Complexes of Conics and the Weddle Surface

1924 ◽  
Vol 22 (3) ◽  
pp. 201-216
Author(s):  
C. G. F. James
Keyword(s):  
Arbitrary Point ◽  
Three Dimensions ◽  
Irreducible Curve ◽  
Quartic Surface ◽  
Ordinary Space ◽  
Four Dimensions ◽  
Interesting Connection ◽  
Linear Complexes ◽  
Pass Through ◽  
Do So

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.

Characteristics of complexes of conics in space of four dimensions

1925 ◽  
Vol 22 (5) ◽  
pp. 621-629
Author(s):  
C. G. F. James
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Singular System ◽  
Arbitrary Point ◽  
Singular Surface ◽  
Singular Curve ◽  
Ordinary Space ◽  
Four Dimensions ◽  
Linear Complexes ◽  
Pass Through

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

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.

Multiple tangents to forms in four dimensions

1930 ◽  
Vol 26 (3) ◽  
pp. 285-304
Author(s):  
L. Roth
Keyword(s):  
Present Method ◽  
Three Dimensions ◽  
Simple Manner ◽  
Mechanical Process ◽  
Ordinary Space ◽  
Four Dimensions ◽  
General Surface ◽  
Complete Set ◽  
Correspondence Theorem ◽  
Single Contact

Formulae for multiple tangents to the general surface in ordinary space were obtained at different times by various writers, but the discussion remained incomplete until the advent of Schubert's enumerative method, which solves the whole problem by a purely mechanical process. Schubert later extended the method to general forms in [n], in the restricted case where the multiple tangents have only a single contact. In the present paper the results for three dimensions have been used to build up the formulae for forms in [4], by means of the correspondence theorem on which much of Schubert's work is based. It would no doubt be possible to evolve a complete set of incidence formulae for four dimensions and then to proceed as in Schubert's discussion of surfaces; but the present method is preferable for two reasons. In the first place, all the results have been obtained in a very simple manner; and secondly, a large number of minor results have been found in the process.

Configurations defined by six lines in space of three dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 52-68 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Linear System ◽  
Classical Problem ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Lines In Space ◽  
Pass Through ◽  
Quartic Surfaces

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.

scholarly journals On the Stereometric Generation of the De Jonquières Transformation

1918 ◽  
Vol 37 ◽  
pp. 48-58
Author(s):  
J. F. Tinto
Keyword(s):  
Three Dimensions ◽  
Plane Figure ◽  
Cremona Transformation ◽  
Quartic Surface ◽  
Cremona Transformations ◽  
Intrinsic Interest ◽  
Four Dimensions ◽  
Single Instance

In the geometry of the plane the logical interrelations of figures may often be rendered clearer by considering the plane to be a part of space of three dimensions. Thus, by taking the plane figure as part of a more extensive configuration in space of three dimensions, the elucidation of its properties, and in particular its relation with other figures, are often facilitated. Similarly, the figures of space of three dimensions can sometimes be treated more advantageously and compendiously by considering them as parts of figures in a space of four dimensions, and so on. As a single instance we may take Segre's elegant and powerful mode of treatment of the quartic surface which possesses a nodal conic. This surface he obtains as a projection in space of four dimensions of the quartic surface which constitutes the base of a pencil of quadratic varieties. In the following paper this mode of treatment has been applied to the interesting variety of the Cremona transformation in the plane known as the De Jonquieres transformation, a transformation which possesses some intrinsic interest in view of the fundamental rôle which it plays in the theory of Cremona Transformations. By the aid of a surface in space of three dimensions, a variety in space of four dimensions, etc., simple constructions are given for the De Jonquières transformation between two planes, between two spaces of three dimensions, etc., respectively.

scholarly journals LARGE QUANTUM GRAVITY EFFECTS: CYLINDRICAL WAVES IN FOUR DIMENSIONS

