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, 1965 This textbook introduces non-Euclidean geometry, and the third edition adds a new chapter, including a description of the two families of 'mid-lines' between two given lines and an elementary derivation of the basic formulae of spherical trigonometry and
Coxeter, HSM, E. A. Maxwell
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IS SPACE FLAT? NINETEENTH-CENTURY ASTRONOMY AND NON-EUCLIDEAN GEOMETRY
Journal of Astronomical History and Heritage, 2012The geometrical structure of space entered astronomy in the second half of the nineteenth century, but slowly and hesitantly. Although in this period non-Euclidean geometry became a very important branch of mathe- matics, it aroused little interest among
H. Kragh
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Position: Beyond Euclidean - Foundation Models Should Embrace Non-Euclidean Geometries
arXiv.orgIn the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental limitations.
Neil He +8 more
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1997
Abstract This chapter focuses on the connections between non-Euclidean geometries and the complex numbers. Euclid began with just five axioms, the first four of which never aroused controversy. However, the status of the fifth axiom (the so-called parallel axiom) was less clear, and it became the subject of investigations that ultimately
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Abstract This chapter focuses on the connections between non-Euclidean geometries and the complex numbers. Euclid began with just five axioms, the first four of which never aroused controversy. However, the status of the fifth axiom (the so-called parallel axiom) was less clear, and it became the subject of investigations that ultimately
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The Mathematics Teacher, 1922
About 2200 years ago there was published in Greek one of the most remarkable books of all times, Euclid's “Elements of Geometry”. It contains a systematic exposition of the leading propositions of elementary geometry and the elementary theory of numbers. It was at once adopted by the Greeks as the standard text book on pure mathematics.
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About 2200 years ago there was published in Greek one of the most remarkable books of all times, Euclid's “Elements of Geometry”. It contains a systematic exposition of the leading propositions of elementary geometry and the elementary theory of numbers. It was at once adopted by the Greeks as the standard text book on pure mathematics.
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Euclidean and Non-Euclidean Geometry
Visual Differential Geometry and Forms, 2021semanticscholar +1 more source
1984
It is unlikely that Euclid ever held his five postulates to be self-evident. Mathematicians sharing the Aristotelian conviction that only manifest truths may be admitted without proof in geometry usually did not find the fifth postulate quite so obvious as the other four.
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It is unlikely that Euclid ever held his five postulates to be self-evident. Mathematicians sharing the Aristotelian conviction that only manifest truths may be admitted without proof in geometry usually did not find the fifth postulate quite so obvious as the other four.
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2011
In this chapter we discuss the non-Euclidean geometry of curved surfaces, using the sphere as our primary example. We find that all the information about the geometry of the surface is contained in the expression for the distance between two nearby points in some coordinate system, called the metric. For example, the distance between two distant points
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In this chapter we discuss the non-Euclidean geometry of curved surfaces, using the sphere as our primary example. We find that all the information about the geometry of the surface is contained in the expression for the distance between two nearby points in some coordinate system, called the metric. For example, the distance between two distant points
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2014
The fruitless attempts to prove Euclid’s parallel postulate, in particular the theory of limit parallels, lead eventually the mathematicians of the nineteenth century to consider that the negation of this postulate could possibly be taken as an axiom.
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The fruitless attempts to prove Euclid’s parallel postulate, in particular the theory of limit parallels, lead eventually the mathematicians of the nineteenth century to consider that the negation of this postulate could possibly be taken as an axiom.
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