Coordination sequences for lattices

1995 ◽  
Vol 210 (12) ◽  
Author(s):  
M. O'Keeffe
Keyword(s):  
Three Dimensions ◽  
3 Dimensional ◽  
Higher Dimensional ◽  
Closest Packing ◽  
Geometric Density

AbstractCoordination sequences for five 3-dimensional, ten 4-dimensional and eleven higher-dimensional lattices have been determined and all but one can be expressed as simple polynomials. Some regularities in these polynomials are observed. The correlation between topological and geometric density is demonstrated for 4-dimensional lattices. It is conjectured that hexagonal closest packing is topologically the densest packing in three dimensions.

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.

Does the Body Image Exist in Three Dimensions? The Study of Visual Mental Representation of a Body and a Nonbody Object

2001 ◽  
Vol 92 (1) ◽  
pp. 223-233
Author(s):  
D. P. McCabe ◽  
D. I. Ben-Tovim ◽  
M. K. Walker ◽  
D. Pomeroy
Keyword(s):  
Spatial Information ◽  
Mental Image ◽  
The Body ◽  
Three Dimensions ◽  
Long Term Memory ◽  
Depth Information ◽  
Mental Images ◽  
Term Memory ◽  
3 Dimensional

Do the mental Images of 3-dimensional objects recreate the depth characteristics of the original objects' This investigation of the characteristics of mental images utilized a novel boundary-detection task that required participants to relate a pair of crosses to the boundary of an image mentally projected onto a computer screen. 48 female participants with body attitudes within expected normal range were asked to image their own body and a familiar object from the front and the side. When the visual mental image was derived purely from long-term memory, accuracy was better than chance for the front (64%) and side (63%) of the body and also for the front (55%) and side (68%) of the familiar nonbody object. This suggests that mental images containing depth and spatial information may be generated from information held in long-term memory. Pictorial exposure to views of the front or side of the objects was used to investigate the representations from which this 3-dimensional shape and size information is derived. The results are discussed in terms of three possible representational formats and argue that a front-view 2½-dimensional representation mediates the transfer of information from long-term memory when depth information about the body is required.

Dynamical Variation of Weierstrass-Mandelbrot Function in Higher Dimensional Space

2013 ◽  
Vol 470 ◽  
pp. 767-771
Author(s):  
L. Zhang ◽  
Shu Tang Liu
Keyword(s):  
Hurst Exponent ◽  
Dimensional Space ◽  
Dynamical Behavior ◽  
Fractal Dimensions ◽  
Statistical Characteristics ◽  
3 Dimensional ◽  
Higher Dimensional ◽  
Complex Phenomena ◽  
Real Complex

Many real complex phenomena are related with Weierstrass-Mandelbrot function (WMF). Most researches focus on the systems as parameters fixed, such as calculations of its different fractal dimensions or the statistical characteristics of its generalized form and so on. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these systems as parameters change is important. However, there is few results about this aim. In this paper, we propose simulated results for the effects of parameters changeably on the graph of WMF in higher dimensional space. In addition, the relationships between the Hurst exponent of WMF and its parameters dynamically in 2-and 3-dimensional spaces are also given.

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 A framework for quantifying properties of 3-dimensional movement-based signals

2010 ◽  
Vol 56 (3) ◽  
pp. 327-336 ◽  
Author(s):  
Shaun Td New ◽  
Richard A Peters
Keyword(s):  
Three Dimensions ◽  
Tail Flick ◽  
Signal Design ◽  
3D Animation ◽  
3D Display ◽  
Single Plane ◽  
3 Dimensional ◽  
Tail Tip ◽  
Detector Model ◽  
New Framework

