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Cayley–Klein geometries and projective-metric geometry
Cayley and Klein discovered in the 19th century that Euclidean and non-Euclidean geometries can be introduced as geometries living inside of a...
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Cayley--Klein Geometries
In Chap. 12 we introduced the Poincaré half-plane as a model of the plane hyperbolic geometry. It is...
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Cayley-Klein Spaces
In Klein’s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for,...
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Special Cases of Hyperbolic Parallelograms on the Lobachevsky Plane
In this paper, we consider particular cases of hyperbolic parallelograms obtained by transferring characteristic properties of rectangles and squares...
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Central Projection of Quadrics and Möbius Geometry
In this section we study the general construction of central projection of a quadric from a point onto its polar hyperplane, see, e.g., [Kle1928,...
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Burmester theory in Cayley–Klein planes with affine base
In this paper, we study the Burmester theory in Euclidean, Galilean and pseudo-Euclidean planes and extend the classical Burmester theory to the...
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Elementary Plane Geometry
The present subject matter is usually part of a separate lecture course. Early exposure to it is desirable because it is complementary to the rest of...
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Diameter, width and thickness in the hyperbolic plane
In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and...
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Absolute isotropic geometry
The term ‘absolute geometry’ was coined by János Bolyai to characterize the part of Euclidean geometry that does not depend on the parallel...
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Description of Rotation of a Rigid Body About a Fixed Point
Although in motion of a rigid body about a fixed point, we deal with continuous change of position, i.e. with a sequence of infinitesimal rotations,...
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Minkowski: The Universe Is a 4-Dimensional Manifold
In 1908 in a lecture Minkowski gave a brilliant reformulation of Einstein’s theory. He views the World as a 4-dimensional manifold that can be...
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Thought Experiments in Mathematics: From Fiction to Facts
As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice,...
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The geometry of discrete asymptotic-geodesic 4-webs in isotropic 3-space
The geometry of webs has been investigated over more than a century driven by still open problems. In our paper we contribute to extending the...
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Interpretation of Geometry on Manifolds as a Geometry in a Space with Projective Metric
In this paper, we give essential concepts of geometry of three-dimensional spaces in vector formulation in an affine vector space A n .
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Felix Klein’s early contributions to anschauliche Geometrie
Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of...
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Thought Experiments in Mathematics: From Fiction to Facts
As in science and philosophy, thought experiments in mathematics link a problem to new epistemic resources that are unavailable in a given practice,...
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Catenaries and Singular Minimal Surfaces in the Simply Isotropic Space
This paper investigates the hanging chain problem in the simply isotropic plane and its 2-dimensional analog in the simply isotropic space. The...
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Overview
For over 170 years, the connection between group theory and the fields of geometry and topology has held central in mathematics—the connection is...
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Lobachevsky Geometry and Stellar Parallaxes
Using Beltrami–Poincaré models in the Euclidean semiplane and semispace for two-dimensional and three-dimensional Lobachevsky spaces, we give...
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Abstract Algebra and Number Theory
The word algebra takes on a very different meaning from how it is used in school. Algebra means much more than the rules of shuffling around symbols...