An Executable Notation, With Illustrations From Elementary Crystallography

1994 ◽  
Author(s):  
Donald B. Mclntyre
Keyword(s):  
Plate Tectonics ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Multiple Data ◽  
Special Cases ◽  
Unit Cells ◽  
Crystallographic Axes

Elementary crystallography is an ideal context for introducing students to mathematical geology. Students meet crystallography early because rocks are made of crystalline minerals. Moreover, morphological crystallography is largely the study of lines and planes in real three-dimensional space, and visualizing the relationships is excellent training for other aspects of geology; many algorithms learned in crystallography (e.g., rotation of arrays) apply also to structural geology and plate tectonics. Sets of lines and planes should be treated as entities, and crystallography is an ideal environment for introducing what Sylvester (1884) called "Universal Algebra or the Algebra of multiple quantity." In modern terminology, we need SIMD (Single Instruction, Multiple Data) or even MIMD. This approach, initiated by W.H. Bond in 1946, dispels the mysticism unnecessarily associated with Miller indices and the reciprocal lattice; edges and face-normals are vectors in the same space. The growth of mathematical notation has been haphazard, new symbols often being introduced before the full significance of the functions they represent had been understood (Cajori, 1951; Mclntyre, 1991b). Iverson introduced a consistent notation in 1960 (e.g., Iverson 1960, 1962, 1980). His language, greatly extended in the executable form called J (Iverson, 1993), is used here. For information on its availability as shareware, see the Appendix. Publications suitable as tutorials in , J are available (e.g., Iverson. 1991; Mclntyre, 1991 a, b; 1992a,b,c; 1993). Crystals are periodic structures consisting of unit cells (parallelepipeds) repeated by translation along axes parallel to the cell edges. These edges define the crystallographic axes. In a crystal of cubic symmetry they are orthogonal and equal in length (Cartesian). Those of a triclinic crystal, on the other hand, are unequal in length and not at right angles. The triclinic system is the general case; others are special cases. The formal description of a crystal gives prominent place to the lengths of the axes (a, b, and c) and the interaxial angles ( α, β, and γ). A canonical form groups these values into a 2 x 3 table (matrix), the first row being the lengths and the second the angles.

Design of Three-Dimensional, Triply Periodic Unit Cell Scaffold Structures for Additive Manufacturing

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Mazher Iqbal Mohammed ◽  
Ian Gibson
Keyword(s):  
Additive Manufacturing ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Unit Cell ◽  
Final Size ◽  
Fused Filament Fabrication ◽  
Cell Structures ◽  
Triply Periodic ◽  
Unit Cells ◽  
Scaffold Structure

Highly organized, porous architectures leverage the true potential of additive manufacturing (AM) as they can simply not be manufactured by any other means. However, their mainstream usage is being hindered by the traditional methodologies of design which are heavily mathematically orientated and do not allow ease of controlling geometrical attributes. In this study, we aim to address these limitations through a more design-driven approach and demonstrate how complex mathematical surfaces, such as triply periodic structures, can be used to generate unit cells and be applied to design scaffold structures in both regular and irregular volumes in addition to hybrid formats. We examine the conversion of several triply periodic mathematical surfaces into unit cell structures and use these to design scaffolds, which are subsequently manufactured using fused filament fabrication (FFF) additive manufacturing. We present techniques to convert these functions from a two-dimensional surface to three-dimensional (3D) unit cell, fine tune the porosity and surface area, and examine the nuances behind conversion into a scaffold structure suitable for 3D printing. It was found that there are constraints in the final size of unit cell that can be suitably translated through a wider structure while still allowing for repeatable printing, which ultimately restricts the attainable porosities and smallest printed feature size. We found this limit to be approximately three times the stated precision of the 3D printer used this study. Ultimately, this work provides guidance to designers/engineers creating porous structures, and findings could be useful in applications such as tissue engineering and product light-weighting.

