Non-periodic Space Structures

1987 ◽  
Vol 2 (2) ◽  
pp. 93-108 ◽  
Author(s):  
Haresh Lalvani
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Infinite Class ◽  
Space Filling ◽  
Higher Dimensional ◽  
Centrally Symmetric ◽  
Interesting Class ◽  
Periodic Space ◽  
Three Dimensional Space

An interesting class of two- and three-dimensional space structures can be derived from projections of higher-dimensional structures. Regular polygons and regular-faced polyhedra provide the geometry of families of n-stars from which two- and three-dimensional projections of n-dimensional grids can be derived. These projections are rhombic space grids composed of all-space filling rhombi and rhombohedra with edges parallel to n directions. An infinite class of single-, double- and multi-layered grids can be derived from n-sided polygons and prisms, and a finite class of multi-directional grids from the polyhedral symmetry groups. The grids can be periodic, centrally symmetric or non-periodic, and act as skeletons to generate corresponding classes of space-filling, packings and labyrinths.

scholarly journals Celestial Spheres

2018 ◽  
Vol 2 (2) ◽  
pp. 168-186
Author(s):  
Austin M. Freeman
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Good Evidence ◽  
Medieval Period ◽  
The Bible ◽  
Higher Dimensional ◽  
Spatial Dimensions ◽  
Modern Physics ◽  
The Subject ◽  
Three Dimensional Space

Angels probably have bodies. There is no good evidence (biblical, philosophical, or historical) to argue against their bodiliness; there is an abundance of evidence (biblical, philosophical, historical) that makes the case for angelic bodies. After surveying biblical texts alleged to demonstrate angelic incorporeality, the discussion moves to examine patristic, medieval, and some modern figures on the subject. In short, before the High Medieval period belief in angelic bodies was the norm, and afterwards it is the exception. A brief foray into modern physics and higher spatial dimensions (termed “hyperspace”), coupled with an analogical use of Edwin Abbott’s Flatland, serves to explain the way in which appealing to higher-dimensional angelic bodies matches the record of angelic activity in the Bible remarkably well. This position also cuts through a historical equivocation on the question of angelic embodiment. Angels do have bodies, but they are bodies very unlike our own. They do not have bodies in any three-dimensional space we can observe, but are nevertheless embodied beings.

scholarly journals ON THE SAMPLING OF SERIAL SECTIONING TECHNIQUE FOR THREE DIMENSIONAL SPACE-FILLING GRAIN STRUCTURES

2011 ◽  
Vol 19 (2) ◽  
pp. 81 ◽  
Author(s):  
Guoquan Liu ◽  
Haibo Yu
Keyword(s):  
Size Distribution ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Low Carbon Steel ◽  
Steel Specimen ◽  
Low Carbon ◽  
Serial Sectioning ◽  
Space Filling ◽  
Almost All ◽  
Three Dimensional Space

Serial sectioning technique provides plenty of quantitative geometric information of the microstructure analyzed, including those unavailable from stereology with one- and two-dimensional probes. This may be why it used to be and is being continuously served as one of the most common and invaluable methods to study the size and the size distribution, the topology and the distribution of topology parameters, and even the shape of three-dimensional space filling grains or cells. On the other hand, requiring tedious lab work, the method is also very time and energy consuming, most often only less than one hundred grains per sample were sampled and measured in almost all reported practice. Thus, a question is often asked: for typical microstructures in engineering materials, are so many grains or cells sampled adequate to obtain reliable results from this technique? To answer this question, experimental data of 1292 contiguous austenite grains in a low-carbon steel specimen obtained from the serial sectioning analysis are presented in this paper, which demonstrates the effect of sampling on the measurement of various parameters of grain size distribution and of the grain topology distribution. The result provides one of rules of thumb for grain stereology of similar microstructures.

A Mathematical Model for the Morphology of the Pulmonary Acinus

1996 ◽  
Vol 118 (2) ◽  
pp. 210-215 ◽  
Author(s):  
E. Denny ◽  
R. C. Schroter
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Computational Method ◽  
Published Data ◽  
Space Filling ◽  
Pulmonary Acinus ◽  
Path Lengths ◽  
Three Dimensional Space

A computational method is proposed for the construction of a three-dimensional space-filling model of an acinar ventilatory unit. Its geometry consists of truncated octahedra arranged in a cuboidal block. The ducts and alveoli are formed by opening specific common faces between polyhedra. The branching structure is automatically computed using algorithms solely to maximise the number of alveoli and minimise the average path lengths; it is not formed with reference to published experimental data. Properties of the model such as the total alveolar and ductal volumes, the distribution of individual path lengths to the alveolar sacs, and the average number of ducts per generation are calculated. The predicted morphology of the model compares well with published data for rat lungs.

scholarly journals Recognizability of Tetrahedral Picture Languages

2019 ◽  
Vol 8 (4) ◽  
pp. 7379-7383
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Vital Role ◽  
Generative Models ◽  
Two Dimensions ◽  
Generative Model ◽  
Space Filling ◽  
Sequential Space ◽  
Picture Languages ◽  
Three Dimensional Space

In two dimensions, tiling the plane plays a vital role. Many picture generative models were attempted to tile the three dimensional space. A. Dharani et al. [6] introduced a new theoretical picture generative model to tile a three dimensional space using tetrahedral tile in two different ways namely Sequential Space Filling Grammar (SSFG) and Parallel Space Filling Grammar (PSFG). Local and recognizable tetrahedral picture languages are introduced in this paper and some of its properties are studied.

scholarly journals A Hierarchical Anti-Hebbian Network Model for the Formation of Spatial Cells in Three-Dimensional Space

2018 ◽  
Author(s):  
Karthik Soman ◽  
Srinivasa Chakravarthy ◽  
Michael M. Yartsev
Keyword(s):  
Network Model ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Grid Cell ◽  
Cognitive Maps ◽  
3D Space ◽  
Learning Rules ◽  
Higher Dimensional ◽  
New Type ◽  
Three Dimensional Space

AbstractThree dimensional (3D) spatial cells in the mammalian hippocampalformationare believed to support the existence of 3D cognitive maps. Modeling studies are crucial to comprehend the neural principles governing the formation of these maps, yet to date very few have addressed this topic in 3D space. Here, we present a hierarchical network model for the formation of 3D spatial cells using anti-hebbian network. Built on empirical data, the model accounts for the natural emergence of 3D place, border and grid-cells as well as a new type of previously undescribed spatial cell type which we call plane cells. It further explains the plausible reason behind the place and grid-cell anisotropic coding that has been observed in rodents and the potential discrepancy with the predicted periodic coding during 3D volumetric navigation. Lastly, it provides evidence for the importance of unsupervised learning rules in guiding the formation of higher dimensional cognitive maps.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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