Fractal Shape Research Articles

Design 3D Wallpaper Motifs from Sierpinski Carpet Fractals Using Mathematica Applications

The development of geometric studies is very rapid, including fractal geometry. A fractal is a geometric shape that fulfills the properties of the fractal dimension. If you look at a fractal at first glance, it has an irregular shape, but if you look further, there is a regularity. One of the properties of fractals is self-similarity which occurs when a fractal is enlarged. The fractal form has a pattern obtained from iterating a function with infinite repetition. Among the fractal shapes there is the Sierpinski carpet shape. Fractals in 2D form can be transformed using geometric concepts into 3D fractals. Fractal shapes can be applied for cultural development, namely to form wallpaper motifs. Fractal geometric motifs are obtained from iterating a function with the help of Wolframe Mathematica.

Neuro-evolutionary evidence for a universal fractal primate brain shape.

The cerebral cortex displays a bewildering diversity of shapes and sizes across and within species. Despite this diversity, we present a universal multi-scale description of primate cortices. We show that all cortical shapes can be described as a set of nested folds of different sizes. As neighbouring folds are gradually merged, the cortices of 11 primate species follow a common scale-free morphometric trajectory, that also overlaps with over 70 other mammalian species. Our results indicate that all cerebral cortices are approximations of the same archetypal fractal shape with a fractal dimension of df = 2.5. Importantly, this new understanding enables a more precise quantification of brain morphology as a function of scale. To demonstrate the importance of this new understanding, we show a scale-dependent effect of ageing on brain morphology. We observe a more than fourfold increase in effect size (from two standard deviations to eight standard deviations) at a spatial scale of approximately 2 mm compared to standard morphological analyses. Our new understanding may, therefore, generate superior biomarkers for a range of conditions in the future.

Neuro-evolutionary evidence for a universal fractal primate brain shape

The cerebral cortex displays a bewildering diversity of shapes and sizes across and within species. Despite this diversity, we present a universal multi-scale description of primate cortices. We show that all cortical shapes can be described as a set of nested folds of different sizes. As neighbouring folds are gradually merged, the cortices of 11 primate species follow a common scale-free morphometric trajectory, that also overlaps with over 70 other mammalian species. Our results indicate that all cerebral cortices are approximations of the same archetypal fractal shape with a fractal dimension of df = 2.5. Importantly, this new understanding enables a more precise quantification of brain morphology as a function of scale. To demonstrate the importance of this new understanding, we show a scale-dependent effect of ageing on brain morphology. We observe a more than fourfold increase in effect size (from two standard deviations to eight standard deviations) at a spatial scale of approximately 2 mm compared to standard morphological analyses. Our new understanding may, therefore, generate superior biomarkers for a range of conditions in the future.

Local quenches in fracton field theory: Lieb-Robinson bound, noncausal dynamics and fractal excitation patterns

We study the out-of-equilibrium dynamics induced by a local perturbation in fracton field theory. For the Z4- and Z8-symmetric free fractonic theories, we compute the time dynamics of several observables such as the two-point Green’s function, ⟨ϕ2⟩ condensate, energy density, and the dipole momentum. The time-dependent considerations highlight that the free fractonic theory breaks causality and exhibits instantaneous signal propagation, even if an additional relativistic term is included to enforce a speed limit in the system. We show that it is related to the fact that the Lieb-Robinson bound does not hold in the continuum limit of the fracton field theory, and the effective bounded speed of light does not emerge. For the theory in finite volume, we show that the fracton wave front acquires fractal shape with nontrivial Hausdorff dimension and argue that this phenomenon cannot be explained by a simple self-interference effect. Published by the American Physical Society 2024

Exploring swine oviduct anatomy through micro-computed tomography: a 3D modeling perspective.