Abstract Understanding how signal properties are optimized for the reliable transmission of information requires accurate description of the signal in time and space. For movement-based signals where movement is restricted to a single plane, measurements from a single viewpoint can be used to consider a range of viewing positions based on simple geometric calculations. However, considerations of signal properties from a range of viewing positions for movements extending into three-dimensions (3D) are more problematic. We present here a new framework that overcomes this limitation, and enables us to quantify the extent to which movement-based signals are view-specific. To illustrate its application, a Jacky lizard tail flick signal was filmed with synchronized cameras and the position of the tail tip digitized for both recordings. Camera alignment enabled the construction of a 3D display action pattern profile. We analyzed the profile directly and used it to create a detailed 3D animation. In the virtual environment, we were able to film the same signal from multiple viewing positions and using a computational motion analysis algorithm (gradient detector model) to measure local image velocity in order to predict view dependent differences in signal properties. This approach will enable consideration of a range of questions concerning movement-based signal design and evolution that were previously out of reach.

scholarly journals Structure determinations for random-tiling quasicrystals

2000 ◽  
Vol 215 (10) ◽  
Author(s):  
C.L. Henley ◽  
V. Elser ◽  
M. Mihalkovic
Keyword(s):  
Structural Information ◽  
Direct Method ◽  
Dimensional Space ◽  
Waller Factor ◽  
Data Set ◽  
3 Dimensional ◽  
Random Tiling ◽  
Debye Waller Factor ◽  
Higher Dimensional ◽  
Structure Determinations

How, in principle, could one solve the atomic structure of a quasicrystal, modeled as a random tiling decorated by atoms, and what techniques are available to do it? One path is to solve the phase problem first, obtaining the density in a higher dimensional space which yields the averaged scattering density in 3-dimensional space by the usual construction of an incommensurate cut. A novel direct method for this is summarized and applied to an i(AlPdMn) data set. This averaged density falls short of a true structure determination (which would reveal the typical unaveraged atomic patterns.) We discuss the problematic validity of inferring an ideal structure by simply factoring out a "perp-space" Debye-Waller factor, and we test this using simulations of rhombohedral tilings. A second, "unified" path is to relate the measured and modeled intensities directliy, by adjusting parameters in a simulation to optimize the fit. This approach is well suited for unifying structural information from diffraction and from minimizing total energies derived ultimately from ab-initio calculations. Finally, we discuss the special pitfalls of fitting random-tiling decagonal phases.

Constituent Particles and Dispersoids in an Al-Mn-Fe-Si Alloy Studied in Three-Dimensions by Serial Sectioning

2013 ◽  
Vol 765 ◽  
pp. 451-455 ◽  
Author(s):  
Liam Dwyer ◽  
Joseph Robson ◽  
Joao Quinta da Fonseca ◽  
Nicolas Kamp ◽  
Teruo Hashimoto ◽  
...  
Keyword(s):  
Accurate Determination ◽  
Three Dimensions ◽  
Process Conditions ◽  
Second Phase ◽  
Serial Sectioning ◽  
3 Dimensional ◽  
Intermetallic Particles ◽  
Second Phase Particles ◽  
Proper Consideration

Second phase particles in wrought aluminium alloys are crucial in controlling recrystallization and texture. In Al-Mn-Fe-Si (3xxx) alloys, the size, spacing, and distribution of both large constituent particles and small dispersoids are manipulated by heat treatment to obtain the required final microstructure and texture for operations such as can-making. Understanding how these particles evolve as a function of process conditions is thus critical to optimize alloy performance. In this study, a novel 3-dimensional technique involving serial sectioning in the scanning electron microscope (SEM) has been used to analyse the intermetallic particles found in an as-cast and homogenized Al-Mn-Fe-Si alloy. This has allowed an accurate determination of the size and shape of the constituent particles and dispersoids derived from a 3-dimensional dataset. It is demonstrated that a proper consideration of the 3-dimensional microstructure reveals important features that are not obvious from 2-dimensional sections alone.

scholarly journals 3D Water Vapor Field in the Atmospheric Boundary Layer Observed with Scanning Differential Absorption Lidar

2016 ◽  
Author(s):  
F. Späth ◽  
A. Behrendt ◽  
S. K. Muppa ◽  
S. Metzendorf ◽  
A. Riede ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Water Vapor ◽  
Atmospheric Boundary Layer ◽  
Elevation Angle ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Differential Absorption ◽  
Differential Absorption Lidar ◽  
3 Dimensional ◽  
Western Germany

