Acute Triangles Research Articles

Conditions for Inscribed Triangle to Be Similar to Given Triangle

Abstract: In this research paper, I will derive the conditions for similarity between a triangle and its inscribed triangle. I will go through the inverse similarity conditions of an inscribed triangle for acute angled triangle and right angled triangle. I will use the basic laws of geometry to deduce the conclusion. Apart from mid-point triangle, we will analyze the conditions of similarity for other inscribed triangle.

Geometric approach to the proof of Fermat’s last theorem

A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle.
 The proof in this case is carried out by introducing a new auxiliary function f(k,p)=kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p).
 Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation f(k, p)=0 and the number corresponding to this supposed solution are considered.
 The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved

The oblique triangle configuration of three parallel screws for femoral neck fracture fixation using computer-aided design modules

Closed reduction and internal fixation with three cannulated compression screws is a common method for treating femoral neck fractures in young and middle-aged patients. Protocols including the inverted triangle configuration and dispersion of the screws still needed further supports. The purpose of this study was to explore a novel oblique triangle configuration (OTC) of three screws in fixing femoral neck fractures based on the morphology of the femoral neck isthmus (FNI). The computer-aided design modules were used to explore the ideal spatial configuration with largest triangle by three parallel screws. A univariate evaluation model was established based on the oval-like cross-section of the FNI. When the three screws were positioned by the OTC, Inverted Equilateral Triangle Configuration (IETC), and the Maximum Area Inverted Isosceles Triangle Configuration (MA-IITC) respectively, the proportion of area and circumference in the cross-section of FNI and the changing trend of proportion were compared under various torsion angles, eccentricity, and cross-sectional area of FNI. The area and circumference ratios of the parallel screws using the OTC method were significantly higher than in the IETC and MA-IITC groups. In the univariate evaluation model, the OTC area ratio and circumference ratio remained stable under the different femoral neck torsion angles, FNI cross-sectional area, and eccentricity. The OTC method provided an ideal spatial configuration for the FNA fixation with the largest area using three parallel screws. The position of the posterior screw was also away from the metaphyseal artery, potentially reducing the possibility of vascular injury and screw penetrating.

Sexual Predilection of Lingual Morphology - A Cross-sectional Comparative Study.

Background:Proof of identity acts as a key title role in any criminal investigation. Training of tongue prints has been documented and acknowledged as the gold standard for personal identification worldwide nowadays. Tongue printing has proved as an innovative biometric tool and can be relied upon as an authoritative forensic tool. Research works have been carried out on tongue print independently. However, studies focusing on the morphology or shape of the tongue have not been commenced to a prodigious scope.Aim:The contemporary trial was undertaken upon the people to train and scrutiny the morphological shape of the tongue and validate their magnitude as assistance in sexual dimorphism.Materials and Methods:Hundred dental students in the age group of 18–25 years were included in the trial. For the trial clinical scrutiny, photographs and lingual impressions were recorded of the tongue.Results:In the current trial of the shape of the tongue for gender identification, we used two classifications. According to TCM criteria, five tongue shapes were used for gender identification: rectangle, acute triangle, obtuse triangle, square, and circle. Among the 100 subjects, the square-shaped tongue was chiefly observed, trailed by the circle and obtuse triangle. In the second criteria, Stefanescu et al. classified tongues into two shapes: U- and V-shaped. When the tongue shape was scrutinized, U shape was seen in 79 persons and V shape in 21 persons out of 100 subjects.Conclusion:Hence, the trial concluded that tongue prints exhibited a qualitative result among diverse sex groupings. It can be preached that the human tongue possibilities to distribute a point of exceptionality in morphological shape; besides, it is apt for utilization in gender acknowledgment.

Symmetry in Motion: Stellae Octangulae, and Equifaced Polyhedra

We discuss three seemingly independent topics concerning polyhedra, which finally show some relations. The first topic concerns generalisations of the classical Stella Octangula, whereby the pair of indirect congruent tetrahedra, which generates such a Stella, allows forced motions of one tetrahedron along the other. Symmetry arguments are used to identify pairs of right triangular pyramids and pairs of indirect congruent tetrahedra as candidates for such “movable” Stellae Oc- tangulae. The second topic discusses equifaced octahedra, as they occur as the common body of equifaced Stellae Octangulae. It turns out that, in general, there exist four octahedra with the same acute face triangles. They differ in the way their diagonals intersect and in the number of symme- tries and are, in this paper, distinguished “type A-octahedra” and “type B-octahedra”. For obtuse face triangles the is no type A-octahedron. For general right triangles there are two type B- octahedra, which for isosceles right triangles coincide and become the single Rodrigues- octahedron. The third topic concerns polyhedra with congruent isosceles right triangular faces, thus generalising the Rodrigues-octahedron. This chapter also the aims at providing material for educational purposes.

