Equilateral Research Articles

Packing a triangle by equilateral triangles of harmonic sidelengths

Equilateral triangles of sidelengths 1/3, 1/4,1/5,1/6,1/7,… can be packed into an equilateral triangle of sidelength (π2/6−5/4)1/2+1/270.

Deciphering the influence of shape on vortex-induced vibrations: Insight from diamond and equilateral triangle cylinder simulations

In this study, we present a comprehensive numerical investigation on the impact of geometric shapes on vortex-induced vibrations (VIV). We deploy the OpenFOAM computational fluid dynamics toolbox to simulate undamped transverse flow-induced vibrations in diamond and equilateral triangular cylinders, operating at a Reynolds number of 100 in a uniform flow. Both cylinders possess an identical mass ratio of 10 and operate within a reduced velocity range of 1–13. Our findings reveal a substantial shift in VIV branching behavior when transitioning from a diamond to a triangular geometry, with both cylinders exhibiting solely VIV responses. Intriguingly, the triangular cylinder does not exhibit a lock-out feature. Furthermore, the triangular cylinder showcases rich dynamical behavior, the occurrence of beating. Coinciding with this geometric transition is a surge in fluid forces and heightened flow asymmetry. While the diamond cylinder predominantly exhibits the P + S mode of vortex shedding, the triangular cylinder displays an unconventional 2P vortex arrangement, contributing to the observed asymmetry. As the geometry transitions from diamond to triangular, we note a phase alignment between the lift and transverse displacement. Remarkably, the triangular cylinder exhibits a higher energy conversion efficiency than its diamond counterpart. This research underscores the significant influence of geometry on vortex-induced vibrations, providing pivotal insight for optimizing the design and performance of structures subjected to fluid flows.

Design of two-dimensional piezoelectric laser scanner system for precision laser beam steering.

In this work, a large aperture two-dimensional (2D) piezoelectric laser scanner system with an onboard sensor to detect the scanner's deflection angle has been designed and prototyped. The proposed 2D laser scanner is driven using three amplified piezoelectric actuators assembled in an equilateral triangle configuration to provide structural stability and compactness. To overcome the nonlinearity that derives from the hysteresis effect of the piezoelectric actuators, the tip-tilt principle and the control schemes of the 2D scanner are analyzed and compared. Thanks to the combination of onboard sensing of the deflection angle of the scanner with the closed-loop control capability, this 2D scanner's features include a large aperture and high accuracy. The experimental results demonstrate that closed-loop control achieves higher control accuracy than the open-loop control approach, leading to a reduction in the relative error from 2%-4% to ∼0.5%, while the deflection angle tracking accuracy lies approximately within the 40 μrad range.

Demographic characteristics influencing the stem subsidence in total hip arthroplasty: an imaging study

IntroductionThe present study evaluated whether patient demographic characteristics influence the subsidence of the stem in total hip arthroplasty (THA). The following characteristics were evaluated: age, height, weight, and sex. The association between the time elapsed from the THA implantation and the amount of stem subsidence was also investigated.MethodsThe records of patients who underwent THA in the period between 2016 and 2023 were accessed. All patients underwent two-staged bilateral THA using cementless DePuy collarless Corail (DePuy Synthes, Raynham, MA, USA) stems. The following parameters were measured and compared to assess stem subsidence: distance from the proximal femur at the stem bone interface and the medial apex of the regular triangle built within the trochanter minor (point A); distance from the medial apex of the regular triangle built within the trochanter minor and the distal portion of the femoral stem (point B).ResultsOverall, 294 patients were included. 62% (182 of 294 patients) were women. 45% (134 of 296 THAs) were on the right side. The mean age was 64.9 ± 10.4 years. The mean BMI was 28.3 ± 5.1 kg/m2. The mean length of the follow-up was 14.4 ± 11.0 months. The mean subsidence in point A was 2.1 mm (P < 0.0001), and that in point B was 3.1 mm (P < 0.0001). There was evidence of a weak positive association between patient weight (P < 0.0001), age (P = 0.03), follow-up (P = 0.002) and the amount of stem subsidence. Patient height did not demonstrate any association with the amount of stem subsidence (P = 0.07). There was no difference in stem subsidence between women and men (P = 0.9).ConclusionStem subsidence in THA using cementless DePuy collarless Corail implants is approximately 2.6 mm after 14.4 months. Greater patient weight, age, and longer time elapsed from THA implantation were associated with greater stem subsidence. Patient height and sex did not demonstrate any influence on the amount of stem subsidence. These results must be considered in light of the limitations of the present study.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

