Geometry In Nature Research Articles

Experimental realization of fractal fretwork metasurface for sound anomalous modulation

Natural creatures and ancient cultures are full of potential sources to provide inspiration for applied sciences. Inspired by the fractal geometry in nature and the fretwork frame in ancient culture, here we design the acoustic metasurface to realize sound anomalous modulation, which manifests itself as an incident-dependent propagation behavior: sound wave propagating in the forward direction is allowed to transmit with high efficiency while in the backward direction is obviously suppressed. We quantitatively investigate the dependences of asymmetric transmission on the propagation direction, incident angle and operating frequency by calculating sound transmittance and energy contrast. This compact fractal fretwork metasurface for acoustic anomalous modulation would promote the development of integrated acoustic devices and expand versatile applications in acoustic communication and information encryption.

Spiral‐Shape Fast‐Moving Soft Robots

AbstractSoft robots typically exhibit limited agility due to inherent properties of soft materials. The structural design of soft robots is one of the key elements to improve their mobility. Inspired by the Archimedean spiral geometry in nature, here, a fast‐moving spiral‐shaped soft robot made of a piezoelectric composite with an amorphous piezoelectric vinylidene fluoride film and a layer of copper tape is presented. The soft robot demonstrates a forward locomotion speed of 76 body length per second under the first‐order resonance frequency and a backward locomotion speed of 11.26 body length per second at the third‐order resonance frequency. Moreover, the multitasking capabilities of the soft robot in slope climbing, step jumping, load carrying, and steering are demonstrated. The soft robot can escape from a relatively confined space without external control and human intervention. An untethered robot with a battery and a flexible circuit (a payload of 1.665 g and a total weight of 1.815 g) can move at an absolute speed of 20 mm s−1 (1 body length per second). This study opens a new generic design paradigm for next‐generation fast‐moving soft robots that are applicable for multifunctionality at small scales.

SPATIAL SENSE ABILITY INSTRUMENT FOR PRIMARY SCHOOL STUDENTS

This research is motivated by the importance of spatial sense skills possessed by elementary school students. The research aims to develop an instrument of spatial sense ability suitable for elementary school students. The indicators developed in the preparation of questions include: 1) classifying real objects into 3D form (building space); 2) investigating, exploring, and describing geometry in nature and the real world; 3) exploring the directional relationship of objects in space; 4) analyzing the nature of 3D shapes and interpreting 3D shapes into 2D forms. Research and Development was chosen as the method of this research. A total of 33 grade VI students of SDN 1 Nagri Kidul, Purwakarta Regency, West Java Province, were the samples in this study. The sample was selected by purposive sampling. The instrument used is an essay test consisting of seven questions regarding the geometry of cubes and blocks. The expert (Professor in the field of mathematics education) validated the seven questions that had been prepared. Expert considerations are used as the basis for the improvement of the compiled questions. Then the questions that have been corrected are tested on the research sample. The results of the trial were calculated for validity, reliability, difficulty index, and discriminatory power. Of the seven questions compiled, six were declared valid, reliable, and had high discriminatory power, and the level of difficulty of the questions consisted of medium and difficult. Thus, 6 of the seven items developed can be used by teachers or researchers as an instrument to measure the spatial sense abilities of elementary school students, primarily fifth-grade students.

Fire development study on physical models of transport tunnels

The work gives the study results for 1:10 scale physical models of the road tunnels. The tunnel slope changes from 0 to 10% with a 2% spacing. As the models show, the fire strength varies within the range of 5 to15 MW with 5 MW increments. The properties of the tunnel geometry in nature are as follows: cross-sectional area: 46.7m2, length: 120 m, width: 8.5m, height: 5.5m and width and height ratio: 1.54. The models are made of 2 mm-thick stainless-steel sheets. We studied the nature of the spread of smoke, the variation of critical velocity depending on the tunnel slope and the characteristic value of such variation - the grade correction factor. The observation over such values is done depending on the air temperature variability in the tunnel model. The measurements are made with K-type thermocouples with open and closed detectors. Maximum temperature measurement is 800°C. The thermocouples are located in the holes provided in the ceiling of the tunnel model. The distance between the holes is 5 and 10 cm what corresponds to the natural length of the tunnel of 5 and 10 m. Fire is modeled by using the natural gas. The indications of thermocouples and air and gas meters are measured simultaneously and the primary analysis and the digital data transmission are provided with a Data Taker DT85 data logger. The gained results may be used to develop ventilation and emergency management projects for transport tunnels as well as to train personnel and rescuers.

