Results 231 to 240 of about 213,055 (273)
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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 the ...
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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 the ...
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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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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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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
Abstract Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably ...
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Abstract Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably ...
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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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2000
Certainly one of the greatest mathematical discoveries of the nineteenth century was that of non-Euclidean geometry: seen but not revealed by Gauss, and developed in all its glory by Bolyai and Lobachevsky. The purpose of this chapter is to give an account of this theory, but we do not always follow the historical development. Rather, with hindsight we
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Certainly one of the greatest mathematical discoveries of the nineteenth century was that of non-Euclidean geometry: seen but not revealed by Gauss, and developed in all its glory by Bolyai and Lobachevsky. The purpose of this chapter is to give an account of this theory, but we do not always follow the historical development. Rather, with hindsight we
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Supertwisted spirals of layered materials enabled by growth on non-Euclidean surfaces
Science, 2020Yuzhou Zhao +2 more
exaly
Using smartphone photographs of the Moon to acquaint students with non-Euclidean geometry
American Journal of Physics, 2021Rodrigo A Carrasco
exaly