The oviduct plays a crucial role in the reproductive process, serving as the stage for fertilization and the early stages of embryonic development. When the environment of this organ has been mimicked, it has been shown to enhance in vitro embryo epigenetic reprogramming and to improve the yield of the system. This study explores the anatomical intricacies of two oviduct regions, the uterotubal junction (UTJ) and the ampullary-isthmic junction (AIJ) by using micro-computed tomography (MicroCT). In this study, we have characterized and 3D-reconstructed the oviduct structure, by measuring height and width of the oviduct's folds, along with the assessments of fractal dimension, lacunarity and shape factor. Results indicate distinct structural features in UTJ and AIJ, with UTJ displaying small, uniformly distributed folds and high lacunarity, while AIJ shows larger folds with lower lacunarity. Fractal dimension analysis reveals values for UTJ within 1.189-1.1779, while AIJ values range from 1.559-1.770, indicating differences in structural complexity between these regions. Additionally, blind sacs or crypts are observed, akin to those found in various species, suggesting potential roles in sperm sequestration or reservoir formation. These morphological differences align with functional variations and are essential for developing an accurate 3D model. In conclusion, this research provides information about the oviduct anatomy, leveraging MicroCT technology for detailed 3D reconstructions, which can significantly contribute to the understanding of geometric-morphological characteristics influencing functional traits, providing a foundation for a biomimetic oviduct-on-a-chip.

Multilayer multiband hybrid fractal antenna for public safety and 5G Sub-6 GHz bands

In this work, a multiband hybrid fractal antenna is developed for public safety and 5G Sub-6 GHz bands. Proposed antenna design is designed, analyzed, and optimized using HFSS simulator. Radiating element of the proposed antenna consists of Hybrid Fractal shape which is designed using Koch and Hilbert curve fractal geometry. Hybrid Fractal Antenna (HFA) is designed on a Multilayer FR4 substrate which has an overall size of 0.22 × 0.47 × 0.044 ƛ3. Parametric analysis of the proposed design is done to get the optimum results. Proposed HFA resonates at 2.64 GHz (2.34–2.80 GHz), 3.87, 4.54, 5.02, 5.35 GHz (3.73–5.55 GHz), and 6.42 GHz (5.70–6.72 GHz) with a gain of 4.2, 0.9, 2.9, 3.3, 4.6, and 3.7 dBi respectively. The proposed HFA is useful for 5G bands such as: N7 band, N38 band, N40 band, N41 band, N47 band, N53 band, N79 band, N90 band and N93 band, and also useful for public safety band (4.9 GHz). Prototype of the proposed HFA is fabricated and tested in lab for the verification of simulated results. The results show good performance in terms of reflection coefficient, gain, and bandwidth. Measured results are in good agreement with the simulated results.

Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs

Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between the large value to the small value. A Fibonacci sequence is a sequence of numbers that has a unique shape. The first term of this sequence of numbers is 1, then the second term is also 1, then for the third term it is determined by adding the two previous terms so that a sequence of numbers with a certain pattern is obtained. Pascal's triangle is a geometric rule of binomial coefficients in a triangle. Shapes that resemble fractal geometry, golden ratios and Fibonacci numbers are found in many places in this realm, for example the shapes of various plants, animals, as well as in humans themselves, even the universe. This fractal geometry can also be used to design batik motif creations, architectural design or musical art. This paper describes how a fractal object can be made with geometric transformations that can be used to develop batik motif creations. And also look at the relationship between fractal geometry, the golden ratio, Fibonacci numbers, and Pascal's triangles and the shapes of these shapes that exist in this nature.

Pairwise‐Interaction Model Unifies Different Asymptotic Shapes of UHI Intensity

AbstractCity size is a primary determinant of the urban heat island (UHI) intensity, with its effects further nuanced by the urban form. But how to factor in the urban form into the UHI assessment remains unresolved. We propose an every‐pair‐interaction model that meaningfully incorporates urban size and fractal dimension to characterize the UHI intensity. Regression on the summertime surface UHI intensity of 5,000 European cities shows that the model outperforms the simple linear combination of logarithmic size and fractal dimension. Subject to the interplay between the range of the every‐pair interaction and the urban fractal shape, the model also represents a generalization as it includes power‐law, logarithmic, and saturating size dependence of UHI—all three possibilities have been reported empirically in the literature. Our theoretical framework indicates that the surface UHI intensity saturates with urban size, opening up new research perspectives around UHI intensity.

