scholarly journals Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse

2021 ◽  
Author(s):  
Asli Pinar Tan
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Mathematical Basis ◽  
Multiple Dimensions ◽  
Special Cases ◽  
Position Data ◽  
Gravitational Pull ◽  
Elliptical Motion ◽  
The Universe ◽  
Moving Bodies

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the “distance between points on any two different circles in three-dimensional space” is equivalent to the “distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.” Then, it is further mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Finally, two special cases when the “distance between points on two different circles in multi-dimensional space” become mathematically equivalent to distances in “parabolic” or “near-hyperbolic” trajectories are investigated. Concepts of “vector ellipse”, “vector hyperbola”, and “vector parabola” are also mathematically defined. The mathematical basis derived in this Article is utilized in the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe” in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.

scholarly journals How Can I Grab That?

i-com ◽  
2020 ◽  
Vol 19 (2) ◽  
pp. 67-85
Author(s):  
Matthias Weise ◽  
Raphael Zender ◽  
Ulrike Lucke
Keyword(s):  
Virtual Reality ◽  
User Experience ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Advantages And Disadvantages ◽  
Multiple Dimensions ◽  
Application Developers ◽  
Three Dimensional Space

AbstractThe selection and manipulation of objects in Virtual Reality face application developers with a substantial challenge as they need to ensure a seamless interaction in three-dimensional space. Assessing the advantages and disadvantages of selection and manipulation techniques in specific scenarios and regarding usability and user experience is a mandatory task to find suitable forms of interaction. In this article, we take a look at the most common issues arising in the interaction with objects in VR. We present a taxonomy allowing the classification of techniques regarding multiple dimensions. The issues are then associated with these dimensions. Furthermore, we analyze the results of a study comparing multiple selection techniques and present a tool allowing developers of VR applications to search for appropriate selection and manipulation techniques and to get scenario dependent suggestions based on the data of the executed study.

An Executable Notation, With Illustrations From Elementary Crystallography

1994 ◽  
Author(s):  
Donald B. Mclntyre
Keyword(s):  
Plate Tectonics ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Multiple Data ◽  
Special Cases ◽  
Unit Cells ◽  
Crystallographic Axes

Elementary crystallography is an ideal context for introducing students to mathematical geology. Students meet crystallography early because rocks are made of crystalline minerals. Moreover, morphological crystallography is largely the study of lines and planes in real three-dimensional space, and visualizing the relationships is excellent training for other aspects of geology; many algorithms learned in crystallography (e.g., rotation of arrays) apply also to structural geology and plate tectonics. Sets of lines and planes should be treated as entities, and crystallography is an ideal environment for introducing what Sylvester (1884) called "Universal Algebra or the Algebra of multiple quantity." In modern terminology, we need SIMD (Single Instruction, Multiple Data) or even MIMD. This approach, initiated by W.H. Bond in 1946, dispels the mysticism unnecessarily associated with Miller indices and the reciprocal lattice; edges and face-normals are vectors in the same space. The growth of mathematical notation has been haphazard, new symbols often being introduced before the full significance of the functions they represent had been understood (Cajori, 1951; Mclntyre, 1991b). Iverson introduced a consistent notation in 1960 (e.g., Iverson 1960, 1962, 1980). His language, greatly extended in the executable form called J (Iverson, 1993), is used here. For information on its availability as shareware, see the Appendix. Publications suitable as tutorials in , J are available (e.g., Iverson. 1991; Mclntyre, 1991 a, b; 1992a,b,c; 1993). Crystals are periodic structures consisting of unit cells (parallelepipeds) repeated by translation along axes parallel to the cell edges. These edges define the crystallographic axes. In a crystal of cubic symmetry they are orthogonal and equal in length (Cartesian). Those of a triclinic crystal, on the other hand, are unequal in length and not at right angles. The triclinic system is the general case; others are special cases. The formal description of a crystal gives prominent place to the lengths of the axes (a, b, and c) and the interaxial angles ( α, β, and γ). A canonical form groups these values into a 2 x 3 table (matrix), the first row being the lengths and the second the angles.

