Geometry in three dimensions over GF (3)

1954 ◽  
Vol 222 (1149) ◽  
pp. 262-286 ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Galois Field ◽  
The Other ◽  
Three Dimensions ◽  
Common Vertex ◽  
Line Geometry ◽  
Fundamental Properties ◽  
The Common ◽  
Three Dimensional Space

The object of this paper is to give some account of the geometry of the three-dimensional space S wherein the co-ordinates belong to a Galois field K of 3 marks. A description of the fundamental properties of quadrics is sufficiently long for one paper, and so an account of the line geometry is deferred. The early paragraphs (§§ 1 to 4) are necessarily concerned with geometry on a line or in a plane. A line consists of 4 points; these are self-projective under all 4! permutations. A plane consists of 13 points and has the same number, 234, of triangles, quadrangles, quadri-laterals and non-singular conics. A diagram is helpful, especially when we consider sections by planes in S . The space S has 40 points. Non-singular quadrics are of two kinds: either ruled, when we call them hyperboloids, or non-ruled, when we call them ellipsoids. A hyperboloid H consists of 16 points and has a pair of reguli; the 24 points of S not on H are the vertices of 6 tetra-hedra that form two allied desmic triads. The ellipsoid F is introduced in § 12; it consists of 10 points, the other 30 points of S being separated into two batches of 15 between which there is a symmetrical (3, 3) correspondence. Either batch can be arranged as a set of 6 pentagons, each of the 15 points being the common vertex of 2 of these. The pentagons of either set have all their edges tangents of F and, with their polar pentahedra, have significant properties and interrelations. By no means their least important attribute is that they afford, with F , so apposite a domain of operation for the simple group of order 360. In §§ 23 to 26 are described the operations of the group in this setting. Thereafter the 36 separations of the 10 points of F into complementary pentads are discussed, no 4 of either pentad being coplanar. During the work constructions for an ellipsoid are encountered; one is in § 16, another in § 30.

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.

scholarly journals On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation

1926 ◽  
Vol 110 (756) ◽  
pp. 700-708
Keyword(s):  
Present Note ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Four Dimensions ◽  
Cubic Surfaces ◽  
Mutual Relations ◽  
Quartic Surfaces ◽  
Three Dimensional Space

There exists in space of four dimensions an interesting figure of 15 lines and 15 points, first considered by Stéphanos (‘Compt. Rendus,’ vol. 93, 1881), though suggested very clearly by Cremona’s discussion of cubic surfaces in three-dimensional space. In connection with the figure of 15 lines there arises a quartic surface, the intersection of two quadrics, which is of importance as giving rise by projection to the Cyclides, as Segre has shown in detail (‘Math. Ann.,’ vol. 24, 1884). The symmetry of the figure suggests, howrever, the consideration of 15 such quartic surfaces; and it is natural to inquire as to the mutual relations of these surfaces, in particular as to their intersections. In general, two surfaces in space of four dimensions meet in a finite number of points. It appears that in this case any two of these 15 surfaces have a curve in common; it is the purpose of the present note to determine the complete intersection of any two of these 15 surfaces. Similar results may be obtained for a system of cubic surfaces in three dimensions, corresponding to those here given for this system of quartic surfaces in four dimensions, since the surfaces have one point in common, which may be taken as the centre of a projection.

scholarly journals A quadratic transformation of the three-dimensional space of all conics through two points of a projective plane

2009 ◽  
Vol 42 (3) ◽  
Author(s):  
Giorgio Donati
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Quadratic Transformation ◽  
Pencil Of Conics ◽  
Dimensional Projective Space ◽  
Three Dimensional Space

AbstractUsing the Steiner’s method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.

Installing Technology of Spherical Welding Lattice Shell in Three-Dimensional Space

2013 ◽  
Vol 838-841 ◽  
pp. 273-279
Author(s):  
Xiao Bo Xu ◽  
Qian Zhao ◽  
Hui Ying Li
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Shell Structures ◽  
Engineering Practice ◽  
High Quality ◽  
Good Efficiency ◽  
The Common ◽  
Lattice Shell ◽  
Technical Measures ◽  
Three Dimensional Space

Spherical welding lattice shell structures are usually used in stadiums and public buildings. The main difficult problems in construction are positioning of welding members and controlling welding deformations in three-dimensional space. The common positioning methods are poor in operability and accuracy, which cannot meet the demands of precise construction. In this paper, a three-dimensional positioner was developed according to the spherical latitude and longitude lines intersect positioning principle based on the Kitwitt monolayer welding lattice shell in Guangzhou Conghua Liuxi Square project. In addition, the welding deformations were controlled effectively by innovative technical measures. Good efficiency has been achieved in engineering practice with this technology and the installation is of high quality.

On the Interpretation of Vowel “Quality”: The Dimension of Rounding

1989 ◽  
Vol 19 (1) ◽  
pp. 24-30 ◽  
Author(s):  
Leigh Lisker
Keyword(s):  
Vocal Tract ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Vowel Space ◽  
Tongue Position ◽  
Two Measures ◽  
The One ◽  
Jaw Position ◽  
Three Dimensional Space

The usual description of vowels in respect to their “phonetic quality” requires the linguist to locate them within a so-called “vowel space,” apparently articulatory in nature, and having three dimensions labeled high-low (or close-open), front-back, and unrounded-rounded. The first two are coordinates of tongue with associated jaw position, while the third specifies the posture of the lips. It is recognized that vowels can vary qualitatively in ways that this three-dimensional space does not account for. So, for example, vowels may differ in degree of nasalization, and they may be rhotacized or r-colored. Moreover, it is recognized that while this vowel space serves important functions within the community of linguists, both the two measures of tongue position and the one for the lips inadequately identify those aspects of vocal tract shapes that are primarily responsible for the distinctive phonetic qualities of vowels (Ladefoged 1971). With all this said, it remains true enough that almost any vowel pair of different qualities can be described as occupying different positions with the space. Someone hearing two vowels in sequence and detecting a quality difference will presumably also be able to diagnose the nature of the articulatory shift executed in going from one vowel to the other.

