scholarly journals The relations connecting the angle-sums and volume of a polytope in space of n dimensions

1927 ◽  
Vol 115 (770) ◽  
pp. 103-119 ◽  
Keyword(s):  
Three Dimensional ◽  
Spherical Geometry ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Euler Formula ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Elliptic Geometry ◽  
Space Constant ◽  
Different Dimensions

1.1. In two-dimensional spherical or elliptic geometry we have the familiar relation between the area of a triangle and its angle-sum, ∆ = k 2 (∑ α ‒ π ), (1.11) 2 π being the measure of the whole angle at a point, and k the space-constant, or, in spherical geometry, the radius of the sphere. It is well known that in three-dimensional spherical or elliptic geometry there is no corresponding relation involving the volume of a tetrahedron. For elliptic or hyperbolic space of four dimensions it was proved by Dehn that the volume of a simplex can be expressed linearly in terms of the sums of the dihedral angles (angles at a face), angles at an edge, and angles at a vertex, but for space of five dimensions the linear relations do not involve the volume. He indicates also, in a general way, the extensions of these results for spaces of any odd or even dimensions. He shows further that these results are connected with the form of the Euler polyhedral theorem, which is expressed by a linear relation connecting the numbers of boundaries of different dimensions, and which for space of odd dimensions is not homogeneous, e. g ., N 2 — N 1 + N 0 = 2 in three dimen­sions, but for space of even dimensions is homogeneous, e. g ., N 1 — N 0 = 0 in two dimensions, N 3 — N 2 + N 1 — N 0 = 0 in four. The connection, as Dehn points out, was made use of by Legendre in a proof which he gave for the Euler formula in three dimensions. 1.2. Dehn extends this connection in detail for four and five dimensions, and states the following general results in space of dimensions R n for simplexes and for polytopes bounded entirely by simplexes : (1) In R n there are ½ n + 1 or ½( n + 1) relations (according as n is even or odd) between the numbers of boundaries of a polytope bounded by simplexes.

scholarly journals Nervous Activity of the Brain in Five Dimensions

Biophysica ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 38-47
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
Norbert Jausovec ◽  
Arjuna P. H. Don ◽  
Sheela Ramanna ◽  
...  
Keyword(s):  
Fourier Analysis ◽  
Nervous Activity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Data Sets ◽  
Spatial And Temporal Patterns ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Scale Free ◽  
The Brain

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

scholarly journals Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

2021 ◽  
Vol 932 ◽  
Author(s):  
Idan S. Wallerstein ◽  
Uri Keshet
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Four Dimensions ◽  
Potential Flows ◽  
Prolate Spheroids ◽  
Flow Potential ◽  
Inverse Radius ◽  
Rayleigh Expansion ◽  
Simply Connected

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

THE FREEDMAN-TOWNSEND MODEL IN TWO AND THREE DIMENSIONS

1992 ◽  
Vol 07 (09) ◽  
pp. 2005-2020 ◽  
Author(s):  
D.G.C. MCKEON
Keyword(s):  
Configuration Space ◽  
Three Dimensional ◽  
Transverse Component ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Sigma Model ◽  
Four Dimensions ◽  
Space Formalism ◽  
Vector Theory ◽  
Definition Of

Using a mechanism similar to that employed by Freedman and Townsend in four dimensions, we discuss a variety of two- and three-dimensional gauge theories. The simplest of these models is equivalent to the nonlinear sigma model; another corresponds to a massive vector theory. In three dimensions, there exists a gauge invariance associated with the auxiliary vector ϕμa; when we quantize, this is accommodated using both the Batalin-Vilkovisky configuration space formalism and the Batalin-Fradkin-Vilkovisky configuration space formalism. Explicit one-loop calculations in the simplest two-dimensional model are carried out. The regularization used is a variant of operator regularization, allowing one to remain in two dimensions, hence circumventing problems associated with definition of the antisymmetric tensor εμν. This model is renormalizable, with renormalization mixing the scalar field ϕa and the transverse component of the vector field Vμa.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Turbulent three-dimensional dielectric electrohydrodynamic convection between two plates

2012 ◽  
Vol 696 ◽  
pp. 228-262 ◽  
Author(s):  
A. Kourmatzis ◽  
J. S. Shrimpton
Keyword(s):  
Spatial Data ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Budgets ◽  
Turbulent Regime ◽  
Scalar Flux ◽  
Wide Range ◽  
Scalar Variance

AbstractThe fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low (${\mathit{Re}}_{E} = 1$) and high electric Reynolds numbers (up to ${\mathit{Re}}_{E} = 120$). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field ${ E}_{i}^{\ensuremath{\prime} } $ in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that ${ E}_{i}^{\ensuremath{\prime} } $ is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

scholarly journals On the forces that cable webs under tension can support and how to design cable webs to channel stresses