Abstract. The scanning differential absorption lidar (DIAL) of the University of Hohenheim (UHOH) determines fields of the atmospheric water vapor number density with a temporal resolution of a few seconds and spatial resolution of up to a few tens of meters. We present three case studies which show that this high resolution combined with 2- and 3-dimensional scans allows for new insights in the 3-dimensional structure of the water vapor field in the atmospheric boundary layer (ABL). In spring 2013, the UHOH DIAL was operated within the scope of the HD(CP)2 Observational Prototype Experiment (HOPE) in western Germany. HOPE was part of the project High Definition of Clouds and Precipitation for advancing Climate Prediction (HD(CP)2). Range-height indicator (RHI) scans of the UHOH DIAL show the water vapor heterogeneity within a range of a few kilometers and its impact on the formation of clouds at the ABL top. The uncertainty of the measured data was assessed by extending a technique, which was formerly applied to vertical time series, to scanning data. Typically, even during daytime, the accuracy of the DIAL measurements is between 0.5 and 0.8 g m−3 (or < 6 %) within the ABL, so that now the performance of an RHI scan from the surface to an elevation angle of 90 degrees becomes possible within 10 min. In summer 2014, the UHOH DIAL participated in the Surface-Atmosphere-Boundary-Layer-Exchange (SABLE) campaign in south-western Germany. Volume scans show the water vapor field in three dimensions. In this case, multiple humidity layers were present. Differences in their heights in different directions can be attributed to different surface elevation. With low elevation scans in the surface layer, the humidity profiles and gradients related to different land use and surface stabilities were also revealed.

scholarly journals Perfect hypercomplex algebras: Semi-tensor product approach

2021 ◽  
Vol 1 (4) ◽  
pp. 177-187
Author(s):  
Daizhan Cheng ◽  
◽  
Zhengping Ji ◽  
Jun-e Feng ◽  
Shihua Fu ◽  
...  
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Characteristic Functions ◽  
Hypercomplex Numbers ◽  
3 Dimensional ◽  
Zero Sets ◽  
Higher Dimensional ◽  
Product Approach ◽  
Necessary And Sufficient

<abstract><p>The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.</p></abstract>

scholarly journals TRAINING AUTODESK FUSION 360 FOR TEENAGE OF SENIOR HIGH SCHOOL GRADUATES IN IMPROVING ABILITY IN DISASTERS

ICCD ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 285-289 ◽  
Author(s):  
Resa Trauna Suhada ◽  
Silvi Ariyanti ◽  
Anggun Vionieta Fajar ◽  
Aam Komalasari
Keyword(s):  
High School ◽  
Junior High School ◽  
High School Graduates ◽  
Three Dimensions ◽  
3 Dimensional ◽  
App Store ◽  
Design Collaboration ◽  
Training Materials ◽  
Service Activity ◽  
Design Activities

Autodesk launches Autodesk Fusion 360 - a cloud-based device that combines industrial design and 3D mechanics, collaboration, machinery in one package - the first solution available for purchase on Mac App Store worldwide including Indonesia. The Fusion 360 combines design, collaboration and machine tools into one package and is now available with the security and simplicity of the Mac App Store. With Fusion 360, designers and engineers can create products that have shapes and functions and prepare them for the fabrication process by using a single device. This community service activity is in the form of Autodesk Fusion 360 application training, with a total of 26 participants. Training activities include: introduction, visual demonstration and practice. The training materials are as follows: introduction of Autodesk Fusion 360, drawing 3-dimensional shapes, 3-dimensional drawing modifications, Solid modeling, Plotting. The purpose of this devotional activity is: Introduce and explain the basic theory of drawing using the computer, especially the use of Autodesk Fusion 360 program in the form of three dimensions. The benefits of this devotional activity are the junior high school graduates can recognize and apply computer-based drawing techniques. Improving the skills of high school graduates especially skills in the field of Fusion 360 applications. The results of this activity are the participants can carry out all product design activities and declared to pass and are entitled to receive a certificate issued directly by the company that produces Autodesk Fusion 360 Software.

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