Absolute Geometry Research Articles

On trialities and their absolute geometries

Abstract We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type (I σ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram for the groups G 2(k) and 3 D 4(k), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L 2(q 3), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.

Synthesis, structural studies and Hirshfeld surface analysis of the substituted isoindole-1,3‑dione derivatives

In this study, we have performed the ring-opening reaction of 5-benzyl/ethyl-2-hydroxyhexahydro-4H-oxireno[2,3-e] isoindole-4,6(5H)‑dione (10) with HBr to the corresponding bromodiols, regioselectively. Structures of the synthesized compounds are elucidated through spectral methods (1H NMR,13C NMR and HRMS) Afterwards, the racemic products 11a-b were purified and the absolute geometries of 5‑bromo-2-ethyl-4,6-dihydroxyhexahydro-1H-isoindole 1,3(2H)‑dione (11a) and 2-benzyl-5‑bromo-4,6-dihydroxyhexahydro-1H-isoindole-1,3(2H)‑dione (11b) was determined by X-ray diffraction analysis. Effective O−H∙∙∙O (2.738–2.874 Å) hydrogen bonds between diol and ketone units are important interactions between the molecules. Hirshfeld surface analyses of the compounds (11a-b) have provided further insights into molecular, crystal structure and intermolecular interactions. The results indicated that the most important contributions for the crystal packings are from H∙∙∙H, O∙∙∙H/H∙∙∙O, C∙∙∙H/H∙∙∙C and Br∙∙∙H/H∙∙∙Br interactions. Finally, energy frameworks were analyzed to a better understanding of the packing of crystal structure and the supramolecular rearrangements for the structures (11a-b).

Absolute Geometry: From Basics to the π-rule of the Ф-invariant Physics

This is to clarify in more detail some basic aspects of absolute geometry and discuss what is the π-rule in physics unified by the universal Ф-invariance

Design Approaches on Inner Bodies of Gears with Methods Topology Optimization and Lattice Structures

Gears have wide range application areas in various industries which no matter they are small-scaled or large-scaled. As a part of their design, gears inner bodies are full filled with their solid material and significantly increase the weight of systems which they used in. If the weight of gears’ body mass can be decrease during their design process, the mechanical properties that expected from the systems can be achieved with the minimum cost of material via additive manufacturing comparing to the traditional manufacturing processes. First ideas which comes to mind presents two choices for the way should follow. This study focused on design optimization of material layout rather than material selection. Generative design technic also known as topology optimization can create new designs via mathematical methods that optimize material layouts within a specific design space. Absolute geometry is depending on various parameters such as given set of loads, boundary conditions and constraints. Alternatively, lattice structures are designs which inspired from bio-entities based on repeating unit cells. As a result of static analysis of a helical gear’ s uniform lattice structure, output parameters have been used for varying unit cells’ beam thickness and optimize lattice design. End of this study which used nTopology as engineering software for whole implicit design and analysis process, the analysis of generative designed geometry and pattern of lattice structure gave different results. These outputs compared on point of weight savings.

Physical Universe as Mathematical Machine

This is an introduction to the absolute geometry of space, time and matter which automatically leads to the superunified field theory.

Cayley–Klein geometries and projective-metric geometry

Cayley–Klein geometries and projective-metric geometry

Metric Characteristics of Hyperbolic Polygons and Polyhedra

In this paper, we consider some properties of hyperbolic polyhedra, both common with Euclidean and specific. Asymptotic behavior of metric characteristics of polyhedra in the n-dimensional hyperbolic space is examined in the cases where parameters of the polyhedra change and the dimension of the space unboundedly increases; in particular, the radius of the inscribed sphere of a polyhedron is estimated and its asymptotic behavior is obtained. In connection with this, the problem of estimating the minimal number of faces of the described polyhedron in the n-dimensional hyperbolic space depending on the radius of the inscribed sphere is posed. We also consider some properties of hyperbolic polygons that belong to both absolute geometry or only to hyperbolic geometry.

