MRI volume was encoded into a 16-dimensional latent space using a 3D convolutional autoencoder. Synthetic morphological trajectories, generated through bidirectional growth–shrinkage transformations of tumor masks, enabled training of a contraction-regularized Neural Ordinary Differential Equation (Neural ODE) to model continuous-time latent morphodynamics. Morphological complexity was quantified using fractal dimension (DF), and local dynamical stability was measured via a Lyapunov-like exponent (λ). Robustness analyses assessed the stability of DF–λ regimes under multi-scale perturbations, synthetic-order reversal (directionality; sign-aware comparison) and stochastic noise, including cross-generator generalization against a time-shuffled negative control. Results: The DF–λ morphodynamic phase map revealed three characteristic regimes: (1) stable morphodynamics (λ < 0), associated with compact, smoother boundaries; (2) metastable dynamics (λ ≈ 0), reflecting weakly stable or transitional behavior; and (3) unstable or chaotic dynamics (λ > 0), associated with divergent latent trajectories. Latent-space flow fields exhibited contraction-induced attractor-like basins and smoothly diverging directions. Kernel-density estimation of DF–λ distributions revealed a prominent population cluster within the metastable regime, characterized by moderate-to-high geometric irregularity (DF ≈ 1.85–2.00) and near-neutral dynamical stability (λ ≈ −0.02 to +0.01). Exploratory clinical overlays showed that fractal dimension exhibited a modest negative association with survival, whereas λ did not correlate with clinical outcome, suggesting that the two descriptors capture complementary and clinically distinct aspects of tumor morphology. Conclusions: Glioblastoma morphology can be represented as a continuous dynamical process within a learned latent manifold. Combining Neural ODE–based dynamics, fractal morphometry, and Lyapunov stability provides a principled framework for dynamic radiomics, offering interpretable morphodynamic descriptors that bridge fractal geometry, nonlinear dynamics, and deep learning. Because BraTS is cross-sectional and the synthetic step index does not represent biological time, any clinical interpretation is hypothesis-generating; validation in longitudinal and covariate-rich cohorts is required before prognostic or treatment-monitoring use. The resulting DF–λ morphodynamic map provides a hypothesis-generating morphodynamic representation that should be evaluated in covariate-rich and longitudinal cohorts before any prognostic or treatment-monitoring use. Full article

During drill-and-blast construction, complex and variable rock masses are frequently encountered. Owing to the transient nature of the explosion process and the randomness of crack propagation, the response of different rock masses to explosive loading is highly intricate. This study primarily investigates the dynamic response of rock masses with varying strengths under two different charge configurations. First, four cement mortar specimens of differing strengths were prepared then subjected to general blasting and slit charge blasting, respectively. High-speed cameras and digital image correlation techniques were employed to capture and analyse stress wave propagation and crack propagation during detonation. Fractal dimension analysis was subsequently employed to quantify and compare the extent of damage in the specimens. Findings indicate that rock strength influences stress wave attenuation patterns: lower-strength rocks exhibit higher peak strains but faster decay rates. Crack propagation velocity was calculated by deploying monitoring points along fracture paths and defining fracture initiation thresholds. Higher rock strength correlates with both peak and average crack propagation velocities. Slit charge blasting effectively optimizes damage distribution, concentrating it within the intended directions while reducing chaotic fracturing. These findings provide scientific justification for blasting operations in complex rock formations. Full article

Multi-focus images are essential in various computer vision applications. To mitigate artifacts and information loss in multi-focus image fusion, we propose a novel algorithm based on AGPCNN and fractal dimension in the NSCT domain. The source images are decomposed into low- and high-frequency sub-bands via NSCT; the low-frequency components are fused using an averaging rule, while the high-frequency components are fused through fractal dimension and the AGPCNN model, followed by consistency verification to refine the results. Experiments on the Lytro and MFI-WHU datasets show that the proposed method outperforms existing approaches in terms of both visual quality and quantitative metrics. Furthermore, its successful application to multi-sensor and multi-modal image fusion tasks demonstrates the algorithm’s robustness and generality. Full article

In this article, we investigate the dynamical analysis and soliton solutions of the microtubule equation. First, the Lie symmetry method is applied to the considered model to reduce the governing partial differential equation into an ordinary differential equation. Next, the multivariate generalized exponential rational integral function method is employed to derive exact soliton solutions. Finally, the bifurcation analysis of the corresponding dynamical system is discussed to explore the qualitative behavior of the obtained solutions. When an external force influences the system, its behavior exhibits chaotic and quasi-periodic phenomena, which are detected using chaos detection tools. We detect the chaotic and quasi-periodic phenomena using 2D phase portrait, time analysis, fractal dimension, return map, chaotic attractor, power spectrum, and multistability. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab version 24.2. These results offer a powerful mathematical framework for addressing various nonlinear wave phenomena. Finally, conservation laws are explored. Full article