ASSESSMENT OF DEAD ZONE OF JOINTLY OPERATING RADARS

Research relevance. With the emergence of unmanned aerial vehicles, the assignments of combating them have also become significantly more relevant. Today, operational countermeasures against UAVs are of great importance. Modern UAVs are capable not only of conducting video reconnaissance in a certain area, but also of observing a specific object for a long time and striking it. After detecting and identifying UAVs by RLS, it is necessary to take measures to neutralize them. In the paper, a mathematical model of the evaluation of the dead zone of RLS working together for the effective detection of unmanned aerial vehicles was established and a comparison of the dead zones was made on concrete examples. The purpose of the research work is to solve the problem of effective placement of jointly operating RLS by mathematical methods for nonobserved remaining part of them had minimal volume and to select the necessary placement scheme. The following tasks are solved in the article: analysis of the characteristics of the radar field; development of a mathematical model for evaluating the dead zone of joint radar stations; evaluation of some accommodation options. The following research methods are used to solve the issues: theoretical analysis, mathematical modeling, comparative analysis. The following results were obtained: it was observed that the dead zone was reduced by 1.6 times in the four-station square arrangement scheme, and by 1.3 times in the three-station triangular arrangement scheme. Conclusions: From the comparison of the obtained numbers, it can be concluded that the creation of a radiolocation area by placing RLSs on the vertices of equilateral triangles (in the form of a checkerboard) is more economically profitable, as it requires fewer stations.

Effects of different surface textures on tribological performances of the cylinder liner-piston ring: An experimental study

The tribological performance of internal combustion engines (ICEs) has an important impact on fuel efficiency and power consumption, and the wear of cylinder liner-piston ring (CLPR) system affects the operation reliability. The surface texture of CLPR can significantly improve the tribological performance of ICEs. Aiming at the friction and wear of CLPR, the different textures (dimple, equilateral triangle and square) are generated on the CLPR using laser processing, and the different tests are conducted on the UMT-TriboLab tester. The coefficient of friction, wear topography, surface roughness parameters and profiles are measured. The results show that, compared with the non-textured surface, the dimple texture surface of piston ring or cylinder liner has better tribological performance. If the textures coexist on the cylinder liner and piston ring, the tribological performances are better than single surface texture. Furthermore, the mixed textures can significantly improve the surface lubrication performance and the ability to store abrasive particles, and better surface quality can be obtained at the same operating conditions to enhance the tribological performance. This study can provide guidance for reducing the friction and wear of CLPR system of ICEs, and achieve energy saving and emission reduction of ICEs.

Molecular Dynamics in Rydberg Tweezer Arrays: Spin-Phonon Entanglement and Jahn-Teller Effect.

Atoms confined in optical tweezer arrays constitute a platform for the implementation of quantum computers and simulators. State-dependent operations are realized by exploiting electrostatic dipolar interactions that emerge, when two atoms are simultaneously excited to high-lying electronic states, so-called Rydberg states. These interactions also lead to state-dependent mechanical forces, which couple the electronic dynamics of the atoms to their vibrational motion. We explore these vibronic couplings within an artificial molecular system in which Rydberg states are excited under so-called facilitation conditions. This system, which is not necessarily self-bound, undergoes a structural transition between an equilateral triangle and an equal-weighted superposition of distorted triangular states (Jahn-Teller regime) exhibiting spin-phonon entanglement on a micrometer distance. This highlights the potential of Rydberg tweezer arrays for the study of molecular phenomena at exaggerated length scales.

Dispersion reduction in Feng and Wu’s IPDG method

We give an analytical justification of the determination of an important parameter involved in Feng and Wu’s solution of the Helmholtz equation by an Interior Penalty Discontinuous Galerkin (IPDG) method (Feng and Wu, 2009). This parameter was determined in this reference from a large number of numerical tests for a mesh composed of equal equilateral triangles at a fixed frequency. It plays an essential role in the reduction the dispersion error for a polynomial approximation of degree 1. We show how this justification makes it possible to slightly improve it and to handle other structured meshes. The analytical results are evaluated by several numerical experiments, showing in particular that the parameter determined in principle for equal equilateral triangles also eliminates dispersion for unstructured, well-smoothed meshes.