Design, synthesis, and biological evaluation of polyphenol derivatives as DYRK1A inhibitors. The discovery of a potentially promising treatment for Multiple Sclerosis

Green tea and its natural components are known for their usefulness against a variety of diseases. In particular, the activity of main catechin Epigallocatechin gallate (EGCG) against Dual-specificity tyrosine-(Y)-phosphorylation Regulated Kinase-1A (DYRK1A) has been reported; here we are showing a structure–activity relationship (SAR) for EGCG against this molecular target. We have studied the influence of all four rings on the activity and the nature of its absolute geometry. This work has led to the identification of the more potent and stable trans fluoro-catechin derivative 1f (IC50 = 35 nM). This molecule together with a novel delivery method showed good efficacy in vivo when tested in a validated model of multiple sclerosis (EAE).

Traceable determination of non-static XCT machine geometry: New developments and case studies

It is fundamental to determine the machine geometry accurately for dimensional X-ray computed tomography (XCT) measurements. When performing high-accuracy scans, compensation of a non-static geometry, e.g. due to rotary axis errors or drift, might become necessary. Here we provide an overview of methods to determine and account for such deviations on a per projection basis. They include characterisation of stage error motions, in situ geometry measurements, numerical simulations, and reconstruction-based optimization relying on image quality metrics and will be discussed in terms of their metrological performance. Since a radiographic calibration is always required to provide an initial absolute geometry, this method will be presented as well. The improvements of the XCT geometry correction methods are presented by means of case studies. The methods can be applied individually or in combination and are intended to provide a toolbox for XCT geometry compensation.

Commentary to Book I of the Elements. Hartshorne and beyond

(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.

The Holy Trinity of Geometry: Beauty, Virtue & Truth

The virtue of absolute geometry is its accessibility to the ordinary human mind coupled with certain degree of intuition which is kind of connectedness of the human brain to the universal logico-mathematical machinery. In the end all the problem of existence reduces to the elementary operations within the continuum of natural numbers. Nontriviality and consistence of the system make the beauty of geometry. Geometry’s last truths prove to be written in beautiful and mnemonical aphorisms.

The Quantum Gyroscope Versus the CERN Supercollider

When civilization stagnates, it erects something great but useless in the form of either Pharaoh’s pyramid, or Chinese wall. The CERN collider is the best characteristic of our times. The Chinese wall was never a worthy defense against nomadic forces. I cannot believe that the CERN superconducting supercollider will not give up before the absolute geometry of space, time and matter

The Ubiquitous Axiom

This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.

Quantum Gravitation Solutions

This is intended to clarify the so called problem of quantum gravitation. In absolute geometry space-time is intrinsically quantum-gravitational implying that there is no need to try to quantize gravity.

Absolute isotropic geometry

The term ‘absolute geometry’ was coined by Janos Bolyai to characterize the part of Euclidean geometry that does not depend on the parallel postulate. In the framework of Cayley–Klein geometries the parallel axiom characterizes three classical geometries, namely Euclidean, Galilean and Minkowskian planes. We study the part of Galilean geometry that does not depend on the parallel postulate (briefly called absolute isotropic geometry) and their models (isotropic planes). After an axiomatic foundation of absolute isotropic geometry, we develop the basic theory of isotropic planes, prove the theorem of Saccheri for the angle sum of a triangle, and construct models of the different types of planes (over fields and skew fields of characteristic $$\ne 2$$ ). Surprisingly, these models have counterparts in the theory of Hilbert planes. The article closes with a comparison between Hilbert planes and isotropic planes.

Dynamic Evaluation of Traffic Noise through Standard and Multifractal Models

Traffic microsimulation models use the movement of individual driver-vehicle-units (DVUs) and their interactions, which allows a detailed estimation of the traffic noise using Common Noise Assessment Methods (CNOSSOS). The Dynamic Traffic Noise Assessment (DTNA) methodology is applied to real traffic situations, then compared to on-field noise levels from measurement campaigns. This makes it possible to determine the influence of certain local traffic factors on the evaluation of noise. The pattern of distribution of vehicles along the avenue is related to the logic of traffic light control. The analysis of the inter-cycles noise variability during the simulation and measurement time shows no influence from local factors on the prediction of the dynamic traffic noise assessment tool based on CNOSSOS. A multifractal approach of acoustic waves propagation and the source behaviors in the traffic area are implemented. The novelty of the approach also comes from the multifractal model’s freedom which allows the simulation, through the fractality degree, of various behaviors of the acoustic waves. The mathematical backbone of the model is developed on Cayley–Klein-type absolute geometries, implying harmonic mappings between the usual space and the Lobacevsky plane in a Poincaré metric. The isomorphism of two groups of SL(2R) type showcases joint invariant functions that allow associations of pulsations–velocities manifolds type.