FUNDAMENTAL APPROACH TO THE COSMOLOGICAL CONSTANT ISSUE

2002 ◽  
Vol 17 (29) ◽  
pp. 4219-4228 ◽  
Author(s):  
MOSHE CARMELI
Keyword(s):  
Cosmological Constant ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Phase Evolution ◽  
Positive Pressure ◽  
Field Equations ◽  
Accelerating Expansion ◽  
Three Phase ◽  
The Universe ◽  
Three Dimensional Space

We use a Riemannian four-dimensional presentation for gravitation in which the coordinates are distances and velocity rather than the traditional space and time. We solve the field equations and show that there are three possibilities for the Universe to expand. The theory describes the Universe as having a three-phase evolution with a decelerating expansion, followed by a constant and an accelerating expansion, and it predicts that the Universe is now in the latter phase. It is shown, assuming Ωm = 0.245, that the time at which the Universe goes over from a decelerating to an accelerating expansion, occurs at 8.5 Gyr ago, at which time the cosmic radiation temperature was 146K. Recent observations show that the Universe's growth is accelerating. Our theory confirms these recent experimental results. The theory predicts also that now there is a positive pressure in the Universe. Although the theory has no cosmological constant, we extract from it its equivalence and show that Λ = 1.934 × 10-35 s-2. This value of Λ is in excellent agreement with measurements. It is also shown that the three-dimensional space of the Universe is Euclidean, as the Boomerang experiment shows.

scholarly journals Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematics and revealed by a three dimensional computational framework

2021 ◽  
Author(s):  
Debabrata Auddya ◽  
Xiaoxuan Zhang ◽  
Rahul Gulati ◽  
Ritvik Vasan ◽  
Krishna Garikipati ◽  
...  
Keyword(s):  
Symmetry Breaking ◽  
Lipid Bilayers ◽  
Morphological Changes ◽  
Dimensional Space ◽  
Wide Spectrum ◽  
Three Dimensional ◽  
Mathematical Basis ◽  
Membrane Deformation ◽  
Membrane Geometry ◽  
Organelle Morphology

AbstractBiomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modeling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modeling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff-Love thin-shell kinematics, Helfrich-energy based mechanics, and state-of-the-art numerical techniques for modeling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modeling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a 3D space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modeling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a “phase diagram” for its symmetric and broken-symmetry states.

scholarly journals Generalized 2N Color Theorem

2020 ◽  
pp. 1-4
Author(s):  
Joseph Edward Brierly ◽  
Keyword(s):  
Quantum Physics ◽  
Dimensional Space ◽  
Bubble Chamber ◽  
Space Station ◽  
Three Dimensional ◽  
Physical Reality ◽  
Four Dimensions ◽  
Color Problem ◽  
Four Color Theorem ◽  
The Universe

2N-Color Theorem This article gives a standard proof of the famous Four-Color theorem and generalizes it be the 2N-Color problem. The article gives a number of possible applications of the 2N-Color problem that is the essence of orientation. Orientation is fundamental to many fields of scientific knowledge. The Fourcolor theorem applies to map making by the knowledge that only four colors are necessary to color a planar map. The Six-color theorem applies to three dimensional space implying that a space station could be ideally designed to have six compartments adjacent to one another allowing a door from any one of the compartments to the other five. The 2N-color generalization applies to the physical reality of quantum physics. Bubble chamber investigations suggest that the universe is four or more dimensions. Thus the 2N-color theorem applies to the N dimensional universe. At this time string theorists have suggested that the universe could be greater than four dimensions. Physics has not as of yet proven the exact dimension of the universe that could even be infinite as a possibility

scholarly journals Seeking a Reference Frame for Cartographic Sonification

2018 ◽  
Author(s):  
Megen Brittell
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Visual Display ◽  
Data Cube ◽  
Geospatial Data ◽  
Data Sets ◽  
Extensive Experience ◽  
Map Design ◽  
Position Data ◽  
New Perspective

Sonification of geospatial data must situate data values in two (or three) dimensional space. The need to position data values in space distinguishes geospatial data from other multi-dimensional data sets. While cartographers have extensive experience preparing geospatial data for visual display, the use of sonification is less common. Beyond availability of tools or visual bias, an incomplete understanding of the implications of parameter mappings that cross conceptual data categories limits the application of sonification to geospatial data. To catalyze the use of audio in cartography, this paper explores existing examples of parameter mapping sonification through the framework of the geographic data cube. More widespread adoption of auditory displays would diversify map design techniques, enhance accessibility of geospatial data, and may also provide new perspective for application to non-geospatial data sets.

