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.

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.

The Triangulation Reduction Analysis of Acute Triangle

Positioning on the surface of the earth using the triangulation method can be done in two ways, namely terrestrial method for example by measuring the angle and distance using the total station tool and extraterrestrial methods for example by using satellite-based positioning technology. Extraterrestrial positioning method using the global navigation satellite system combined with terrestrial methods by measuring angles and distances using the total station tool is one alternative to positioning well. In this study position determination will only be calculated using two intermediate calculation plane, on the ellipsoid projection and in the plane projection. Of course, in a process of measuring position for the same point but using different methods it will produce different levels of accuracy. From the results of the comparison in determining the definitive coordinate value generated by the count on the ellipsoid projection with the Gauss-Helmert method, the definitive value that is closer to the reference value measured by static measurements methods rather than definitive coordinates produced through calculations in the plane projection.

Existence of special rainbow triangles in weak geometries

We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

KETAKSAMAAN JUMLAHAN SINUS PANGKAT 2^n YANG BERLAKU PADA SEGITIGA LANCIP

Let \alpha, \beta, \gamma are angles in acute triangle ABC and a,b,c are the length of the triangle. By using the sine of angles as the relationship between the length of triangle and the radius of the circle circumscribed about a plane triangle, will be proven the sum inequality of quadratic sine in acute triangle. Then, by using the quadratic sum inequality of the sides of triangle will be extended for the case of the sum inequality of sine of order 2^n in acute triangle.

Labyrinth Seal Leakage Degradation Due to Various Types of Wear

This paper systematically presents a complete leakage comparison for various types of wear experienced by labyrinth seals. Labyrinth seals used in turbine engines are designed to work at a clearance during steady-state engine operations. The tooth tip rubs the stator and wears either itself or the stator surface during transient operations, depending on the material properties of the tooth and stator. Any type of wear that increases clearance or deforms the tooth tip will cause permanent and unpredictable leakage degradation. This negatively affects the engine's overall efficiency, durability, and life. The teeth have been reported to wear into a mushroom profile or into a rounded profile. A rub-groove on the opposing surface may form in several shapes. Based on a literature survey, five rub-groove shapes are considered in this work. They are rectangle, trapezoid (isosceles and acute), triangle, and ellipse. In this work, leakage degradation due to wear is numerically quantified for both mushroomed and rounded tooth wear profiles. It also includes analyses on rounded teeth with the formation of five rub-groove shapes. All parameters are analyzed at various operating conditions (clearance, pressure ratio, number of teeth, and rotor speed). Computational fluid dynamics (CFD) analyses are carried out by employing compressible turbulent flow in a 2D axisymmetrical coordinate system. CFD analyses show that the following tooth-wear conditions affect leakage from least to greatest: unworn, rounded, and mushroomed. These are for an unworn flat stator. It is also observed that rub-groove shapes considerably affect the leakage depending on the clearance. Leakage increases with the following groove profiles: triangular, rectangular, acute trapezoidal, isosceles trapezoidal, and elliptical. The results show that any type of labyrinth seal wear has significant effects on leakage. Therefore, leakage degradation due to wear should be considered during the engine design phase.

Labyrinth Seal Leakage Degradation due to Various Types of Wear

This paper systematically presents a complete leakage comparison for various types of wear experienced by labyrinth seals. Labyrinth seals used in turbine engines are designed to work at a clearance during steady-state engine operations. The tooth tip rubs the stator and wears either itself or the stator surface during transient operations, depending on the material properties of the tooth and stator. Any type of wear that increases clearance or deforms the tooth tip will cause permanent and unpredictable leakage degradation. This negatively affects the engine’s overall efficiency, durability, and life. The teeth have been reported to wear into a mushroom profile or into a rounded profile. A rub-groove on the opposing surface may form in several shapes. Based on a literature survey, five rub-groove shapes are considered in this work. They are: rectangle, trapezoid (isosceles and acute), triangle, and ellipse. In this work, leakage degradation due to wear is numerically quantified for both mushroomed and rounded tooth wear profiles. It also includes analyses on rounded teeth with the formation of five rub-groove shapes. All parameters are analyzed at various operating conditions (clearance, pressure ratio, number of teeth, rotor speed). CFD analyses are carried out by employing compressible turbulent flow in a 2-D axi-symmetrical coordinate system. CFD analyses show that the following tooth-wear conditions affect leakage from least to greatest: unworn, rounded, and mushroomed. These are for an unworn flat stator. It is also observed that rub-groove shapes considerably affect the leakage depending on the clearance. Leakage increases with the following groove profiles: triangular, rectangular, acute trapezoidal, isosceles trapezoidal, and elliptical. The results show that any type of labyrinth seal wear has significant effects on leakage. Therefore, leakage degradation due to wear should be considered during the engine design phase.

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Acute triangles in the n-ball

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝ n , with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.