NUMERICAL SIMULATION OF PASSIVE MICROMIXER WITH NOVEL OBSTACLE STRUCTURE

The micromixer is an important prerequisite for realizing the function of the microfluidic chip system. Micromixers are mainly used in protein crystallization, biological reactions, drug delivery and other fields. Efficient micromixers enable fast bioreaction processes. In this paper, according to the principle of plane geometry, micromixers with three different structural obstacles, circular, equilateral triangle and square, are designed to study how to improve the mixing efficiency of the micromixer. In this paper, the influence of the obstacle structure on the mixing efficiency and pressure drop of the micromixer is analyzed. Through numerical simulation, the results show that when Re = 100, the mixing efficiency of the circular obstacle micromixer is up to 97.4%, and the pressure drop is 15,789.597 Pa. In addition, this paper analyzes that the use of cylindrical obstacles can make the fluid generate horizontal and vertical vortex flow, break the usual laminar flow, and significantly improve the mixing efficiency.

Equilateral triangles have minimal area and perimeter among all triangles containing a given circle in Hilbert planes

Equilateral triangles have minimal area and perimeter among all triangles containing a given circle in Hilbert planes

On the cubic equation with its Siebeck–Marden–Northshield triangle and the quartic equation with its tetrahedron

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulæfor the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic x4+ax3+bx2+cx+d is proposed for which the conditions are imposed on the individual coefficients a, b, c, and d. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of a and b. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than (3/4)3a2−8b from −a/4. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on a and b and are roots of quadratic equations — which makes this localization helpful for quartic equations with complicated parametric coefficients.

Development of a 3D Computer Simulation Model Using C++ Methods

The article presents modelling of spherical elements based on the developed computer model. We recorded the main combinations of spherical particles during filling, which are formed in the hopper. It was found that the most likely combination that occurs when modelling spherical elements consists of three balls. It should be noted that in the cross-section of such a combination passing through the center of the balls, an equilateral triangle is formed. And in the cross-section of the structure, which consists of four spherical balls, a rhombus is formed, if you connect the centers of these spherical elements. It is worth noting that from this formed combination of spherical elements, it can be seen that the rhombus forms two smaller equilateral triangles that fix the process of pushing the spherical balls apart. In turn, the process of pushing spherical elements apart made it possible to fix the contact between spherical elements, as well as to state the stable position of each (individual) particle. This paper also presents the main fragments of encoding the source text of a 3D computer model for modelling spherical elements, which made it possible to optimize the model parameters. It was found that from the obtained data on the distribution of coordination numbers for different volume fillings of spherical elements, it follows that the largest filling was 72 %, which corresponds to the state when 112 lobules have an average coordination number of 3,92.

Investigation of various absorber surface shapes for performance improvement of solar chimney power plant

Solar chimney power plants are important for the development of renewable energy. The absorber surface shape can play a critical role in the solar chimney’s performance. In the relevant literature, there is no comprehensive study on the effect of various absorber surface shapes on the performance of solar chimney. The present study aims to numerically investigate the effect of absorber surface shapes on the power generated by a solar chimney. The investigated absorber shapes include a right triangle, an equilateral triangle, a rectangle, and a semi-circle. In addition, the effect of the absorber surface height is studied. The modelling of heat transfer and fluid flow in the chimney is based on conservation equations. The finite volume method is used to solve the set of equations using ANSYS Fluent software. The model is verified by comparison with experimental data from the literature, and good agreement is obtained. The outcomes of this study show that the maximum effect is found in the case of a rectangular-shaped absorber surface. Furthermore, increasing the height of the modified absorber surface increases the generated power. The output power improves by up to 67.4% in the case of a rectangular absorber with a height of 7.5 cm when compared to the baseline flat-chimney. These findings are useful for designing efficient solar chimney.

Reverse Isoperimetric Inequality for the Lowest Robin Eigenvalue of a Triangle

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

Experimental study on energy harvesting maximization from the flow-induced vibration of a highly confined bluff body

