Precision of the Nebra Disc in Astronomical and Geometric Aspect

The astronomical knowledge on the disc has been coded on two planes: horizontal and meridian. The range of sunrise and sunset directions during the year has been described on the horizontal plane. In turn, on the meridian (vertical) plane, the range of changes in the horizontal height and declination of the Sun in the upper culmination during the year and the Moon in its 18.61-year cycle were described. The relationships between the latitude of the place of observation, the horizontal height of the celestial body and its declination were described by means of geometric constructions. The presented article is a continuation of two publications [1] and [2], which describe the decryption of the Nebra disc. These publications were based on the interpretation of the results of angular measurements, made using a protractor with a scale of 0.5 degrees, without the use of a computer. The presented publication is based on a digital disc image obtained by means of its digitization. The obtained data was used for further calculations based on analytical geometry and graphic programs. This allowed to obtain results in a linear measure with a precision of less than 1 mm. PRECYZJA DYSKU Z NEBRY W ASPEKCIE ASTRONOMICZNYM I GEOMETRYCZNYM Wiedza astronomiczna na dysku została zakodowana na dwóch płaszczyznach: horyzontalnej i południkowej. Na płaszczyźnie horyzontalnej (poziomej) opisano zakres kierunków wschodu i zachodu Słońca w ciągu roku. Z kolei na płaszczyźnie południkowej (pionowej) opisano zakres zmiany wysokości horyzontalnej i deklinacji Słońca w kulminacji górnej w ciągu roku oraz Księżyca w jego 18,61-rocznym cyklu. Za pomocą konstrukcji geometrycznych opisano związki między szerokością geograficzną miejsca obserwacji, wysokością horyzontalną ciała niebieskiego i jego deklinacją. Prezentowany artykuł jest kontynuacją dwóch publikacji: [1], [2], w których opisano deszyfrację dysku z Nebry. Publikacje te były oparte na interpretacji wyników pomiarów kątowych, wykonanych za pomocą kątomierza o podziałce 0,5 stopnia, bez użycia komputera. Prezentowana publikacja bazuje na cyfrowym obrazie dysku, uzyskanym za pomocą jego digitalizacji. Uzyskane dane wykorzystano do dalszych obliczeń opartych na geometrii analitycznej oraz programach graficznych.

Using digital tasks when teaching mathematics at school

The article discusses various types of digital math tasks to be performed on computers or tablets. The experience of creating digital tasks in mathematics with complex answers from the point of view of automatic verifcation: algebraic expressions, geometric constructions, graphs of functions, etc. is discussed. As an example of the use of digital tasks, the use of digital materials for preparing schoolchildren for the Unifed State Exam in mathematics of a profle level is considered. The article also discusses various forms of using simulators and tests in the educational process: supporting the independent activities of students in the full-time educational process, supporting online learning with a teacher in the context of the coronavirus pandemic, independent online learning without a teacher based on specially prepared and structured training courses. In the latter case, in addition to simulators, it is proposed to use other digital educational resources to increase the visibility of the educational material and organize self-control of learning outcomes: videos, slides, dynamic models, training exercises for problems with a detailed answer to the options for the USE control and measuring materials.

Features of Distance Learning in Geometric and Graphic Disciplines Using Methods of Constructive Geometric Modeling

The article is devoted to the peculiarities of teaching the discipline "Descriptive geometry" in the conditions of distance learning, it examines the application of information technologies in the educational process in geometric and graphic disciplines. Increasing the speed of information processes, reducing the number of hours for mastering the discipline. the conditions of distance learning set new tasks for teachers and dictate their requirements for teaching graphic disciplines and the use of teaching experience in a new reality; there is a need to introduce and develop new forms of education without losing the quality of education. Geometric-graphic disciplines occupy one of the important places in technical education, the complexity of the study of which lies in the development of a graphical representation of phenomena, objects and processes by methods of constructive geometric modeling. The knowledge and skills acquired by students contribute to the development of spatial, imaginative and rational thinking, which is necessary for future professional activities. Descriptive geometry is a discipline that is not easy to master on your own without a conscious understanding of the logic and sequence of geometric constructions, without deep knowledge of theoretical foundations and constant, repeated implementation of practical tasks. The acquisition of practical skills in mastering the methods of discipline has become more difficult in the current epidemiological situation. In modern conditions of distance learning, the use of the Simplex geometric modeling system made it possible to develop and propose a new concept of geometric-graphic interaction, which significantly reduced the time for completing and checking educational tasks in real time. The proposed technology reveals the deep informational essence of the studied discipline "Descriptive Geometry" and becomes a powerful research tool for students. The integration of traditional teaching methods in the graphic preparation of students with computer and communication facilities increases the possibilities of communication and improves the quality of teaching.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Basic Geometric Constructions

Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Static and Rotordynamic Characteristics for Two Novel Hole-Pattern Annular Liquid Seals With Circumferentially- /Axially- Oblique Hole Cavities

Non-contacting annular damper seals, such as hole-pattern seals are gradually used in the multiple-stage centrifugal pumps, as a replacement of the conventional labyrinth seal to reduce the fluid leakage and stabilize the rotor-bearing system.The hole-pattern seal (HPS) possesses numerous radial hole cavities on the seal stator, and the geometric constructions of the hole cavity (such as the hole depth and diameter) have been demonstrated to have significant influences on the leakage and rotordynamic characteristic for hole-pattern seals. Due to the inevitable manufacturing variability, particulate impurity deposition and abrasion during operation, these hole cavities can be non-radial, which may affect the performance of the hole-pattern seal. However, the effects of the non-radical hole cavities on the performance of the hole-pattern seal are still unknown due to the lack of numerical or experimental research. Thus, in this paper, two types of novel hole-pattern seals possessing circumferentially- or axially-oblique hole cavities (C-HPS, A-HPS) with various oblique angles were designed and assessed to better understand the influences of the non-radial hole cavities. To assess the leakage and rotordynamic characteristics of the novel liquid hole-pattern seals, a proposed 3D transient CFD-based perturbation method was utilized for the predictions of seal rotordynamic forces coefficients, based on the multi-frequency one-dimensional rotor whirling model and mesh deformation technique. The accuracy and reliability of the present steady and transient numerical methods were demonstrated based on published experimental data of leakage and rotordynamic force coefficients for an experimental hole-pattern seal with radial hole cavities. The leakage and rotordynamic force coefficients were presented for the novel hole-pattern seals with various circumferentially-oblique angle (α = −30°∼30°) or axially-oblique angle (β = −30°∼30°) at various rotational speeds (n = 0.05, 2.0, 4.0, 6.0 krpm), and compared with the ideal hole-pattern seal with radial hole cavities. Numerical results show that the non-radial hole cavity can result in a modest deviation (∼10%) from the design value for the seal leakage, and the oblique direction is crucial for the sealing performance. The flow field in hole cavities and the pressure distribution in the seal clearance suggest that the oblique hole cavities with positive α or β can strengthen the vortex-dissipation of kinetic energy in the hole cavities, thus reduce the leakage (about 5% ∼ 10%). The non-radial cavity with a positive oblique angle results in a modest increase (∼15% for the circumferentially-oblique angle α = 30°, ∼ 6% for the axially-oblique angle β = 30°) in the effective stiffness of the hole-pattern seal, but shows very weak influence (< 4.0%) on the effective damping of the hole-pattern seals, especially for the circumferentially-oblique hole. Therefore, in view of the inevitable manufacturing variability and abrasion effects, a designed non-radial hole with suitable positive α or β (10°∼20°) is beneficial to be applied to new designs in early design phases for the robust design of hole-pattern seals.

A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.