Drone Shaped Fractal Antenna with Defected Ground Structure for RF Energy Harvesting Applications

A proposed multiband drone shaped fractal antenna (Antenna-1) is introduced in this paper that targets the most required frequencies used for radio frequency energy harvesting applications, including WLAN-2.4G, WLAN-5G, WLAN-6G, Mobile DCS1700, Mobile LTE and WIMAX services. A novel fractal shape is designed and combined with a standard square geometrical shape in order to implement the front patch of the antenna. The antenna is simulated with CST Studio Suite software and fabricated on FR-4 substrate. Different antenna stages with parametric study and optimization were implemented to get the best performance from the drone shape fractal antenna. The multiband (Antenna-1) supports six resonate frequencies (1.7155 GHz, 2.424 GHz, 3.36 GHz, 3.789 GHz, 5.843 GHz, 6.886 GHz) with peak gain of (2.83 dBi) at the frequency 2.45 GHz and a maximum bandwidth of (0.4281 GHz) at the frequency 5.8 GHz that make it a suitable antenna for radio frequency energy harvesting applications.

ETHNOMODELLING: FRACTAL GEOMETRY ON THE DOOR ORNAMENT OF THE SUMENEP PALACE USING THE LINDENMAYER SYSTEM

The Sumenep Palace area is one of the cultural heritage relics of the Duke of Sumenep, which is still preserved today. This research aims to reveal the fractal model of door ornament in the Koneng Office of Sumenep Palace. A qualitative method with an ethnography type is used. Data were collected through initial observation, literature study, interviews, and documentation. Analysis of the fractal form is done with the Lindenmayer system method and constructed with the help of the L-Studio application. The results of the study revealed the existence of fractal structures on the door ornament of the Sumenep palace office. The length, angle, and ratio of each part of the ornament influence the fractal shape. The findings of this study can be utilized in learning mathematics in lectures, such as fractal geometry and computational geometry.

A Metasurface Based High Gain Patch Antenna for Future Multiband Wireless Communication

This paper presents a novel, compact, and high gain metasurface (MS) patch antenna. The antenna structure consists of a slotted square patch in a fractal shape, integrated with a metasurface layer stacked on Taconic RF-30 dielectric material. The individual cells are arranged periodically, creating a 5 x 5-layer configuration. The proposed antenna is designed with physical dimensions of 290 x 290 mm² and achieves a maximum gain of 13.3 dBi at a frequency of 2.2 GHz. Moreover, it achieves a satisfactory minimum reflection coefficient of -10.9 dB. The design process and analysis were performed using the CST MWS simulation software. Based on the performance results obtained from this study, the suggested antenna can be considered suitable for applications in 5G communication.

Effect of Using Fractals as Grain Configuration on the Performance of Solid Rocket Motors

Grain design is a very crucial consideration for determining the performance of solid propellant rocket motors. A rocket’s performance and combustion characteristics (thrust profile, chamber pressure variation, propellant mass flow rate, and sliver among other important performance parameters) depend very strongly on the grain configuration. We propose fractal shapes (Koch star, Koch snowflake, Gosper, and Cesaro) as new port geometries and theoretically predict their performance using simulations based on the fast-marching method implemented in the open-source software Open-Motor. The initial thrust for fractal geometry grains was found to be significantly higher than the commonly used industry standard BATES and Finosyl ports. The fractal grains also showed a significant decrease in sliver as compared to the Finocyl port that is generally used to increase performance as compared to the circular BATES ports. Especially, The Koch snowflake port had a drastically lower sliver. These new grain geometries proposed in this work can be used for applications in which low initiation and ignition times along with high initial thrust are crucial. Moreover, these geometries can improve the performance of the solid rocket motor (SRM) owing to their better regression characteristics which promise complete utilization of the entire grain web.

