Some Helical Structures Generated by Reflexions

1985 ◽  
Vol 38 (3) ◽  
pp. 299 ◽  
Author(s):  
AC Hurley
Keyword(s):  
Dimensional Space ◽  
Helical Structure ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Helical Structures ◽  
Higher Dimensional ◽  
Translation Component ◽  
Explicit Formulae ◽  
Different Dimensions ◽  
Early Discussion

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2

scholarly journals Visualising higher-dimensional space-time and space-scale objects as projections to ℝ3

2017 ◽  
Vol 3 ◽  
pp. e123 ◽  
Author(s):  
Ken Arroyo Ohori ◽  
Hugo Ledoux ◽  
Jantien Stoter
Keyword(s):  
Dimensional Space ◽  
Space Time ◽  
Three Dimensions ◽  
Time And Space ◽  
3D Objects ◽  
Levels Of Detail ◽  
Higher Dimensional Space Time ◽  
Space Scale ◽  
Higher Dimensional ◽  
Dimensional Space Time

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object’s shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from ℝ4to ℝ3. We present three projections that we believe are particularly intuitive for this purpose: (i) a simple ‘long axis’ projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3) followed by a stereographic projection to ℝ3, which results in an inwards-outwards fourth axis. Our focus is in using these projections from ℝ4to ℝ3, but they are formulated from ℝnto ℝn−1so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

scholarly journals AN ALTERNATIVE DIMENSIONAL REDUCTION PRESCRIPTION

1996 ◽  
Vol 11 (13) ◽  
pp. 1037-1045 ◽  
Author(s):  
J.D. EDELSTEIN ◽  
C. NÚÑEZ ◽  
F.A. SCHAPOSNIK ◽  
J.J. GIAMBIAGI
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Green Functions ◽  
Two Dimensional ◽  
Spatial Coordinate ◽  
Higher Dimensional

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to dropping the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular two-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher-dimensional space.

Estimating Three-Dimensional Space from Multiple Two-Dimensional Views

1993 ◽  
Vol 2 (1) ◽  
pp. 44-53 ◽  
Author(s):  
Kristinn R. Thorisson
Keyword(s):  
Visual Search ◽  
Eye Movement ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Three Dimensions ◽  
Limiting Factor ◽  
Single Subject ◽  
Search Performance ◽  
Two Dimensional

The most common visual feedback technique in teleoperation is in the form of monoscopic video displays. As robotic autonomy increases and the human operator takes on the role of a supervisor, three-dimensional information is effectively presented by multiple, televised, two-dimensional (2-D) projections showing the same scene from different angles. To analyze how people go about using such segmented information for estimations about three-dimensional (3-D) space, 18 subjects were asked to determine the position of a stationary pointer in space; eye movements and reaction times (RTs) were recorded during a period when either two or three 2-D views were presented simultaneously, each showing the same scene from a different angle. The results revealed that subjects estimated 3-D space by using a simple algorithm of feature search. Eye movement analysis supported the conclusion that people can efficiently use multiple 2-D projections to make estimations about 3-D space without reconstructing the scene mentally in three dimensions. The major limiting factor on RT in such situations is the subjects' visual search performance, giving in this experiment a mean of 2270 msec (SD = 468; N = 18). This conclusion was supported by predictions of the Model Human Processor (Card, Moran, & Newell, 1983), which predicted a mean RT of 1820 msec given the general eye movement patterns observed. Single-subject analysis of the experimental data suggested further that in some cases people may base their judgments on a more elaborate 3-D mental model reconstructed from the available 2-D views. In such situations, RTs and visual search patterns closely resemble those found in the mental rotation paradigm (Just & Carpenter, 1976), giving RTs in the range of 5-10 sec.

scholarly journals Visualising higher-dimensional space-time and space-scale objects as projections to \(\mathbb{R}^3\)