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180781 ◽  
Author(s):  
Guy Bouchitté ◽  
Ornella Mattei ◽  
Graeme W. Milton ◽  
Pierre Seppecher
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Structural Engineering ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

scholarly journals The House of Marie Shannon

2021 ◽  
Author(s):  
◽  
Sally Margaret Apthorp
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Domestic Architecture ◽  
Three Dimensions ◽  
Camera Obscura ◽  
Architectural Space ◽  
Domestic Lives ◽  
Architectural Tradition ◽  
Light And Shadow ◽  
Insight Into

<p>This thesis creatively explores the architectural implications present in the photographs by New Zealand photographer Marie Shannon. The result of this exploration is a house for Shannon. The focus is seven of Shannon's interior panoramas from 1985-1987 in which architectural space is presented as a domestic stage. In these photograph's furniture and objects are the props and Shannon is an actress. This performance, with Shannon both behind and in front of her camera, creates a double insight into her world; architecture as a stage to domestic life, and a photographers view of domestic architecture. Shannon's view on the world enables a greater understanding to our ordinary, domestic lives. Photography is a revealing process that teaches us to see more richly in terms of detail, shading, texture, light and shadow. Through an engagement with photographs and understanding architectural space through a photographer's eye, the hidden, secret or unnoticed aspects to Shannon's reality will be revealed. This insight into another's reality may in turn enable a deeper understanding of our own. The methodology was a revealing process that involved experimenting with Shannon's panoramic photographs. Models and drawing, through photographic techniques, lead to insights both formally in three dimensions and at surface level in two dimensions. These techniques and insights were applied to the site through the framework of a camera obscura. Shannon's new home is created by looking at her photographs with an architect's 'eye'. Externally the home acts as a closed vessel, a camera obscura. But internally rich and intriguing forms, surfaces, textures and shadings are created. Just as the camera obscura projects an exterior scene onto the interior, so does the home. Shannon will inhabit this projection of the shadows which oppose 30 O'Neill Street, Ponsonby, Auckland; her past home and site of her photographs. Photographers, and in particular Shannon, look at the architectural world with fresh eyes, free from an architectural tradition. Photography and the camera enable an improved power of sight. More is revealed to the camera. Beauty is seen in the ordinary, with detail, tone, texture, light and dark fully revealed. As a suspended moment, a deeper understanding and opportunity is created to observe and appreciate this beauty. Through designing with a photographer's eye greater insight is gained into Shannon's 'reality'. This 'revealing' process acts as a means of teaching us how to see pictorial beauty that is inherent in our ordinary lives. This is the beauty that is often hidden in secret, due to our unseeing eyes. This project converts the photographs beauty back into three dimensional architecture.</p>

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

Radiolaria: validating the Turing theory

2017 ◽  
Author(s):  
Bernard Richards
Keyword(s):  
Diffusion Mechanism ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Black And White ◽  
Alan Turing ◽  
School Sports ◽  
The University ◽  
University Of Manchester

In his 1952 paper ‘The chemical basis of morphogenesis’ Turing postulated his now famous Morphogenesis Equation. He claimed that his theory would explain why plants and animals took the shapes they did. When I joined him, Turing suggested that I might solve his equation in three dimensions, a new problem. After many manipulations using rather sophisticated mathematics and one of the first factory-produced computers in the UK, I derived a series of solutions to Turing’s equation. I showed that these solutions explained the shapes of specimens of the marine creatures known as Radiolaria, and that they corresponded very closely to the actual spiny shapes of real radiolarians. My work provided further evidence for Turing’s theory of morphogenesis, and in particular for his belief that the external shapes exhibited by Radiolaria can be explained by his reaction–diffusion mechanism. While working in the Computing Machine Laboratory at the University of Manchester in the early 1950s, Alan Turing reignited the interests he had had in both botany and biology from his early youth. During his school-days he was more interested in the structure of the flowers on the school sports field than in the games played there (see Fig. 1.3). It is known that during the Second World War he discussed the problem of phyllotaxis (the arrangement of leaves and florets in plants), and then at Manchester he had some conversations with Claude Wardlaw, the Professor of Botany in the University. Turing was keen to take forward the work that D’Arcy Thompson had published in On Growth and Form in 1917. In his now-famous paper of 1952 Turing solved his own ‘Equation of Morphogenesis’ in two dimensions, and demonstrated a solution that could explain the ‘dappling’—the black-and-white patterns—on cows. The next step was for me to solve Turing’s equation in three dimensions. The two-dimensional case concerns only surface features of organisms, such as dappling, spots, and stripes, whereas the three-dimensional version concerns the overall shape of an organism. In 1953 I joined Turing as a research student in the University of Manchester, and he set me the task of solving his equation in three dimensions. A remarkable journey of collaboration began. Turing chatted to me in a very friendly fashion.

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