Energy Harvesting Fractal Antenna for Charging Low Power Devices

Radio frequency energy harvesting (RFEH) is a prospective technique that uses electromagnetic waves which waste in the air to create energy. This cutting-edge technology offers the ability to wirelessly charge battery-free gadgets, making it a potential alternative energy source for use in the future. Fractal antenna systems are a type of antenna that is characterized by their self-similarity and ability to operate over a wide range of frequencies. The proposed design combined two popular fractal shapes in one design. This one design combines the advantages of the other two designs. This research presents a novel multi-band fractal slot antenna design and optimization method for RFEH systems. The fractal antenna designs that were simulated using the CAD design suite, as well as the final design and performance that captured multiple bands are described. The rectifying and matching networks are next evaluated, and the ADS (Advanced Construct System) software is used to construct and simulate each circuit. Additionally, the voltage regulator, energy storage, and application are addressed along with the required voltage to completely function a wireless sensor node, which may be utilized in many applications including wearable technology and intelligent traffic systems (ITS) that control roads to improve environmental quality. Finally, experimental results demonstrated good agreement with simulations that had a high radiation efficiency and gain.

A novel women's ovulation prediction through salivary ferning using the box counting and deep learning

There are several methods to predict a woman's ovulation time, including using a calendar system, basal body temperature, ovulation prediction kit, and OvuScope. This is the first study to predict the time of ovulation in women by calculating the results of detecting the fractal shape of the full ferning (FF) line pattern in salivary using pixel counting, box counting, and deep learning for computer vision methods. The peak of a woman's ovulation every month in her menstrual cycle occurs when the number of ferning lines is the most numerous or dense, and this condition is called FF. In this study, the computational results based on the visualization of the fractal shape of the salivary ferning line pattern from the pixel-counting method have an accuracy of 80%, while the fractal dimensions achieved by the box-counting are 1.474. On the other hand, using the deep learning image classification, we obtain the highest accuracy of 100% with a precision value of 1.00, recall of 1.00, and F1-score 1.00 on the pre-trained network model ResNet-18. Furthermore, visualization of the ResNet-34 model results in the highest number of patches, i.e., 586 patches (equal to 36,352 pixels), by applying fern-like lines pattern detection with windows size 8x8 pixels.

Applications of the Information Dimension in Detecting Border Perturbations

This study utilizes the concept of information dimension based on Shannon and Tsallis' entropy to analyze the contours of flat objects. Our objective is to employ the information dimension for detecting perturbations in borders. We create examples of squares with slight border perturbations, possessing the same mass and sharing identical box-counting dimensions yet exhibiting distinct information dimensions. This construction was devised with the understanding that entropy is responsive to the image frequency within each box. Consequently, the information dimension provides a more precise index of fractal shapes when compared to the box-counting dimension.

Study of fractality nature in VO2 films and its influence on metal-insulator phase transition

The mechanisms underlying the origin of fractal shape of inclusions of a new phase in VO2 films during metal-insulator phase transition are discussed. The obtained results show that hysteresis of the temperature dependence of resistance R(T) significantly depends on the film morphology and texture. Moreover, some fractal features are observed. To determine the fractal dimension D of the structural elements of the studied films from their images, different fractal analysis approaches were preliminary compared and discussed. As a result of the film image treatments, the boundaries of the structural elements were found to have fractal dimensions of 1.3 to 1.5 or higher and to correlate with the shape of R(T). The fractal boundaries indicate the dominant role of elastic stress on the phase transition of films, which is confirmed by numerical modeling. Based on these results, an analytical model is proposed that relates the free energy of a film to the fractal dimension of its constituents. Depending on the ratio of the elastic and interface specific energies, the position of the free energy minimum F corresponds to a certain fractal dimensionality D. A small interface energy leads to a higher fractal dimension making the initial phase more stable. This conclusion explains well all the effects observed experimentally in VO2. The obtained results provide a better understanding of the influence of structure and morphology on other properties of the studied films.