The role of nonlocal response in second harmonic generation at metasurfaces with triangular metaatoms

Metasurface consisting of square array of obtuse isosceles triangle metaatoms is found to exhibit much more efficient second harmonic generation (SHG) compared to that of split ring resonators and its heptagonal modification, experimentally as well as numerically. All of them are designed to lack inversion symmetry in horizontal direction but to have vertical symmetry, and to meet the double resonance condition for vertically-polarized fundamental and horizontally-polarized SHG waves. SHG efficiencies were estimated numerically for each structure by evaluating an overlap integral of second order nonlinear polarization and field distribution at the SHG wavelength. The large overlap integral for the triangle metasurface is ascribed to the nonlocal response at the SHG wavelength. Current induced at the center of the triangle near the obtuse angle by horizontally polarized light at SHG wavelength flows toward two corners of acute angles. As a result, polarization charges oscillate in time at the surfaces of two acute corners, which results in strong field oscillation away from the center where the electric field is applied to majority of free electrons. Experimentally observed wavelength dependence of SHG efficiencies for the three metasurfaces are reasonably reproduced by a numerical estimation.

How to Lay Bricks: Local-Global Alignment Predicts Preference for Shifted Tile Patterns.

We examine the aesthetic characteristics of row tile patterns defined by repeating strips of polygons. In experiment 1 participants rated the perceived beauty of equilateral triangle, square and rectangular tilings presented at vertical and horizontal orientations. The tiles were shifted by one-fourth increments of a complete row cycle. Shifts that preserved global symmetry were liked the most. Local symmetry by itself did not predict ratings but tilings with a greater number of emergent features did. In a second experiment we presented row tiles using all types of three- and four-sided geometric figures: acute, obtuse, isosceles and right triangles, kites, parallelograms, a rhombus, trapezoid, and trapezium. Once again, local polygon symmetry did not predict responding but measures of correspondence between local and global levels did. In particular, number of aligned polygon symmetry axes and number of aligned polygon sides were significantly and positively correlated with beauty ratings. Preference was greater for more integrated tilings, possibly because they encourage the formation of gestalts and exploration within and across levels of spatial scale.

A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a4 + b4 = c4 forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

An Investigational Approach for Vowels of the Salar Language Based on a Database of Speech Acoustic Parameters

According to relevant specifications, this article divides, marks, and extracts the acquired speech signals of the Salar language, and establishes the speech acoustic parameter database of the Salar language. Then, the vowels of the Salar language are analyzed and studied by using the parameter database. The vowel bitmap (average value at the beginning of words), the vowel bitmap (average value at the abdomen of words), the vowel bitmap (average value at the ending of words), and the vowel bitmap (average value) are obtained. Through the vowel bitmaps, we can observe the vowel in different positions of the word, the overall appearance of an obtuse triangle. The high vowel [i], [o], and low vowel [a] occupy three vertices, respectively. Among the three lines, [i] to [o] are the longest, [i] to [a] are the second longest, and [a] to [o] are the shortest. The lines between [a] to [o] and [a] and [i] are asymmetric. Combining with the vowel bitmap, the vowels were discretized, and the second formant (F2) frequency parameter was used as the coordinate of the X axis, and the first formant (F1) frequency was used as the coordinate of the Y axis to draw the region where the vowel was located, and then the vowel pattern was formed. These studies provide basic data and parameters for the future development of modern phonetics such as the database of Sarah language speech, speech recognition, and speech synthesis. It also provides the basic parameters of speech acoustics for the rare minority acoustic research work of the national language project.

STUDENTS’ GEOMETRIC THINKING ON TRIANGLES: MUCH IMPROVEMENT IS NEEDED

A look into students’ misconceptions help explain the very low geometric thinking and may assist teachers in correcting errors to aid students in reaching a higher van Hiele geometric thinking level. In this study, students’ geometric thinking was described using the van Hiele levels and misconceptions on triangles. Participants (N=30) were Grade 9 students in the Philippines. More than half of the participants were in the van Hiele’s visualization level. Most students had imprecise use of terminologies. A few had misconceptions on class inclusion, especially when considering isosceles right triangles and obtuse triangles. Very few students correctly recognized the famous Pythagorean Theorem. Implications for more effective geometry teaching are considered.