THREE-D SINGLETONS AND 2-D C.F.T.

1992 ◽  
Vol 07 (10) ◽  
pp. 2193-2206 ◽  
Author(s):  
A.M. HARUN AR-RASHID ◽  
C. FRONSDAL ◽  
M. FLATO
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Alternative Formulation ◽  
Boundary Values ◽  
Witten Theory ◽  
Space Time Structure ◽  
Dimensional Space Time ◽  
Left And Right ◽  
Three Dimensional Space

Two-dimensional Wess-Zumino-Novikov-Witten theory is extended to three dimensions, where it becomes a scalar gauge theory of the singleton type. The three-dimensional formulation involves a scalar field valued in a compact group G, a Nakanishi-Lautrup field valued in Lie (G) and Faddeev-Popov ghosts. The physical sector, characterized by the vanishing of the Nakanishi-Lautrup field, coincides with the WZNW theory of the group G. Three-dimensional space-time structure involves a generalized metric, but only its boundary values are of consequence. An alternative formulation in terms of left and right movers (in three dimensions!) is also possible.

A notation for vectors and tensors

1955 ◽  
Vol 51 (3) ◽  
pp. 449-453
Author(s):  
F. C. Powell
Keyword(s):  
Scalar Product ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Usual Form ◽  
Three Dimensional Space ◽  
Scalar Moment

The vector notation commonly employed in elementary physics cannot be applied in its usual form to spaces of other than three dimensions. In plane dynamics, for instance, it cannot be used to represent the velocity (– ωx2, ωx1) at the point (x1, x2) due to a rotation ω about the origin, or the (scalar) moment about the origin of the force (F1, F2) acting at (x1, x2). In relativity physics the symbol ⋅ is often used to denote the scalar product of two vectors, it is true, and the tensor aαbβ – aβbα is sometimes denoted by a × b, but there exists no body of rules for the manipulation of these symbols that enables one to dispense with the suffix notation as in the case of vectors in three-dimensional space.

scholarly journals Imaging mitotic processes in three dimensions with lattice light-sheet microscopy

2021 ◽  
Author(s):  
Yuko Mimori-Kiyosue
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Image Data ◽  
Light Sheet ◽  
Three Dimensions ◽  
Cell Level ◽  
Spatiotemporal Resolution ◽  
New Era ◽  
Light Sheet Microscopy ◽  
Three Dimensional Space

AbstractThere are few technologies that can capture mitotic processes occurring in three-dimensional space with the desired spatiotemporal resolution. Due to such technical limitations, our understanding of mitosis, which has been studied since the early 1880s, is still incomplete with regard to mitotic processes and their regulatory mechanisms at a molecular level. A recently developed high-resolution type of light-sheet microscopy, lattice light-sheet microscopy (LLSM), has achieved unprecedented spatiotemporal resolution scans of intracellular spaces at the whole-cell level. This technology enables experiments that were not possible before (e.g., tracking of growth of every spindle microtubule end and discrimination of individual chromosomes in living cells), thus providing a new avenue for the analysis of mitotic processes. Herein, principles of LLSM technology are introduced, as well as experimental techniques that became possible with LLSM. In addition, issues remaining to be solved for use of this technology in mitosis research, big image data problems, are presented to help guide mitosis research into a new era.

scholarly journals The Third Dimention in a Landscape

Secreta Artis ◽  
2021 ◽  
pp. 74-82
Author(s):  
Daria Vladimirovna Fomicheva
Keyword(s):  
Land Surface ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Landscape Painting ◽  
The Third ◽  
Soviet Art ◽  
Other Hand ◽  
Third Dimension ◽  
Three Dimensional Space

The present study examines the principles of conveying the third dimension in landscape painting. The author analyzes the recommendations provided in J. Littlejohns’ manual entitled “The Composition of a Landscape” [London, 1931]. J. Littlejohns describes four methods of showing depth in a landscape painting, each illustrated with pictorial composition schemes: 1) portrayal of long roads, which allows one to unveil the plasticity of the land surface; 2) creation of a “route” for the viewer by means of a well-thought-out arrangement of natural landforms; 3) introduction of vertically and horizontally flowing streams of water on different picture planes; 4) depiction of cloud shadows on a distinctly hilly landscape. The author of the article compares the schemes contained in the manual of J. Littlejohns with the works of G. G. Nissky, which enables readers to comprehend and reflect on the compositions of the masterpieces created by a prominent figure in Soviet art; on the other hand, Nissky’s landscape paintings open for a deeper understanding of the meaning and effectiveness of the methods proposed by J. Littlejohns. The outlined composition techniques are certainly relevant for contemporary artists (painters, graphic artists, animators, designers, etc.) as they make it possible to achieve the plastic expressiveness of a three-dimensional space in a twodimensional image.

scholarly journals Constructive Geometric Models with Imaginary Objects

2020 ◽  
pp. short27-1-short27-9
Author(s):  
Denis Voloshinov ◽  
Alexandra Solovjeva
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Order ◽  
Second Order ◽  
The Other ◽  
Significant Obstacle ◽  
The Relationship ◽  
Planar Drawing ◽  
Three Dimensional Space

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

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