Homogenization of Vibrating Periodic Lattice Structures

2005 ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Smart Materials ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Periodic Pattern ◽  
Periodic Lattice ◽  
Lattice Structures ◽  
Equivalent Representation ◽  
Detailed Model ◽  
Wide Range ◽  
Unit Cells

The paper describes a homogenization technique for periodic lattice structures. The analysis is performed by considering the irreducible unit cell as a building block that defines the periodic pattern. In particular, the continuum equivalent representation for the discrete structure is sought with the objective of retaining information regarding the local properties of the lattice, while condensing its global behavior into a set of differential equations. These equations can then be solved either analytically or numerically, thus providing a model which involves a significantly lower number of variables than those required for the detailed model of the assembly. The methodology is first tested by comparing the dispersion relations obtained through homogenization with those corresponding to the detailed model of the unit cells and then extended to the comparison of exact and approximate harmonic responses. This comparison is carried out for both one-dimensional and two-dimensional assemblies. The application to three-dimensional structures is not attempted in this work and will be approached in the future without the need for substantial conceptual changes in the theoretical procedure. Hence the presented technique is expected to be applicable to a wide range of periodic structures, with applications ranging from the design of structural elements of mechanical and aerospace interest to the optimization of smart materials with attractive mechanical, thermal or electrical properties.

Über Wirkungsbereichsteilungen von kubischer Symmetrie *

1981 ◽  
Vol 154 (3-4) ◽  
Author(s):  
Peter Engel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Space Group ◽  
High Instability ◽  
Three Dimensional Space

Abstract.Partitions of the three-dimensional space by Dirichlet domains with cubic symmetry have been studied and a method of their derivation is described. Detailed results are given for a section through a zone of high instability in the space groupct. Partitions of the three-dimensional space by Dirichlet domains with cubic symmetry have been studied and a method of their derivation is described. Detailed results are given for a section through a zone of high instability in the space group I4132-O8. 172 types of polyhedra could be found in this section and their three-dimensional fields of existence were determined. Two types of Dirichlet domains with 38 faces and 70 vertices were discovered.

scholarly journals Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse

2021 ◽  
Author(s):  
Asli Pinar Tan
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Mathematical Basis ◽  
Multiple Dimensions ◽  
Special Cases ◽  
Position Data ◽  
Gravitational Pull ◽  
Elliptical Motion ◽  
The Universe ◽  
Moving Bodies

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the “distance between points on any two different circles in three-dimensional space” is equivalent to the “distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.” Then, it is further mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Finally, two special cases when the “distance between points on two different circles in multi-dimensional space” become mathematically equivalent to distances in “parabolic” or “near-hyperbolic” trajectories are investigated. Concepts of “vector ellipse”, “vector hyperbola”, and “vector parabola” are also mathematically defined. The mathematical basis derived in this Article is utilized in the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe” in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.

scholarly journals Fragmentary and incidental behaviour of columns, slabs and crystals

2014 ◽  
Vol 372 (2008) ◽  
pp. 20120032 ◽  
Author(s):  
Walter Whiteley
Keyword(s):  
Periodic Structure ◽  
Periodic Structures ◽  
Three Dimensional ◽  
The Real ◽  
Combinatorial Properties ◽  
Open Questions ◽  
Unit Cells ◽  
Periodic Frameworks

Between the study of small finite frameworks and infinite incidentally periodic frameworks, we find the real materials which are large, but finite, fragments that fit into the infinite periodic frameworks. To understand these materials, we seek insights from both (i) their analysis as large frameworks with associated geometric and combinatorial properties (including the geometric repetitions) and (ii) embedding them into appropriate infinite periodic structures with motions that may break the periodic structure. A review of real materials identifies a number of examples with a local appearance of ‘unit cells’ which repeat under isometries but perhaps in unusual forms. These examples also refocus attention on several new classes of infinite ‘periodic’ frameworks: (i) columns—three-dimensional structures generated with one repeating isometry and (ii) slabs—three-dimensional structures with two independent repeating translations. With this larger vision of structures to be studied, we find some patterns and partial results that suggest new conjectures as well as many additional open questions. These invite a search for new examples and additional theorems.

Über Wirkungsbereichsteilungen von kubischer Symmetrie II. Die Typen von Wirkungsbereichspolyedern in den symmorphen kubischen Raumgruppen

1981 ◽  
Vol 157 (3-4) ◽  
Author(s):  
Peter Engel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Space Groups ◽  
Different Types ◽  
Three Dimensional Space

AbstractPartitions of the three-dimensional space by Dirichlet domains with cubic symmetry have been studied. Within the symmorphic cubic space groups a total of 91 different types of Dirichlet domains were found and their fields of existence were accurately determined. The occurence of the same type of Dirichlet domain in various space groups was investigated. In these space groups the

scholarly journals Application of Photopolymer Materials in Holographic Technologies

Polymers ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2020 ◽  
Author(s):  
Nadezhda Vorzobova ◽  
Pavel Sokolov
Keyword(s):  
3D Printing ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Polymer Materials ◽  
Solar Concentrators ◽  
Three Dimensional Objects ◽  
Wide Range ◽  
Material Fabrication ◽  
Dimensional Object