Strongly obtuse rational lattice triangles

We classify rational triangles which unfold to Veech surfaces when the largest angle is at least $\frac{3\pi}{4}$. When the largest angle is greater than $\frac{2\pi}{3}$, we show that the unfolding is not Veech except possibly if it belongs to one of six infinite families. Our methods include a criterion of Mirzakhani and Wright that built on work of Moller and McMullen, and in most cases show that the orbit closure of the unfolding cannot have rank 1.

Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics.

The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and "ice cream cone" (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index K, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities χ̃V(k) and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.

CFD and Experimental Study of Wind Pressure Distribution on the High-Rise Building in the Shape of an Equilateral Acute Triangle

There is a lack of detailed information about wind flow and distribution of wind pressure around atypically shaped high-rise buildings. The national standard EN 1991-1-4 Eurocode 1 used to determine the effects of wind on the territory of Slovakia (and indeed other countries of the European Union) does not have a procedure for determining the effects of wind on objects of triangular shape. This presents a problem for designers and engineers, as there exist no generally binding/valid rules to follow when performing the wind effect analysis. This paper shows the procedure of identification and results of the external wind pressure coefficient for the triangularly shaped high-rise building. Two methods of calculation have been chosen for this purpose. First, experimental measurements were performed on a scaled model of the building cross-section in the wind tunnel. Subsequently, software simulations were performed on the same scaled model in the CFD (computational fluid dynamics) program ANSYS CFX. Results of wind pressure were obtained for two directions of wind flow measured in 16 sampling points distributed irregularly around the circumference of the model. Results were mutually compared and verified. At the end, the wind flow effects on a real-size triangular high-rise building in the built-up area performed by software simulation are shown.

Calculation of the triangular panels’ orthotropic covers for the retained domes

The experience of design and construction of single-mesh spherical dome coverings based on meshes with triangular cells was studied. Constructive solutions of wooden domes made of triangular panels with orthotropic plywood covers are considered. The method analysis for constructing a bar model of an orthotropic plate in the form of an oblique triangle has been carried out. It is proposed to reduce the calculation of the triangular dome panel’s rod-sheet structure to the calculation of the hinge-bar system. The stiffness matrices of the contour and intra-contour rods of the super cell are obtained.

Fetal sigmoid colon mesentery made visible by routine ultrasound in the first and second trimester of pregnancy

BackgroundUltrasound is a routine procedure performed during pregnancy to monitor the status of the human fetus, including the development of the digestive tract. The aim of this publication was to determine the shape of the fetal sigmoid colon mesentery during the first and second trimester of pregnancy by means of ultrasound. MethodsThe study was performed in 45 pregnant women (age: 36.4 years on average, range 25–45) during a routine ultrasound examination. The fetuses were between 12 and 22 weeks of pregnancy. The shape of the fetal sigmoid colon mesentery was analyzed. ResultsWe visualized the triangular shape of the mesentery in all 45 cases. A prevalence of scalene or isosceles acute triangles was found in both trimesters. At the 12−13 weeks we observed 5 different forms of mesenteric triangles – the scalene or isosceles acute triangles appeared at 33.3% and 28.6%, respectively. The obtuse scalene was present in 23.8 %. In the 2nd trimester (20−22 weeks) 4 types were found, among them 37.5 % acute scalene and 33.3 % acute isosceles. ConclusionsThe fetal sigmoid colon mesentery can be visualized from the 12th week of pregnancy. The triangular shape of the sigmoid colon mesentery is easy to follow during routine ultrasound examinations.

Study on shallow donor-type impurities of GaN epilayer regrown by epitaxial lateral overgrowth technique

p-doped gallium nitride (GaN) regrowth by epitaxial lateral overgrowth using a SiO2 mask is studied. A comparison between SiO2 and Al2O3 masked p-GaN by cathodoluminescence spectroscopy and scanning electron microscopy indicates that donor-type impurities are related to the SiO2 mask. A domain peak of 3.25 eV induced by shallow-donor and acceptor transitions and the dark contrast of obtuse triangles have been detected in SiO2 masked p-type GaN. Secondary ion mass spectroscopy is simultaneously employed for the analysis of SiO2 and Al2O3 masked p-GaN and identifies that the source of donor-type impurities is from Si atoms. Furthermore, the experimental results of cross-sectional microstructures at different regrowth times have been investigated. It is found that the donor-type impurities tend to cluster in semi-polar 112¯2 facets before the coalescence at the bottom of adjacent triangular stripes starts. The explanation for the non-uniform distribution of impurities is that semi-polar 112¯2 facets exhibit more dangling bond densities than the (0001) plane, and the SiO2 mask exposed to the vapor phase would likely introduce more impurities before the coalescence of GaN stripes.