This paper proposes a new simple method for generating fractal-like aggregates in 2D real spaces. The idea is to use an initial fractal aggregate and simulate a Gravity-based attraction from a distant point (using a gravity attractor with a large mass, but without volume). A Diffusion-Limited Aggregation (DLA) procedure is then applied by considering a single particle situated in the gravity attractor, with a minimum distance $R_a$ for deciding between aggregation or no aggregation. The final aggregates obtained are completely new fractal-like aggregates (images or structures). We analyze the fractal-like generated images obtained with the proposed method, considering different configurations and parameters in the simulations, including different initial fractals, different minimum distances $R_a$, etc. We also analyze the fractal dimensions of some of the new aggregates constructed by the proposed Gravity-based DLA simulation method. Full article

Vision Transformers achieve strong performance across computer vision tasks but suffer from quadratic computational complexity with respect to token count, limiting deployment in resource-constrained environments. Existing token pruning methods rely on attention scores to identify important tokens, but attention mechanisms capture query-specific relevance rather than intrinsic information content, potentially discarding tokens that carry information for subsequent layers or different downstream tasks. We propose fractal-guided token pruning, a method that leverages the correlation dimension $D_{\mathrm{corr}}$ of token embeddings as a task-agnostic measure of geometric complexity. Our key insight is that tokens with high $D_{\mathrm{corr}}$ span higher-dimensional manifolds in representation space, indicating complex patterns, while tokens with low $D_{\mathrm{corr}}$ collapse to simpler structures representing redundant information. By computing a local $D_{\mathrm{corr}}$ for each token and pruning those with the lowest values, our method retains geometrically complex tokens independent of attention-based relevance. The correlation dimension quantifies how token embeddings fill the representation space: embeddings from uniform background regions cluster tightly in low-dimensional subspaces (low $D_{\mathrm{corr}}$), while embeddings from complex textures or object boundaries spread across higher-dimensional manifolds (high $D_{\mathrm{corr}}$), reflecting their richer information content. Experiments on CIFAR-10 and CIFAR-100 with fine-tuned ViT-B/16 models show that fractal-guided pruning consistently outperforms random and norm-based pruning across all tested ratios. At forty percent pruning, fractal pruning maintains 92.26% accuracy on CIFAR-10 with only a 0.99 percentage point drop from the 93.25% baseline while achieving 1.17× speedup. Our approach provides a geometry-based criterion for token importance that complements attention-based methods and shows promising generalization between CIFAR-10 and CIFAR-100 datasets. Full article

As urban disasters intensify, the relationship between urban spatial form and disaster risk is increasingly important. Different spatial configurations reflect varying levels of resilience to disasters. However existing research offers limited quantitative evidence linking spatial form indicators and disaster risk indices. This study addresses this gap by developing a quantifiable, city-scale framework to analyze the form–risk relationship across 32 Chinese cities. Urban spatial form is quantified using fractal dimension to measure boundary complexity and compactness to assess internal structure, supplemented by a diagrammatic classification of urban patterns. A comprehensive disaster risk index is developed based on four dimensions: hazards, exposure, vulnerability, and resilience. Regression analysis is then applied to quantify the direction and magnitude of correlations between spatial-form indicators and the comprehensive risk index. The results reveal three major findings: (1) Disaster risk increases with fractal dimension, indicating that cities with more complex and irregular boundaries tend to be more vulnerable. In contrast, compactness has no statistically significant effect on disaster risk. (2) Spatial patterns are strongly associated with risk levels: cluster-type and block-type cities generally experience lower risks than radial-type and constellation-type cities. (3) City size and geography also influence risk, as larger cities typically exhibit lower risks, whereas southern cities face higher risks than those in northern regions. These results highlight the critical role of urban spatial structure in shaping disaster resilience. Managing boundary complexity, fostering polycentric and block-based spatial layouts, and improving road-network redundancy can effectively enhance urban adaptive capacity. These insights provide theoretical foundations and practical guidance for resilience-oriented spatial optimization and disaster-risk reduction in vulnerable cities. Full article

Integration in non-integer-dimensional spaces (NIDS) is actively used in quantum field theory, statistical physics, and fractal media physics. The integration over the entire momentum space with non-integer dimensions was first proposed by Wilson in 1973 for dimensional regularization in quantum field theory. However, self-consistent calculus of integrals and derivatives in NIDS and the vector calculus in NIDS, including the fundamental theorems of these calculi, have not yet been explicitly formulated. The construction of precisely such self-consistent calculus is the purpose of this article. The integral and differential operators in NIDS are defined by using the generalization of the Wilson approach, product measure, and metric approaches. To derive the self-consistent formulation of the NIDS calculus, we proposed some principles of correspondence and self-consistency of NIDS integration and differentiation. In this paper, the basic properties of these operators are described and proved. It is proved that the proposed operators satisfy the NIDS generalizations of the first and second fundamental theorems of standard calculus; therefore, these NIDS operators form a calculus. The NIDS derivative satisfies the standard Leibniz rule; therefore, these derivatives are integer-order operators. The calculation of the NIDS integral over the ball region in NIDS gives the well-known equation of the volume of a non-integer dimension ball with arbitrary positive dimension. The volume, surface, and line integrals in D-dimensional spaces are defined, and basic properties are described. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) are proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are proved. Some basic elements of the calculus of differential forms in NIDS are also proposed. The proposed NIDS calculus can be used, for example, to describe fractal media and the fractal distribution of matter in the framework of continuum models by using the concept of the density of states. Full article