The possibility of the application of acrylate compositions and Bayfol HX photopolymers in holographic technologies is considered. The holographic characteristics of materials, their advantages, and limitations in relation to the tasks of obtaining holographic elements based on periodic structures are given. The conditions for obtaining controlled two and multichannel diffraction beam splitters are determined with advantages in terms of the simplicity of the fabrication process. The diffraction and selective properties of volume and hybrid periodic structures by radiation incidence in a wide range of angles in three-dimensional space are investigated, and new properties are identified that are of interest for the development of elements of holographic solar concentrators with advantages in the material used and the range of incidence angles. A new application of polymer materials in a new method of holographic 3D printing for polymer objects with arbitrary shape fabrication based on the projection of a holographic image of the object into the volume of photopolymerizable material is proposed, the advantage of which, relative to additive 3D printing technologies, is the elimination of the sequential synthesis of a three-dimensional object. The factors determining the requirements for the material, fabrication conditions, and properties of three-dimensional objects are identified and investigated.

A system of metrically invariant relations between the moduli squares of reciprocal-lattice vectors in one-, two- and three-dimensional space

2010 ◽  
Vol 43 (2) ◽  
pp. 269-275 ◽  
Author(s):  
Diedrich Stöckelmann ◽  
Herbert Kroll ◽  
Wolfgang Hoffmann ◽  
Rolf Heinemann
Keyword(s):  
Dimensional Space ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Powder Pattern ◽  
Alternative Route ◽  
Trial And Error ◽  
Special Relations ◽  
Lattice Planes ◽  
Three Dimensional Space

Given the background of trial-and-error methods employed in recent automatic powder pattern indexing, an alternative route is suggested based on a generalization of the original Runge–de Wolff approach. For this purpose, a system of five metrically invariant relations between the squared moduli (Qvalues) of reciprocal-lattice vectors is developed that encompasses the earlier special relations. The five invariant relations correspond to a line, a zone, a bizone, a cone and a pencil configuration of reciprocal-lattice vectors. In particular, the zone configuration relates four vectors being arbitrarily distributed in a plane and as such allows one to identify among a set of measuredQvalues all quadruples that define reciprocal-lattice planes intersecting in space.

Quasicrystals Perspectives and Potential Applications

MRS Bulletin ◽  
1997 ◽  
Vol 22 (11) ◽  
pp. 34-39 ◽  
Author(s):  
Daniel J. Sordelet ◽  
Jean Marie Dubois
Keyword(s):  
Window Glass ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Unit Cell ◽  
Translational Symmetry ◽  
Integer Number ◽  
Unit Cells ◽  
Potential Applications ◽  
Incommensurate Structures ◽  
Three Dimensional Space

For decades scientists have accepted the premise that solid matter can only order in two ways: amorphous (or glassy) like window glass or crystalline with atoms arranged according to translational symmetry. The science of crystallography, now two centuries old, was able to relate in a simple and efficient way all atomic positions within a crystal to a frame of reference in which a single unit cell was defined. Positions within the crystal could all be deduced from the restricted number of positions in the unit cell by translations along vectors formed by a combination of integer numbers of unit vectors of the reference frame. Of course disorder, which is always present in solids, could be understood as some form of disturbance with respect to this rule of construction. Also amorphous solids were naturally referred to as a full breakdown of translational symmetry yet preserving most of the short-range order around atoms. Incommensurate structures, or more simply modulated crystals, could be understood as the overlap of various ordering potentials not necessarily with commensurate periodicities.For so many years, no exception to the canonical rule of crystallography was discovered. Any crystal could be completely described using one unit cell and its set of three basis vectors. In 1848 the French crystallographer Bravais demonstrated that only 14 different ways of arranging atoms exist in three-dimensional space according to translational symmetry. This led to the well-known cubic, hexagonal, tetragonal, and associated structures. Furthermore the dihedral angle between pairs of faces of the unit cell cannot assume just any number since an integer number of unit cells must completely fill space around an edge.

HEISENBERG QUANTIZATION FOR SYSTEMS OF IDENTICAL PARTICLES

1993 ◽  
Vol 08 (21) ◽  
pp. 3649-3695 ◽  
Author(s):  
JON MAGNE LEINAAS ◽  
JAN MYRHEIM
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Classical System ◽  
Two Dimensions ◽  
Identical Particles ◽  
Natural Approach ◽  
Fermion Systems ◽  
Special Cases ◽  
Continuous Interpolation

We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.

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