2017 ◽  
Author(s):  
Ken Arroyo Ohori ◽  
Hugo Ledoux ◽  
Jantien Stoter
Keyword(s):  
Dimensional Space ◽  
Space Time ◽  
Three Dimensions ◽  
Time And Space ◽  
3D Objects ◽  
Levels Of Detail ◽  
Higher Dimensional Space Time ◽  
Space Scale ◽  
Higher Dimensional ◽  
Dimensional Space Time

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object's shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from \(\mathbb{R}^4\) to \(\mathbb{R}^3\). We present three projections that we believe are particularly intuitive for this purpose: (i) a simple `long axis' projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (\(S^3\)) followed by a stereographic projection to \(\mathbb{R}^3\), which results in an inwards-outwards fourth axis. Our focus is in using these projections from \(\mathbb{R}^4\) to \(\mathbb{R}^3\), but they are formulated from \(\mathbb{R}^n\) to \(\mathbb{R}^{n-1}\) so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

scholarly journals Three-dimensional space: locomotory style explains memory differences in rats and hummingbirds

2014 ◽  
Vol 281 (1784) ◽  
pp. 20140301 ◽  
Author(s):  
I. Nuri Flores-Abreu ◽  
T. Andrew Hurly ◽  
James A. Ainge ◽  
Susan D. Healy
Keyword(s):  
Horizontal Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Vertical Component ◽  
Horizontal Component ◽  
Vertical Dimension ◽  
Three Dimensions ◽  
Two Dimensional ◽  
With Memory ◽  
Three Dimensional Space

While most animals live in a three-dimensional world, they move through it to different extents depending on their mode of locomotion: terrestrial animals move vertically less than do swimming and flying animals. As nearly everything we know about how animals learn and remember locations in space comes from two-dimensional experiments in the horizontal plane, here we determined whether the use of three-dimensional space by a terrestrial and a flying animal was correlated with memory for a rewarded location. In the cubic mazes in which we trained and tested rats and hummingbirds, rats moved more vertically than horizontally, whereas hummingbirds moved equally in the three dimensions. Consistent with their movement preferences, rats were more accurate in relocating the horizontal component of a rewarded location than they were in the vertical component. Hummingbirds, however, were more accurate in the vertical dimension than they were in the horizontal, a result that cannot be explained by their use of space. Either as a result of evolution or ontogeny, it appears that birds and rats prioritize horizontal versus vertical components differently when they remember three-dimensional space.

scholarly journals Visualising higher-dimensional space-time and space-scale objects as projections to \(\mathbb{R}^3\)

2017 ◽  
Author(s):  
Ken Arroyo Ohori ◽  
Hugo Ledoux ◽  
Jantien Stoter
Keyword(s):  
Dimensional Space ◽  
Space Time ◽  
Three Dimensions ◽  
Time And Space ◽  
3D Objects ◽  
Levels Of Detail ◽  
Higher Dimensional Space Time ◽  
Space Scale ◽  
Higher Dimensional ◽  
Dimensional Space Time

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object's shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from \(\mathbb{R}^4\) to \(\mathbb{R}^3\). We present three projections that we believe are particularly intuitive for this purpose: (i) a simple `long axis' projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (\(S^3\)) followed by a stereographic projection to \(\mathbb{R}^3\), which results in an inwards-outwards fourth axis. Our focus is in using these projections from \(\mathbb{R}^4\) to \(\mathbb{R}^3\), but they are formulated from \(\mathbb{R}^n\) to \(\mathbb{R}^{n-1}\) so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

scholarly journals Determination of load distribution in a given medium according to the values of the loads at certain points

2021 ◽  
pp. 167-175
Author(s):  
Oleksandr Mostovenko ◽  
Serhii Kovalov ◽  
Svitlana Botvinovska
Keyword(s):  
Boundary Conditions ◽  
Load Distribution ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Geometric Model ◽  
Three Dimensional ◽  
The Body ◽  
Three Dimensions ◽  
Two Dimensional ◽  
One Dimensional

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2]. In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7]. From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes. Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions. If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid. If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes. The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells. As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

