A numerical disproof of a conjecture in projective geometry

1971 ◽  
Vol 323 (1555) ◽  
pp. 443-454 ◽  
Keyword(s):  
Projective Geometry ◽  
Three Dimensions ◽  
Lines In Space ◽  
Straight Lines

A longstanding conjecture concerning the intersection of straight lines in space of three dimensions is disproved by a counterexample. At the same time a related conjecture due to Babbage is confirmed.

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.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

scholarly journals A general vector analysis, with applications to electrodynamical theory

1925 ◽  
Vol 108 (747) ◽  
pp. 418-455 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Cartesian Coordinates ◽  
Straight Line ◽  
Straight Lines ◽  
General Vector

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

scholarly journals Determination of the characteristic parameters in the special collinear space in the general case

2007 ◽  
Vol 5 (1) ◽  
pp. 49-59
Author(s):  
Sonja Krasic ◽  
Miroslav Markovic
Keyword(s):  
Projective Space ◽  
Projective Geometry ◽  
Descriptive Geometry ◽  
Characteristic Parameters ◽  
Straight Lines ◽  
Structural Methods

The projective space consists of the finitely and infinitely distant elements. The special collinear spaces in the general case, are set with five pairs of biunivocally associated points, so the quadrangle in the first space obtained by the three principal and one penetration point of the remaining two through the plane of the first three identical or similar to the associated quadrangle obtained in the same way in the second space. In order to associate two special collinear spaces, it is necessary to determine the following characteristic parameters: vanishing planes, space axes (principal normal lines), foci (apexes of the associated identical bundles of straight lines) and directrix plane (associated identical fields of points). The paper is based on constructive processing of the special collinear spaces in the general case. The structural methods which are used are Descriptive Geometry (a pair of Mange's projections) and Projective geometry.

scholarly journals Quasi-Empirical Fictionalism as an Approach to the Philosophy of Geometry

2021 ◽  
Author(s):  
◽  
Scott Waygood
Keyword(s):  
Projective Geometry ◽  
Predictive Power ◽  
Philosophy Of Mathematics ◽  
Line Drawing ◽  
Physical Reality ◽  
Empirical Science ◽  
Primitive Operation ◽  
Straight Lines ◽  
Final Question ◽  
Central Claim

<p>The central claim of this thesis is that geometry is a quasi-empirical science based on the idealisation of the elementary physical operations that we actually perform with pen and paper. This conclusion is arrived at after searching for a theory of geometry that will not only explain the epistemology and ontology of mathematics, but will also fit with the best practices of working mathematicians and, more importantly, explain why geometry gives us knowledge that is relevant to physical reality. We will be considering all the major schools of thought in the philosophy of mathematics. Firstly, from the epistemological side, we will consider apriorism, empiricism and quasi-empiricism, finding a Kitcherian style of quasi-empiricism to be the most attractive. Then, from the ontological side, we will consider Platonism, formalism, Kitcherian ontology, and fictionalism. Our conclusion will be to take a Kitcherian epistemology and a fictionalist ontology. This will give us a kind of quasiempirical-fictionalist approach to mathematics. The key feature of Kitcher's thesis is that he placed importance on the operations rather than the entities of arithmetic. However, because he only dealt with arithmetic, we are left with the task of developing a theory of geometry along Kitcherian lines. I will present a theory of geometry that parallels Kitcher's theory of arithmetic using the drawing of straight lines as the most primitive operation. We will thereby develop a theory of geometry that is founded upon our operations of drawing lines. Because this theory is based on our line drawing operations carried out in physical reality, and is the idealisation of those activities, we will have a connection between mathematical geometry and physical reality that explains the predictive power of geometry in the real world. Where Kitcher uses the Peano postulates to develop his theory of arithmetic, I will use the postulates of projective geometry to form the foundations of operational geometry. The reason for choosing projective geometry is due to the fact that by taking it as the foundation, we may apply Klein's Erlanger programme and build a theory of geometry that encompasses Euclidean, hyperbolic and elliptic geometries. The final question we will consider is the problem of conventionalism. We will discover that investigations into conventionalism give us further reason to accept the Kitcherian quasi-empirical-fictionalist approach as the most appealing philosophy of geometry available.</p>

scholarly journals Classification of Geometries with Projective Metric

1909 ◽  
Vol 28 ◽  
pp. 25-41 ◽  
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Dihedral Angle ◽  
Plane Geometry ◽  
Three Dimensions ◽  
Angular Measurement ◽  
Plane Angle ◽  
Degenerate Form ◽  
Projective Metric ◽  
The Absolute ◽  
Straight Lines

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

scholarly journals A generalized vector analysis of four dimensions

1923 ◽  
Vol 103 (723) ◽  
pp. 644-663 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Four Dimensions ◽  
Straight Line ◽  
Straight Lines

The vector analysis in use up to the present, as a rule, are concerned with quantities which are represented by straight lines. The straight line AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian co-ordinates ( x, y, z ) with respect to A, we write a = i x + j y + k z , where i, j, k are fundamental vectors. An account will be given of a vector analysis in which the vector is represented by δa = Ʃ n i n δ x n .

scholarly journals XI. On a graphical representation of the twenty-seven lines on a cubic surface

1894 ◽  
Vol 54 (326-330) ◽  
pp. 148-149
Keyword(s):  
Graphical Representation ◽  
Three Dimensions ◽  
Cubic Surface ◽  
Straight Lines

The converse of Pascal’s well-known theorem may be stated thus: if two triangles be in perspective, their non-corresponding sides intersect in six points lying on a conic. An extension of this theorem to three dimensions may be stated thus: if two tetrahedrons be in perspective, their non-corresponding faces intersect in twelve straight lines lying on a cubic surface. This theorem may be deduced from the equation xyzu = ( x + a T) ( y + b T) ( z + c T) ( u + d T), where T = αx + βy + γz + δu ; and a, b, c, d, α, β, γ ,δ are constants. The equations of twelve lines on the surface are evident.

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.

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