Seismic Rating of Multistoried Frame System by Non-Linear Pushover Research

This paper compares multistoried frame structures on plain ground and on hilly areas, first under the action of normal commercial loading and thereafter by considering the effect under the action of heavy loads. Behavior of multistoried frame buildings on plains are completely different from buildings which are constructed on hilly slopes. Buildings constructed on plain is simple and symmetrical, whereas the buildings that are constructed on hilly slopes are complex in geometry and asymmetrical in nature. Similarly, the behavior of commercial buildings with normal loading is distinctively different in comparison to special buildings such as Data centre buildings which are heavily loaded under gravity. Four different types of Symmetrical and Asymmetrical analytical Models are generated for five and ten storey buildings using "ETABS". Seismic evaluation of these models is done separately under the effect of normal and heavy loading by performing Nonlinear Pushover analysis.

Adhesion tilt-tolerance in bio-inspired mushroom-shaped adhesive microstructure

We studied experimentally and theoretically the effect of different tilt angles on the adhesion of mushroom-shaped adhesive microstructures. The marginal measured influence of tilting on pull-off forces is quantitatively well confirmed by numerical and theoretical calculations and was shown to be a direct consequence of an optimized stress distribution. In addition, the presence of a joint-like narrowing under the contact elements, as found in some biological attachment systems, was shown to further contribute to the tilt-tolerance. The results obtained allow us to explain the advantage of the widely observed mushroom-shaped contact geometry in nature for long-term and permanent adhesion.

Dynamical quantum Hall effect in the parameter space

Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.

Moiré-free color halftoning using hexagonal geometry and spot functions

Hexagons are widely observed as a packing or tiling geometry in nature, yet they appear to have been avoided in conventional halftone tiling. A goal of the present study is to understand the potential barriers that have prevented their use and present new halftone geometry options that overcome the issues while offering several potential benefits. While conventional halftone geometries often include the fourth screen (e.g., yellow) in a suboptimal manner, the hexagonal geometry presented here can include a clustered-dot fourth screen moiré-free. Hexagonal screens can appear to have smoother texture. Due to differences in packing geometry and touch point geometry, hexagons have the potential to possess different tone reproduction characteristics, which may be favorable for some marking processes. We also present a parametrically controlled hexagonal halftone spot function that allows for optimization of dot touch points and provides compact growth. The controllable touch points can prevent a tone reproduction discontinuity, while the compact growth throughout the gray range ensures maximum stability. Additionally, we present a three-colorant dot-off-dot halftone configuration using hexagonal geometry. Examples are provided.

On reactive solute transport through a curved pipe

The transport of a reactive solute by diffusion and convection in a thin (or long) curved pipe is considered. Using asymptotic analysis with respect to the pipe’s thickness, the effective model for solute concentration is formally derived. A simple approximation is computed, showing explicitly the effects of the pipe’s geometry in nature and magnitude.

Particle image velocimetry of a flow at a vaulted wall

The assessment of flow along a vaulted wall (with two main finite radii of curvature) is of general interest; in biofluid mechanics, it is of special interest. Unlike the geometry of flows in engineering, flow geometry in nature is often determined by vaulted walls. Specifically the flow adjacent to the wall of blood vessels is particularly interesting since this is where either thrombi are formed or atherosclerosis develops. Current measurement methods have problems assessing the flow along vaulted walls. In contrast with conventional particle image velocimetry (PIV), this new method, called wall PIV, allows the investigation of a flow adjacent to transparent flexible surfaces with two finite radii of curvature. Using an optical method which allows the observation of particles up to a predefined depth enables the visualization solely of the boundary layer flow. This is accomplished by adding a specific dye to the fluid which absorbs the monochromatic light used to illuminate the region of observation. The obtained images can be analysed with the methods of conventional PIV and result in a vector field of the velocities along the wall. With wall PIV, the steady flow adjacent to the vaulted wall of a blood pump was investigated and the resulting velocity field as well as the velocity fluctuations were assessed.

Microwave transmission behavior of the single dissymmetrical hexagonal split-ring resonators

Inspired by the dendritic geometry in nature, we designed the single dissymmetrical hexagonal split ring resonators (SRRs). The SRRs were fabricated on one side of the dielectric substrate using shadow mask/etching technique. By experiments and computer simulations, we have researched the microwave transmission behavior of the SRRs in the frequency band of 7—12 GHz. The results show that, for an individual hexagonal SRR, the resonance frequency is affected by the size of the SRR; for two SRRs, there are two resonance peaks in the transmittion curve when they are close to each other. For three SRRs or the radialized rings, the resonance frequency of the structure with high-level branches shifts towards lower values compared with that with no branches. The simulations are in agreement with the experimental results.