HEISENBERG QUANTIZATION FOR SYSTEMS OF IDENTICAL PARTICLES

1993 ◽  
Vol 08 (21) ◽  
pp. 3649-3695 ◽  
Author(s):  
JON MAGNE LEINAAS ◽  
JAN MYRHEIM
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Classical System ◽  
Two Dimensions ◽  
Identical Particles ◽  
Natural Approach ◽  
Fermion Systems ◽  
Special Cases ◽  
Continuous Interpolation

We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.

Perceptual-cognitive universals as reflections of the world

2001 ◽  
Vol 24 (4) ◽  
pp. 581-601 ◽  
Author(s):  
Roger N. Shepard
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Euclidean Group ◽  
Dimensional Vector Space ◽  
The World ◽  
Geodesic Paths ◽  
Abstract Spaces ◽  
The Universe ◽  
Three Dimensional Space

The universality, invariance, and elegance of principles governing the universe may be reflected in principles of the minds that have evolved in that universe – provided that the mental principles are formulated with respect to the abstract spaces appropriate for the representation of biologically significant objects and their properties. (1) Positions and motions of objects conserve their shapes in the geometrically fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three-dimensional space and the symmetry group of each object. (2) Colors of objects attain constancy when represented as points in a three-dimensional vector space in which each variation in natural illumination is canceled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in an evolutionarily-shaped space of possible objects, as connected regions with associated weights determined by Bayesian revision of maximum-entropy priors.

scholarly journals Possible Observational Evidence for the Existence of a Parallel Universe

Foundations ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 1-5
Author(s):  
Eugene Oks
Keyword(s):  
Flow Velocity ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Bulk Flow ◽  
Data Sets ◽  
Microwave Background ◽  
Detection Systems ◽  
Preferred Direction ◽  
Wide Range ◽  
Gravitational Pull

Many totally different kinds of astrophysical observations demonstrated that, in our universe, there exists a preferred direction. Specifically, from observations in a wide range of frequencies, the alignment of various preferred directions in different data sets was found. Moreover, the observed Cosmic Microwave Background (CMB) quadrupole, CMB octopole, radio and optical polarizations from distant sources also indicate the same preferred direction. While this hints at a gravitational pull from the “outside”, the observational data from the Plank satellite showed that the bulk flow velocity was relatively small: much smaller than was initially thought. In the present paper we propose a configuration where two three-dimensional universes (one of which is ours) are embedded in a four-dimensional space and rotate about their barycenter in such a way that the centrifugal force nearly (but not exactly) compensates their mutual gravitational pull. This would explain not only the existence of a preferred direction for each of the three-dimensional universes (the direction to the other universe), but also the fact that the bulk flow velocity, observed in our universe, is relatively small. We point out that this configuration could also explain the perplexing features of the Unidentified Aerial Phenomena (UAP), previously called Unidentified Flying Objects (UFOs), recorded by various detection systems—the features presented in the latest official report by the US Office of the Director of National Intelligence. Thus, the proposed configuration of the two rotating, parallel three-dimensional universes seems to explain both the variety of astrophysical observations and (perhaps) the observed features of the UAP.

scholarly journals Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff–Love kinematicsand revealed by a three-dimensional computational framework

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Debabrata Auddya ◽  
Xiaoxuan Zhang ◽  
Rahul Gulati ◽  
Ritvik Vasan ◽  
Krishna Garikipati ◽  
...  
Keyword(s):  
Symmetry Breaking ◽  
Lipid Bilayers ◽  
Morphological Changes ◽  
Dimensional Space ◽  
Wide Spectrum ◽  
Three Dimensional ◽  
Mathematical Basis ◽  
Membrane Deformation ◽  
Membrane Geometry ◽  
Three Dimensional Space

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modelling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model-symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modelling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff–Love thin-shell kinematics, Helfrich-energy-based mechanics, and state-of-the-art numerical techniques for modelling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modelling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a three-dimensional space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modelling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three-dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a ‘phase diagram’ for its symmetric and broken-symmetry states.

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