Geometrical interpretation of the pilot wave theory and manifestation of spinor fields

Abstract Using the hydrodynamical formalism of quantum mechanics for a Schrödinger spinning particle developed by Takabayashi, Vigier, and followers, which involves vortical flows, we propose a new geometrical interpretation of the pilot wave theory. The spinor wave in this interpretation represents an objectively real field, and the evolution of a material particle controlled by the wave is a manifestation of the geometry of space. We assume this field to have a geometrical nature, basing on the idea that the intrinsic angular momentum, the spin, modifies the geometry of the space, which becomes a manifold, represented as a vector bundle with a base formed by the translational coordinates and time, and the fiber of the bundle, specified at each point by the field of a tetrad $e^a_{\mu}$, forms from bilinear combinations of the spinor wave function. It has been shown that the spin vector rotates following the geodesic of the space with torsion, and the particle moves according to the geometrized guidance equation. This fact explains the self-action of the spinning particle. We show that the curvature and torsion of the spin vector line is determined by the space torsion of the absolute parallelism geometry.

Superunified Field Theory

This is a briefest possible introduction to the absolute geometry of space, time and matter. Absolute geometry or the post-Euclidean geometry does automatically lead to the superunified theory of quantized fields and fundamental interactions. In general, we have eventually constructed the ultimate system of universal mathematical harmony observed by us as the physical Universe. No work in theoretical physics and pure mathematics directly precedes to this theory we propose. Instead, it accomplishes original Pythagorean (arithmetisation) znd Platonic (geometrization) concepts of natural philosophy integrated afterwards by Jiordano Bruno.

Investigation of the impact of various robot properties on a twin Robot-CT system

ABSTRACT Robot-CT is drawing great attention for its potential to support highly flexible scan trajectories and various scan modes with a single system. However, the quality of CT reconstruction largely depends on accurate knowledge of the scan geometry, i.e. position and orientation of the source and the detector. This imposes major challenges on current serial industrial robots. In order to realistically assess the impact of various robot characteristics on the Robot-CT reconstruction quality and to facilitate the development of a dedicated calibration method, this paper elaborates on a set of simulations based on the performance evaluation of an ABB industrial robot. Simulation results reveal that: (a) synchronous error motions of the twin robots can partially compensate for the robot errors; (b) offline calibration is a possible way to implement high accuracy Robot-CT due to the excellent repeatability of current industrial robots; (c) both robot positioning and orientation errors need to be compensated for to get a high quality reconstruction; (d) compensating for only the relative error cannot improve the final reconstruction quality: absolute positioning and orientation errors need to be measured and compensated for. External sensors or radiograph based absolute geometry calibration are needed to improve the reconstruction quality.

The ‘Cardinality of the Continuum’ Is False in Non-Euclidean Geometries

According Georg Cantor's Cardinality of the Continuum, as proved in Euclidean geometry, any segment (not inclusive of its two end points) has as many points as an entire line. This theorem is an underlying assumption of the Hypothesis, that conjectures that every infinite subset of the continuum of real numbers $\mathbb{R}$ is either equivalent to the set of natural numbers $\mathbb{N}$, or to $\mathbb{R}$ itself. However, the proof in Euclidean geometry uses parallel lines, and therefore assumes that Euclid's Parallel Postulate is true. But in Non-Euclidean geometries, Euclid's Parallel Postulate is false. In Hyperbolic geometry, at any point off of a given line, there are a plurality of lines parallel to the given line. In Elliptic geometry (which includes Spherical geometry), no lines are parallel, so two lines always intersect. Absolute geometry has neither Euclid's parallel postulate nor either of its alternatives. We provide examples in Spherical geometry and in Hyperbolic geometry where the Cardinality of the Continuum is false. Therefore it is also false in Absolute geometry. So it is a false proposition, as is the Hypothesis, that falsely assumes that the Cardinality of the Continuum is true. Concepts in theory, such as the number line of real numbers, cannot be limited to Euclidean geometry.