2000 ◽  
Vol 09 (06) ◽  
pp. 669-686 ◽  
Author(s):  
MARÍA E. ANGULO ◽  
GUILLERMO A. MENA MARUGÁN
Keyword(s):  
Quantum Gravity ◽  
Symmetry Axis ◽  
Point Of View ◽  
Three Dimensions ◽  
Asymptotic Region ◽  
Four Dimensions ◽  
Cylindrical Waves ◽  
Significant Difference ◽  
Gravity Effects ◽  
Linearly Polarized

Linearly polarized cylindrical waves in four-dimensional vacuum gravity are mathematically equivalent to rotationally symmetric gravity coupled to a Maxwell (or Klein–Gordon) field in three dimensions. The quantization of this latter system was performed by Ashtekar and Pierri in a recent work. Employing that quantization, we obtain here a complete quantum theory which describes the four-dimensional geometry of the Einstein–Rosen waves. In particular, we construct regularized operators to represent the metric. It is shown that the results achieved by Ashtekar about the existence of important quantum gravity effects in the Einstein–Maxwell system at large distances from the symmetry axis continue to be valid from a four-dimensional point of view. The only significant difference is that, in order to admit an approximate classical description in the asymptotic region, states that are coherent in the Maxwell field need not contain a large number of photons anymore. We also analyze the metric fluctuations on the symmetry axis and argue that they are generally relevant for all of the coherent states.

On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

1924 ◽  
Vol 22 (2) ◽  
pp. 189-199
Author(s):  
F. Bath
Keyword(s):  
Three Dimensions ◽  
Apparent Contradiction ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Lines In Space ◽  
Quadric Threefold ◽  
Quartic Curves

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

scholarly journals Nervous Activity of the Brain in Five Dimensions

Biophysica ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 38-47
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
Norbert Jausovec ◽  
Arjuna P. H. Don ◽  
Sheela Ramanna ◽  
...  
Keyword(s):  
Fourier Analysis ◽  
Nervous Activity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Data Sets ◽  
Spatial And Temporal Patterns ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Scale Free ◽  
The Brain

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

scholarly journals How Kindergarten Teachers Assess Their Own Professional Competencies

2021 ◽  
Vol 11 (12) ◽  
pp. 769
Author(s):  
Eva Pupíková ◽  
Dalibor Gonda ◽  
Kitti Páleníková ◽  
Janka Medová ◽  
Dana Kolárová ◽  
...  
Keyword(s):  
Classroom Management ◽  
Content Knowledge ◽  
Educational Needs ◽  
Kindergarten Teachers ◽  
Further Education ◽  
Three Dimensions ◽  
Self Assessment ◽  
Teaching Competencies ◽  
Four Dimensions

One of the requirements of Education 4.0 is that students and practitioners should be involved in the creation of the content of study plans. Therefore, in the present research we focused on identifying the further educational needs of kindergarten teachers. Teachers’ educational needs were divided into four dimensions: ‘content knowledge’, ‘diagnostic knowledge’, ‘didactical knowledge’, and ‘classroom management knowledge’. In parallel, we discovered how teachers assess the level of their own teaching competencies. Based on the obtained data, we identified that teachers have the greatest need for further education in the dimension of ‘diagnostic knowledge’ and that the need for their further education in this dimension did not depend on the length of practice. In the other three dimensions, a declining trend in teachers’ educational needs has been recorded with an increasing length of practice, declining significantly in three of the four dimensions examined. The study points to the need to create in-service courses for kindergarten teachers to deepen their ‘diagnostic knowledge’ and thus ensure the sustainability of the quality of pre-school education for children. Teachers‘ self-assessment of their own teaching competencies corresponds to their educational needs, which supports the relevance of the findings on the further educational needs of kindergarten teachers. This study aimed to obtain relevant data on which the improvement of the higher education of future kindergarten teachers might be based. At the same time, this would allow the analysis and tailoring of the content of professional development courses to the needs of in-service kindergarten teachers.

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