scholarly journals Geometry of the 3D Pythagoras' Theorem

2016 ◽  
Vol 8 (6) ◽  
pp. 78 ◽  
Author(s):  
Luis Teia
Keyword(s):  
Three Dimensional ◽  
Transformation Process ◽  
Two Dimensional ◽  
Natural Evolution ◽  
Dimensional Version ◽  
Pythagoras Theorem

This paper explains step-by-step how to construct the 3D Pythagoras' theorem by geometric manipulation of the two dimensional version. In it is shown how $x+y=z$ (1D Pythagoras' theorem) transforms into $x^2+y^2=z^2$ (2D Pythagoras' theorem) via two steps: a 90-degree rotation, and a perpendicular extrusion. Similarly, the 2D Pythagoras' theorem transforms into 3D using the same steps. Octahedrons emerge naturally during this transformation process. Hence, each of the two dimensional elements has a direct three dimensional equivalent. Just like squares govern the 2D, octahedrons are the basic elements that govern the geometry of the 3D Pythagoras' theorem. As a conclusion, the geometry of the 3D Pythagoras' theorem is a natural evolution of the 1D and 2D. This interdimensional evolution begs the question -- Is there a bigger theorem at play that encompasses all three?

scholarly journals Optimal control of the two-dimensional Vlasov-Maxwell system

2020 ◽  
Author(s):  
Jörg Weber
Keyword(s):  
Adjoint Equation ◽  
Collisionless Plasma ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Maxwell System ◽  
Global Classical Solutions ◽  
Global Existence Of Solutions ◽  
Dimensional Version ◽  
The One

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

Self-Reproduction in Three-Dimensional Reversible Cellular Space

2002 ◽  
Vol 8 (2) ◽  
pp. 155-174 ◽  
Author(s):  
Katsunobu Imai ◽  
Takahiro Hori ◽  
Kenichi Morita
Keyword(s):  
Cellular Automata ◽  
Power Dissipation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Large Problem ◽  
Cellular Space ◽  
Reversible Cellular Automaton ◽  
Dimensional Version ◽  
Reversible Cellular Automata ◽  
Physical Phenomena

Due to inevitable power dissipation, it is said that nano-scaled computing devices should perform their computing processes in a reversible manner. This will be a large problem in constructing three-dimensional nano-scaled functional objects. Reversible cellular automata (RCA) are used for modeling physical phenomena such as power dissipation, by studying the dissipation of garbage signals. We construct a three-dimensional self-inspective self-reproducing reversible cellular automaton by extending the two-dimensional version SR8. It can self-reproduce various patterns in three-dimensional reversible cellular space without dissipating garbage signals.

The Three-Dimensional Necker Cube

2017 ◽  
Author(s):  
Nicola Bruno
Keyword(s):  
Necker Cube ◽  
Three Dimensional ◽  
Visual Illusions ◽  
Two Dimensional ◽  
Reversible Figure ◽  
Sensory Signals ◽  
Dimensional Version ◽  
Definition Of ◽  
Theoretical Issues

The Necker cube is a widely known example of a reversible figure. Perceptual reversals were first observed in engravings of crystals by the Swiss geologist Louis Albert Necker in 1832. Although Necker’s engravings were not exactly of regular cubes, the figure as it is used now can be perceived in two alternative arrangements of a three-dimensional (3D) cube. Although less widely known than the popular two-dimensional version, the 3D Necker cube is a surprisingly rich model for psychophysical investigation. This chapter summarizes relevant main results and their implications for diverse theoretical issues such as the definition of visual illusions, the role of global three-dimensional interpretations in the integration of local sensory signals, and the exploratory and multisensory nature of perceptual processes.

