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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
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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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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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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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2020
One of the new frontiers in geometry opened up by calculus was the study of curvature. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding ...
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One of the new frontiers in geometry opened up by calculus was the study of curvature. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding ...
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1990
Abstract Amidst all the complex technical creations of the nineteenth century the most profound one, non-Euclidean geometry, was technically the simplest. This creation gave rise to important new branches of mathematics but its most significant implication is that it obliged mathematicians to revise radically their understanding of ...
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Abstract Amidst all the complex technical creations of the nineteenth century the most profound one, non-Euclidean geometry, was technically the simplest. This creation gave rise to important new branches of mathematics but its most significant implication is that it obliged mathematicians to revise radically their understanding of ...
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1981
In Chapter 3 we saw that the group G which we have been discussing is formed from the transformation $$ y = \frac{{xT + xx'{v_1} + {v_2}}}{{xu{'_2} + xx'b + d}} $$ (1) (And at the same time we have $$ yy'\left( {\frac{{xu{'_1} + xx'a + c}}{{xu{'_2} + xx'b + d}}} \right).) $$ (2) Observe that the matrix $$ M = \left( {\begin ...
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In Chapter 3 we saw that the group G which we have been discussing is formed from the transformation $$ y = \frac{{xT + xx'{v_1} + {v_2}}}{{xu{'_2} + xx'b + d}} $$ (1) (And at the same time we have $$ yy'\left( {\frac{{xu{'_1} + xx'a + c}}{{xu{'_2} + xx'b + d}}} \right).) $$ (2) Observe that the matrix $$ M = \left( {\begin ...
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Using smartphone photographs of the Moon to acquaint students with non-Euclidean geometry
American Journal of Physics, 2021Rodrigo A Carrasco +2 more
exaly