Optimizing Performance of Antenna Arrays with Clustered Fractal Shapes for Multiband Applications

Fractal antennas are mainly used in multiband applications. However, these types of arrays suffer from numerous disadvantages, such as high sidelobe levels, low directivity, poor taper efficiency, and high design computational complexity. In this paper, the conventional fractal procedures are redesigned and efficient clustered subarrays are deployed, such that their multiband properties are maintained while simultaneously achieving significant improvements in radiation characteristics. A genetic optimization algorithm is used to find the optimal clustered fractal shapes and their associated amplitude distributions, such that the sidelobe levels are minimized at the narrower beam width, i.e. maximum feasible directivity. Since the optimization process is carried out at the clustered level, it can be represented by merely a few variables, which solves the problem of time intensity. Simulation results confirm the superiority of the proposed clustered fractal array, where the sidelobe level has been reduced to more than -10 dB over a wide range of frequencies. Directivity and taper efficiency have been improved by more than 6 dB and 50%, respectively, in comparison to the parameters of conventional, original fractal arrays. Moreover, the proposed fractal array pattern offers an additional advantage, as it is capable of wide sidelobe nulling at some undesired directions.

Generation of arbitrarily structured optical vortex arrays based on the epicycle model.

Optical vortex arrays (OVAs) are complex light fields with versatile structures that have been widely studied in large-capacity optical communications, optical tweezers, and optical measurements. However, generating OVAs with arbitrary structures without explicit analytical expressions remains a challenge. To address this issue, we propose an alternative scheme for customizing OVAs with arbitrary structures using an epicycle model and vortex localization techniques. This method can accurately generate an OVA with an arbitrary structure by pre-designing the positions of each vortex. The influence of the number and coordinates of the locating points on customized OVAs is discussed. Finally, the structures of the OVA and each vortex are individually shaped into specifically formed fractal shapes by combining cross-phase techniques. This unique OVA will open up novel potential applications, such as the complex manipulation of multiparticle systems and optical communication based on optical angular momentum.

On the fractal dimension of carbon black particles in pyrolysis flow reactors

This study presents a numerical approach for simulating the internal morphology of carbon black particles formed in a pyrolysis reactor. The simulation process involves a two-step approach using population balance models (PBM) and detailed population balance models (DPBM), solved via sectional and stochastic methods, to simulate the arrangement of primary particles within aggregates to allow determination of their fractal dimension (FD). The outcome is a novel introduction of simulating real flow reactors that for the first time provides the fractal dimension of particles as an output, rather than an assumed input. The results of this study have practical implications for optimizing the synthesis processes in carbon black production. The effects of various production parameters, including aggregation efficiency, temperature, pressure, and acetylene concentrations, on the fractal dimension values of carbon black particles are examined. It is observed that higher temperatures lead to the formation of larger fractal shapes with lower fractal dimensions and larger primary particle diameters. Moreover, increased reactor pressure and higher aggregation efficiency enhance the formation of carbon black aggregates, but also have a time-based effect with higher compactness at longer residence times. The time-based effect reveals the importance of sintering, where high loads of small particles enhance the overall sintering of the aggregates. These findings provide insights into the interplay between temperature, pressure, and particle morphology, highlighting the dynamic nature of carbon black nanoparticles and their response to synthesis process conditions.

Theoretical analysis of dendrite formation generated in an electroosmotic flow with variable shape microelectrodes

In this work, an analytical solution for the hydrodynamic forces that transport silver ions between microelectrodes of axially variable shape is presented. Four different microelectrode shapes were employed to explore the use of passive methods for silver dendritic growth. The results indicate that a fractal microelectrode shape promotes better silver dendritic formation due to lightning rod effect, while in all configurations, a condition of no-contact between dendrites is reached due to an induced pressure field.