Structure analysis of modulated crystals. III. Scattering formalism and symmetry considerations: extension to higher-dimensional space groups

1988 ◽  
Vol 44 (3) ◽  
pp. 235-239 ◽  
Author(s):  
V. Petříček ◽  
P. Coppens
Keyword(s):  
Structure Analysis ◽  
Dimensional Space ◽  
Space Groups ◽  
Higher Dimensional

The decomposition of crystal families

1980 ◽  
Vol 88 (2) ◽  
pp. 245-263 ◽  
Author(s):  
J. D. Jarratt
Keyword(s):  
Dimensional Space ◽  
Canonical Decomposition ◽  
Space Groups ◽  
Higher Dimensional ◽  
Result Theorem ◽  
Free Parameters

In the course of their enumeration of all 4-dimensional space groups, Brown, Bülow, Neubüser, Wondratschek and Zassenhaus have introduced concepts appropriate to the study of n-dimensional crystallography for n ≥ 4 (see (2), (3)). One such concept is that of a crystal family. Families seem to be particularly useful as a framework within which to study higher dimensional crystallography, primarily because they determine a classification of all the standard crystallographic objects and overcome the traditional confusion over crystal systems (see (2); pp. 16–17, (9)). In this paper, techniques are developed for the determination of all rationally decomposable families in a given dimension from the indecomposable families of lower dimensions. These techniques place emphasis on three geometric invariants of families: the decomposition pattern; the canonical decomposition pattern; and the number of free parameters. This, it is felt, further reinforces their position as fundamental objects. The key result (Theorem 5.4) is:a family uniquely determines, and is uniquely determined by, the constituent families in the canonical decomposition.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

Intergrowth polytypoids as modulated structures: a superspace description of the Sr n (Nb,Ti) n O3n + 2 compound series

2001 ◽  
Vol 57 (4) ◽  
pp. 471-484 ◽  
Author(s):  
L. Elcoro ◽  
J. M. Perez-Mato ◽  
R. L. Withers
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Separation Distance ◽  
Space Group ◽  
Space Groups ◽  
Cell Parameters ◽  
Average Structure ◽  
Layer Stacking ◽  
Compound Series ◽  
Three Dimensional Space

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

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 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.

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