Geometry in nature and Persian architecture

Nature displays profound preference for certain specific ratios to design her life-forms. These are geometric relationships that are transcendent and originated from Sacred Geometry. The view that geometry had a ritual origin is a part of a wider view that civilisation itself had a ritual origin, and therefore the history of utilisation of Sacred Geometry by man goes back to many centuries ago. The Pythagorean tradition, and the Egyptian and Babylonian sciences from which it derived, and Persian mathematics, a part of which reflects a Pythagorean intellectuality, are based on the sacred conception of numbers and their symbolism. In the traditional world, geometry was inseparable from the other sciences of the Pythagorean Quadrivium, namely arithmetic (numbers), music and astronomy. Traditional geometry is related to the symbolic configurations of space. Geometric forms such as the triangle, square and various regular polygons, the spiral and the circle are seen in the traditional perspective to be, like traditional numbers, as aspects of the multiplicity of the Unity. Architecture itself has always had a sacred meaning to all traditional civilisations through millennia, by which means man has tried to provide for himself a manifestation of heavens. Persian architecture always emphasised on Beauty, and by means of Sacred Geometry Persians measured the proportions of heaven and reflected them in the dimensions of buildings on the earth. A comprehensive utilisation of proportions in Persian architecture, such as in the design of plans, elevations, geometric and architectural patterns, and mechanical and structural features, can be proved through geometrical analysis of Persian historical buildings. In this paper, the sacred conception of geometry and its symbolism in the Pythagorean tradition, and Sacred Geometry and proportions in natural life-forms will be explained. The use of the science of geometry in design of a number of Persian historical buildings will be presented. The geometric factors upon which the design of these buildings, from both architectural and structural viewpoints, is made will be discussed.

Self-Similar Criticality

Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.

Fractional quantum Hall effects as an example of fractal geometry in nature

A prescription is provided for constructing the Hall curve including both integral (I)- and fractional (F)-quantum Hall effects (QHE) that is based upon the iterative application of particular transformations simultaneously to the Hall resistance (Rxy) and magnetic field (B) axes of a template constructed from the elementary (integral quantum) Hall curve to filling factor v = 1. The construction shows that scaled copies of the elementary Hall curve reappear in various parts of the constructed curve upon increasing the magnification, resulting in FQHE sequences in higher Landau bands, and novel FQHE sequences between main sequence FQHE’s in the lowest Landau band. The self similarity observed in the constructed Hall curve helps to draw a connection between FQHE’s and the classical problem of an electronin-a-periodic-potential-subjected-to-a-magnetic-field (‘Hofstadter’s butterfly’), and suggests that fractional quantum Hall effects constitute another manifestation of fractal geometry in nature—one that might also be viewed as a signature of transport in a Wigner crystal.

The influence of pre-existing thrust faults on normal fault geometry in nature and in experiments

Relations between normal faults and pre-existing thrust faults are classically described in terms of three basic situations: normal faults can cross-cut thrust faults; they can branch out from thrust faults at depth on a de´collement level, or they can entirely reactivate thrust planes. The mechanical aspects of these types of interaction were studied by analogue modelling in which sand simulates the ‘brittle’ rocks and silicone putty an interlayered de´collement. The models underwent compression, producing thrust faults with variable dips, followed by extension. Three possible ways of interaction are described here: (a) no interaction occurs in the case of low-dip thrust faults (<32 ° ± 1 °) and normal faults are developed independently, displaying a listric geometry; (b) branching at depth on the de´collement level occurs when dip of the thrust faults reaches 32 ° ± 1°. In this case, the dip of the normal faults, whose geometry becomes planar, decreases with increasing thrust dip. We suggest that this change in dip of normal faults depends upon the rotation of stress tensor axes along the pre-existing fault zone, where a drop in the friction coefficient is likely to occur; (c) reactivation occurs in brittle material when dip of the pre-existing fault exceeds 41° ± 1°.

Phyllotactic Patterns for Domes

The naked eye misleads us, when we wish to understand geometry and patterns in Nature. The basis of visible natural structures lies in the repeated application of very simple procedures [1],[2],[3]. If we want to work in harmony with Nature, our technicity should strive to work according to her ways and to her rules. In growth, we can distinguish expanding and stacking. In many organisms, we observe a relatively smooth overall expanding. In the contrary, in the snail house, material is added only on the outside. Plants show patterns in the placement of appendages. The author found a very simple algorithm, that leads to these phyllotactic patterns [4]. The Dislodgement Model explains phyllotaxis via an inversion from expanding to stacking. This inversion is allowed, while plants follow the rules of gnomonic growth [5], Now, we have a tool to reproduce natural stackings. The section 'Basic principles for the arrangement of units' in the present article is a revised version of part of 'Creating Phyllotaxis: The Dislodgement Model' [4].

Geometry in nature

Geometry in nature

Mathematics and Beauty: A Sampling of Spirals and 'Strange' Spirals in Science, Nature and Art

Computers with graphics can be used to produce visual representations of complicated physical and mathematical structures and phenomena. This paper presents an overview of spiral geometry in mathematics, nature, physics, civilization and art, with an emphasis on novel spiral forms created by the author. ‘Recipes’ for both traditional and nontraditional spirals are given. The computer-generated patterns are of interest artistically and mathematically, and they reveal a visually striking and intricate class of spirals whose generating algorithms range from electronic circuit equations to random segment methods.