On the quasi non-euclidean geometry of the absolute of arbitrary real order

1957 ◽  
Vol 6 (3) ◽  
pp. 334-342
Author(s):  
Hiroyoshi Sasayama
Keyword(s):  
Euclidean Geometry ◽  
Non Euclidean Geometry ◽  
The Absolute

Graphical Proof of the Main Theorem of Non-Euclidean Geometry

2015 ◽  
Vol 3 (3) ◽  
pp. 18-23
Author(s):  
Сафиулина ◽  
Yu. Safiulina ◽  
Шмурнов ◽  
V. Shmurnov
Keyword(s):  
Euclidean Geometry ◽  
Civil Process ◽  
Parabolic Geometry ◽  
Non Euclidean Geometry ◽  
Scientific World ◽  
Anisotropy Index ◽  
The Absolute ◽  
And Mathematics ◽  
Substantive Law ◽  
Special Case

The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recognized as obvious, or recognized without proof (postulates). Yet the founders of non-Euclidean geometry could not imagine such traditional evidence at all desire, because it had not yet been developed, and most importantly respective starting points (axioms, postulates, and theorems) had not been recognized by mathematicians. The paper outlines the original concept of non-Euclidean geometries. Hyperbolic geometry of Lobachevsky is considered based on viewing the sphere as a surface of zero curvature. In this case, the plane will have a real curvature properties of hyperboloid or a pseudosphere depending on the absolute and space anisotropy index, which replaces the concept of curvature of space; i.e. the notion of the curvature of the surface is converted to purely analytical attributes. Parabolic geometry of Euclid with degenerate absolute becomes a special case of geometries with non-degenerate absolute. The geometry of Riemann having the absolute of imaginary surface with negative Gaussian curvature at all points is declared not real but imaginary, since, according to the authors, it is impossible for plotting. References to textbooks of mechanics and mathematics departments of universities.

scholarly journals Elementary Considerations relating to the Absolute

1909 ◽  
Vol 28 ◽  
pp. 65-72
Author(s):  
Duncan M.Y. Sommerville
Keyword(s):  
Fixed Point ◽  
Special Interest ◽  
Euclidean Geometry ◽  
Criterion A ◽  
Parallel Lines ◽  
Non Euclidean Geometry ◽  
The Absolute ◽  
The Given

Non-Euclidean geometry in the narrowest sense is that system of geometry which is usually associated with the names of Lobachevskij and Bolyai, and which arose from the substitution for Euclid's parallel-postulate of a postulate admitting an infinity of lines through a fixed point not intersecting a given line, the two limits between the intersectors and the non-intersectors being called the parallels to the given line through the fixed point. In a wider sense, any system of geometry which denies one or more of the fundamental assumptions upon which Euclid's system is based is a non-euclidean geometry. Of special interest are, however, those which touch only the question of parallel lines ; and there exists, in addition to Lobachevskij's geometry, another, commonly associated with the name of Riemann, in which the parallels to any line through a fixed point are imaginary. The three geometries, Lobachevskij's, Euclid's, and Riemann's, thus form a trio characterised by the existence of real, coincident, or imaginary pairs of parallels through a given point to a given line. With reference to this criterion, a consistent nomenclature was introduced by Klein, who called these three geometries respectively Hyperbolic, Parabolic, and Elliptic.

scholarly journals “I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Peter Ullrich
Keyword(s):  
Present Article ◽  
19Th Century ◽  
Euclidean Geometry ◽  
Academic Life ◽  
David Hilbert ◽  
Non Euclidean Geometry ◽  
The 19Th Century

AbstractFriedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

scholarly journals Langevin simulations of protoplasmic streaming in non-Euclidean geometry

2021 ◽  
Vol 1730 (1) ◽  
pp. 012037
Author(s):  
Shuta Noro ◽  
Masahiko Okumura ◽  
Satoshi Hongo ◽  
Shinichiro Nagahiro ◽  
Toshiyuki Ikai ◽  
...  
Keyword(s):  
Euclidean Geometry ◽  
Protoplasmic Streaming ◽  
Non Euclidean Geometry

Practical Non-Euclidean Geometry

1925 ◽  
Vol 12 (177) ◽  
pp. 422 ◽  
Author(s):  
T. C. J. Elliott
Keyword(s):  
Euclidean Geometry ◽  
Non Euclidean Geometry

Anti-Geometry and NeutroGeometry Characterization of Non-Euclidean Data

2021 ◽  
pp. 24-33
Author(s):  
Prem Kumar Singh ◽  
Keyword(s):  
Euclidean Geometry ◽  
Current Paper ◽  
Data Sets ◽  
Knowledge Processing ◽  
Non Euclidean Geometry ◽  
Precise Representation

Recently, a problem is addressed while dealing with fourth dimensional or non-Euclidean data sets. These are the data sets does not follow one of the postulates established by Euclid specially the parallel postulates. In this case, the precise representation of these data sets is major issues for knowledge processing tasks. Hence, the current paper tried to introduce some non-Euclidean geometry or Anti-Geometry methods and its examples for various applications.

scholarly journals The Non-Euclidean Geometry Inevitable

The Monist ◽  
1894 ◽  
Vol 4 (4) ◽  
pp. 483-493
Author(s):  
George Bruce Halsted ◽  
Keyword(s):  
Euclidean Geometry ◽  
Non Euclidean Geometry

scholarly journals After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Janet Folina
Keyword(s):  
19Th Century ◽  
Euclidean Geometry ◽  
Symbolic Logic ◽  
Non Euclidean Geometry ◽  
The 19Th Century ◽  
Spatial Intuition ◽  
Mathematical Inference

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics.

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