Effects of Texture Bottom Profile on Static and Dynamic Characteristics of Journal Bearings

The purpose of this paper is to numerically study the effect of texture bottom profile on static, dynamic, and stability performance parameters of hydrodynamic journal bearings. The different performance parameters of square textured journal bearings with different bottom profiles are numerically investigated and compared with those of smooth journal bearing. There are five bottom profiles of this square texture: flat, curved, isosceles triangle (T1), oblique triangle (T2), and oblique triangle (T3). The static and dynamic coefficients are calculated by solving the steady‐state Reynolds equation and the perturbation equations with FDM numerical technique. The performance characteristics under different texture distribution, depth, and bottom profiles are studied, and the current numerical results show that the selection of texture parameters is crucial to improve the static, dynamic, and stability performances of hydrodynamic journal bearing. Meanwhile, it is also found that the square texture with a flat bottom profile has a higher improvement in the values of static performance parameters in comparison to those other bottom profiles. Moreover, the simulation results indicate that the dynamic and stability performances improvement of textured journal bearing is also significant, especially when the eccentricity ratio is smaller.

The Formation of the Post-Impressionist Trend in the Creative Practice of Oleh Harahonych

The article examines the artistic experience during the period of 1980–2000. Oleh Harahonych’s individual painting style was formed at the turn of the 1960s and 1970s under the influence of a large-scale talent of the famous Ukrainian painter Yosyp Bokshay (1891–1975). O. Harahonych visited Bokshay’s Uzhhorod workshop several years in a row. Having escaped the system-defined academic education, the post-impressionist-to-be was experimenting with the new formal means of reproduction. Being primarily a landscapist, the painter began to develop new dynamic angles of the composition, involving the fragments of a high-altitude highway in presentation of his works, which ensured that his landscapes obtained a sense of modernity. Winding Carpathian roads suggested to the artist a zigzag structure of the composition, which was based on the principles of polar dynamics. In O. Harahonych’s landscapes of the mid‑1980s, the contrast of warm and cold color gradations and intensity of colors enhanced. The dominance of pure colors harmoniously combined with simple shapes that focused on an acute triangle. The renewed lyrical sense of the landscape was reproduced by the simplest artistic means which included decorative ornaments, logic of compositional accents, and clarity of silhouettes. It was realized in a series of plein‑air images of the 2000s. In search of new visual means, Oleh Harahonych’s imagination was ahead of the visual technologies of easel painting of that time. Developed in the 1980s, the author’s style visually resembles the digital presentations in Photoshop. The author’s synthesis of color and form reveals an integral connection between post-impressionism and design, which Roger Fry called “Vision and design” in his book a century ago.

Different Modes of Deformation of Soft Triangular Honeycombs at the Sub-5nm Scale.

Patterning on the sub-5nm length scale is a contemporary challenge for further miniaturization of microelectronic circuits. Here, the first soft self-assembled triangular patterns are reported showing transitions between regular and two different kinds of isosceles (acute and obtuse angled) triangles on this length scale, formed by liquid crystalline honeycombs of polyphilic block molecules involving a fluorinated oligo(para-phenylene ethynylene) core. The type of formed triangular pattern depends on the degree and position of fluorination and on temperature. They are the first soft honeycombs combining tilted and nontilted organizations in a uniform nanostructure, where the tilted molecules in only one or two sides of the triangular prismatic cells dominate the shape and the size of the morphology.

Valence optimization and angle improvement for molecular surface remeshing

Molecular surface mesh generation plays a vital role in molecular modeling and visualization. However, meshes extracted directly from Protein Data Bank files have several issues such as small and large triangles, redundant elements, self-intersections, and irregular vertices. The state-of-the-art mesh improvement methods often fail to deal with these issues. In this paper, we present a novel method for valence optimization and angle improvement. For valence optimization, we remove the bad valence vertices with its neighbor triangle making regular holes in the mesh. The holes are filled in a careful manner to improve their valences as well as angle quality. We also use a segmentation-based surface remeshing which segments the mesh into random segments and then each segment is independently remeshed. In addition, a point insertion scheme is applied to minimize the ratio of obtuse triangles. Experimental results show that our method not only improves the maximal and minimal angles to an angle bound of $$[30^{\circ }\;120^{\circ }]$$ but also improves the vertices’ regularity, reduces the ratio of obtuse triangles, preserves the area and volume, and always succeeds with downstream applications.