TERRAIN DECOMPOSITION AND LAYERED MANUFACTURING

2001 ◽  
Vol 11 (06) ◽  
pp. 647-668 ◽  
Author(s):  
SÁNDOR P. FEKETE ◽  
JOSEPH S. B. MITCHELL
Keyword(s):  
Three Dimensional ◽  
Layered Manufacturing ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Polygonal Region ◽  
Axis Parallel ◽  
Dimensional Version ◽  
Three Dimensional Models ◽  
Base Facet

We consider a problem that arises in generating three-dimensional models by methods of layered manufacturing: How does one decompose a given model P into a small number of sub-models each of which is a terrain polyhedron? Terrain polyhedra have a base facet such that, for each point of the polyhedron, the line segment joining the point to its orthogonal projection on the base facet lies within the polyhedron. Terrain polyhedra are exactly the class of polyhedral models for which it is possible to construct the model using layered manufacturing (with layers parallel to the base facet), without the need for constructing "supports" (which must later be removed). In order to maximize the integrity of a prototype, one wants to minimize the number of individual sub-models that are manufactured and then glued together. We show that it is NP-hard to decide if a three-dimensional model P of genus 0 can be decomposed into k terrain polyhedra. We also prove a two-dimensional version of this theorem, for the case in which P is a polygonal region with holes. Both results still hold if we are restricted to isothetic objects and/or axis-parallel layering directions.

An experiment design for schools to demonstrate the Maxwell distribution

2021 ◽  
Vol 57 (2) ◽  
pp. 025009
Author(s):  
Igor V Grebenev ◽  
Petr V Kazarin ◽  
Olga V Lebedeva
Keyword(s):  
Particle Interaction ◽  
Three Dimensional ◽  
Ideal Gas ◽  
Two Dimensional ◽  
Maxwell Distribution ◽  
Physical Content ◽  
Demonstration Experiment ◽  
Dimensional Version ◽  
Dimensional Distribution ◽  
Particle Velocities

Abstract The article describes a new version of a demonstration experiment for the Maxwell distribution. In the first part students analyse the applicability of the Gaussian distribution to the projection of the particle velocities in the suggested experiment. Further, students observe two-dimensional distribution of particles by the modulus of velocity in a mechanical demonstration model and compare the results with theoretical provisions. Demonstration of the two-dimensional version of the Maxwell distribution for particle interaction allows students to independently derive formulas for the three-dimensional Maxwell distribution for particles in an ideal gas. The use of the suggested demonstration ensures active engagement in fundamentally important physical content.

Application of the Ffowcs Williams/Hawkings equation to two-dimensional problems

2000 ◽  
Vol 403 ◽  
pp. 201-221 ◽  
Author(s):  
Y. P. GUO
Keyword(s):  
Turbulent Flows ◽  
Functional Dependence ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
The Body ◽  
Far Field ◽  
Two Dimensional ◽  
Retarded Time ◽  
Practical Applications ◽  
Dimensional Version

This paper discusses the application of the Ffowcs Williams/Hawkings equation to two-dimensional problems. A two-dimensional version of this equation is derived, which not only provides a very efficient way for numerical implementation, but also reveals explicitly the features of the source mechanisms and the characteristics of the far-field noise associated with two-dimensional problems. It is shown that the sources can be interpreted, similarly to those in three-dimensional spaces, as quadrupoles from turbulent flows, dipoles due to surface pressure fluctuations on the bodies in the flow and monopoles from non-vanishing normal accelerations of the body surfaces. The cylindrical spreading of the two-dimensional waves and their far-field directivity become apparent in this new version. It also explicitly brings out the functional dependence of the radiated sound on parameters such as the flow Mach number and the Doppler factor due to source motions. This dependence is shown to be quite different from those in three-dimensional problems. The two-dimensional version is numerically very efficient because the domains of the integration are reduced by one from the three-dimensional version. The quadrupole integrals are now in a planar domain and the dipole and monopole integrals are along the contours of the two-dimensional bodies. The calculations of the retarded-time interpolation of the integrands, a time-consuming but necessary step in the three-dimensional version, are completely avoided by making use of fast Fourier transform. To demonstrate the application of this, a vortex/airfoil interaction problem is discussed, which has many practical applications and involves important issues such as vortex shedding from the trailing edge.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

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