Packing Optimisation Using a Rubberband Analogy

2001 ◽  
Author(s):  
Georges M. Fadel ◽  
Avijit Sinha ◽  
Todd McKee
Keyword(s):  
Convex Hull ◽  
Relative Position ◽  
Dimensional Space ◽  
Packing Problem ◽  
Rubber Band ◽  
Single Component ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Optimal Packing ◽  
3 Dimensional

Abstract Packing is a topic of interest in many fields. During our research on the underhood-packing problem, we observed that to bring components together — without switching their relative position — one could use the analogy of a rubber object stretched around the artifacts. In two-dimensional space, that object is a rubber band, and in three-dimensions, it is a balloon. Using this analogy, the convex hull can be used to determine the direction of forces applied to a single component, and a motion can result from the application of such forces. The objects can then be moved until contact occurs, at which point the forces become moments, and the objects can rotate with respect to each other. This technique can guarantee locally optimal packing, and displays a very intuitive behavior that might lead to further advances in optimization. This paper introduces the methodology for optimizing the packing of 2-dimensional geometric entities (polygons) in a plane and of 3-dimensional objects in space using the Rubber band Analogy.

scholarly journals Crystal structure and Hirshfeld surface analysis of 1,2-bis(2′,6′-diisopropoxy-[2,3′-bipyridin]-6-yl)benzene

2018 ◽  
Vol 74 (10) ◽  
pp. 1475-1479 ◽  
Author(s):  
Ki-Min Park ◽  
Suk-Hee Moon ◽  
Youngjin Kang
Keyword(s):  
Surface Analysis ◽  
Pyridine Ring ◽  
Helical Structure ◽  
Hirshfeld Surface ◽  
Van Der Waals Interactions ◽  
Hirshfeld Surface Analysis ◽  
Supramolecular Network ◽  
Two Dimensional ◽  
Helical Structures ◽  
Π Interactions

The title molecule, C38H42N4O4, displays a helical structure induced by the combination of the C—C—C—C torsion angle [−10.8 (2)°] between two 2,3′-bipyridyl units attached to the 1,2-positions of the central benzene ring and consecutive connections between five aromatic rings through the meta- and ortho-positions. Intramolecular C—H...π interactions between an H atom of a pyridine ring and the centroid of a another pyridine ring contributes to the stabilization of the helical structure. In the crystal, weak C—H...π interactions link the title molecules into a two-dimensional supramolecular network extending parallel to the ac plane, in which the molecules with right- and left-handed helical structures are alternately arranged. Hirshfeld surface analysis and two-dimensional fingerprint plots indicate that the molecular packing is dominated by van der Waals interactions between neighbouring H atoms, as well as by C—H...π interactions. One isopropoxyl group is disordered over two sets of sites [occupancy ratio 0.715 (5):0.285 (5)].

scholarly journals Emergence of complex patterns in a higher-dimensional phyllotactic system

2016 ◽  
Vol 85 (4) ◽  
Author(s):  
Robert M. Beyer ◽  
Jürgen Richter-Gebert
Keyword(s):  
Chaotic Behavior ◽  
Three Dimensions ◽  
Dynamical Model ◽  
Divergence Angle ◽  
Dimensional Case ◽  
Complex Structures ◽  
Two Dimensional ◽  
Higher Dimensional ◽  
Biological Problem ◽  
Space Occupation

A hypothesis commonly known as Hofmeister’s rule states that primordia appearing at the apical ring of a plant shoot in periodic time steps are formed in the position where the most space is available with respect to the space occupation of already-formed primordia. A corresponding two-dimensional dynamical model has been extensively studied by Douady and Couder, and shown to generate a variety of observable phyllotactic patterns indeed. In this study, motivated by mathematical interest in a theoretical phyllotaxis-inspired system rather than by a concrete biological problem, we generalize this model to three dimensions and present the dynamics observed in simulations, thereby illustrating the range of complex structures that phyllotactic mechanisms can give rise to. The patterns feature unexpected additional properties compared to the two-dimensional case, such as periodicity and chaotic behavior of the divergence angle.

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