Labeling ice cracks in Antarctic near-shore sea ice aerial orthophotos is critical for sea ice cargo route development; rapid, accurate identification and labeling of cracks in UAV imagery aids safe goods transfer between icebreakers and expedition stations, and studying ice crack distribution provides a key basis for assessing sea ice route reliability. Ice cracks have complex morphologies that traditional recognition methods struggle to handle, so this study proposes the ICI-YOLOv8 algorithm to improve sea ice crack detection near Antarctica’s Zhongshan Station, using crack density and fractal dimension to characterize spatial distribution and a Monte Carlo-based numerical model to quantify distribution probability. The algorithm achieves 0.628 accuracy and 0.662 mAP@0.5 (outperforming comparable methods in speed and accuracy) and reaches 0.933 accuracy and 0.657 mAP@0.5 with better generalization than similar models when tested on general remote sensing water datasets; a positive correlation exists between fractal dimension and ice crack density, and Monte Carlo simulation and probability distribution models verify their distribution properties. The proposed algorithm is suitable for rapid summer Antarctic near-shore sea ice crack identification, the numerical model effectively quantifies crack distribution to aid route development, and this study is important for understanding polar ice stability and sea ice route development. Full article

Fractal antennas employ self-similar geometries to generate scaled multiple resonances within compact structures, thereby achieving broadband performance. However, many reported designs remain constrained by narrow impedance bandwidths or demonstrate only multiband characteristics. To address these limitations, we present a novel pentagonal fractal antenna with ultra-wideband performance suitable for C, X, Ku and K-band applications. The key innovation lies in a double-pentagonal fractal configuration, created by embedding a secondary pentagonal ring within the conventional pentagonal radiator. This design significantly enhances the impedance bandwidth and enables ultra-wideband operation. The proposed antenna was validated through both electromagnetic simulations and experimental measurements. Results show a measured −10 dB impedance bandwidth of 3.84–22.4 GHz, corresponding to a fractional bandwidth of 141.5%. The antenna dimensions are only 0.384 × 0.525 × 0.01λ₀³. A peak gain of 10.2 dBi was achieved, with the gain varying between 2.88 and 10.2 dBi across the operating frequency range. Owing to these characteristics, the proposed antenna is well-suited for diverse wireless communication systems, including Wi-Fi, ultra-wideband communication, 5G mid-band and emerging 6G technologies. Full article

This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case $D=0$ has been little explored. Starting from the question “what happens to a fractal when its complexity collapses completely?”, Moran’s similarity equation becomes tautological ($r^s=1$ with solution $s={dim}_M=0$) and that only the Hausdorff and box-counting definitions allow an exact calculation. Based on Banach’s fixed point theorem and these definitions, we prove that the attractor of a degenerate IFS is a singleton with ${dim}_H={dim}_B=0$. We develop a reproducible computational methodology to visualize the collapse in dimensions 1–3 (the Iterated Line Contraction—1D/Iterated Square Contraction—2D/Iterated Cube Contraction—3D families), including deterministic and stochastic variants, and we provide a Python script 3.9. The theoretical and numerical results show that the covering box-counting retains unity across all generations, confirming the zero-dimension element and the stability of the phenomenon under moderate perturbations. We conclude that degenerate fractals are an indispensable benchmark for validating fractal dimension estimators and for studying transitions to attractors with positive dimensions. Full article

This study aims to address the research gap in the digital analysis of traditional patterns by proposing an image-processing-driven parametric modeling method that combines graphic primitive function modeling with topological reconstruction. The image is processed using a dual-channel image processing algorithm (Canny edge detection and grayscale mapping) to extract and vectorize graphic primitives. These primitives are uniformly represented using B-spline curves, with variations generated through parametric control. A topological reconstruction approach is introduced, incorporating mapped geometric parameters, topological combination rules, and geometric adjustments to output topological configurations. The generated patterns are evaluated using fractal dimension analysis for complexity quantification and applied in cultural heritage imaging practice. The proposed image processing pipeline enables flexible parametric control and continuous structural integration of the graphic primitives and demonstrates high reproducibility and expandability. This study establishes a novel computational framework for traditional patterns, offering a replicable technical pathway that integrates image processing, parametric modeling, and topological reconstruction for digital expression, stylistic innovation, and heritage conservation. Full article

Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allows for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and, in particular, extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting. Full article