Flow-induced vibrations of rigid prisms supported elastically were studied experimentally in a free-surface water channel with a high blockage (2/5). The study focused on finding the prism cross-sectional shape that maximizes the efficiency of energy harvesting. Seven cross-sectional shapes were tested: square, circular, 45° tilted square, equilateral triangle, isosceles 120° triangle, D-section, and C-section. All other dimensionless parameters of the problem, mass ratio, damping, blockage ratio, reduced velocity range, and the Reynolds (Re) number (characteristic velocity times characteristic length divided by kinematic viscosity) range (400–1070), were kept unchanged. By doing so, the effect of the cross-sectional shape was isolated. D-section proved to be the geometry with the highest values of energy transfer efficiency. A hysteresis loop was present in its oscillatory response (dimensionless oscillation amplitude vs reduced velocity). This loop was characterized by two branches, (+) and (−), meaning a bi-valued amplitude response for each reduced velocity. Regarding temporal patterns of wake topology and body motion, it was found that synchronization occurs in the (+) branch, but not in the (−). Regarding vortex shedding modes, particle image velocimetry was used for identification purposes, and it was found that the 2P mode is the dominant mode in the (+) branch, while the 2S mode pervades the (−). Finally, a new relative reduced velocity definition was introduced, and, when re-plotting the experimental results, it was found that the hysteresis loop disappears, thereby providing a more compact mathematical description of the observed phenomena.

Shear wave modes in an equilateral triangular prismatic bar.

Shear wave excitation of an equilateral triangular solid bar shows the existence of horizontally polarized (SH) and vertically polarized (SV) shear waves. The SH wave modes are Lamé solutions of the Neumann problem for displacement and Dirichlet problem for stress in the equilateral triangle. It is shown that Lamé's solutions are obtained from superposition of three equal plane waves and their reflections. The Neumann SH modes have cutoff wavenumber 4mπ/(3a), where m=1,2,3,…, a being the side length of triangular cross section. The first Dirichlet SH mode has a higher cutoff wavenumber 4π7/(3a). The SV modes are related to the 30°-60°-90° and 30°-120°-30° sub triangles. Their cutoff wavenumbers lie between those of the first (m = 1) and second (m = 2) Neumann SH modes. The standing wave condition in each sub triangle is related to three unequal plane waves which decompose via symmetrical component analysis into two different sets of equal plane waves. A nonlinear product solution based on the new plane wave set leads to modal SV wave solutions. Dispersion curves calculated using COMSOL confirm the correct cutoff wavenumbers of both SH and SV waves. Normal displacement mode shapes are calculated analytically and verified experimentally using laser vibrometer with good agreement for all modes.

Ethnomathematics: Utilization and Introduction of Geometry Buildings Using Janur (Coconut Leaf)

One area of mathematics that requires a good understanding and depiction of abstract concepts is geometry. Therefore it is necessary to apply realistic learning. One type of realistic learning is ethnomathematics. Ethnomathematics is a bridge between mathematics and culture by linking culture and students' everyday experiences. Traditional crafts that can be lifted are creations from coconut leaves (janur) which can be formed into various kinds of craft creations that contain geometric shapes. The type of research used in this research is descriptive qualitative research. Data collection was used through literature and interviews. The results of the research show that creations from coconut leaves (janur) can be geometric shapes that can be used as ethnomathematics objects. These forms include (1) coins that can be made into hexagons, (2) kerises that have elements of equilateral triangles, right triangles, and parallelograms, (3) diamond bricks that have elements of square geometry lengths and blocks, (4) balls (balans) which have square and cube geometry elements, (5) sinta diamonds which have rhombus geometry elements and rhombus upright prisms. Other geometric elements contained in the janur creations are congruence, angles, and symmetry.&#x0D; Keywords: ethnomathematics, geometry, coconut leaves.

A new window function for the spectral analysis of effect of celestial body on Taiji mission below 0.1 mHz

The Taiji project is a space gravitational wave detection mission consisting of three satellites that form a large equilateral triangle with a side length of approximately 3×109 m in heliocentric orbit. It may have the potential to detect gravitational waves below 0.1 mHz. Gravitational wave signals in the μHz range have a strain sensitivity as large as 2.9×10-14, or approximately 8.7×10-5 m variation in inter-satellite distance. However, inter-satellite distances are affected by the celestial bodies in the solar system, introducing changes that may reach magnitudes of 3×107 m; thus, the effect of celestial bodies on these distances may exceed the gravitational signals below 0.1 mHz. Therefore, it is necessary to accurately estimate the effect of the gravity of celestial bodies below 0.1 mHz. In this study, based on the inter-satellite distances in the Taiji mission, a window function design method is proposed and a six-term cosine window function is designed. It has a high decay rate of h-12 and a good peak sidelobe level of −88 dB. It improves the attenuation of spectral leakage while maintaining an acceptable minimum number of signal periods, which is beneficial for signals where it is difficult to obtain more periods, such as for the Taiji mission. Based on the analytical results of the states of satellites in cases when the satellites move only in the gravitational field of the Sun, the ability of the new window function is validated.