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Geometry: A Very Short Introduction最新文献

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5. Projective geometry 5. 射影几何
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0005
M. Dunajski
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引用次数: 0
6. Other geometries 6. 其他几何图形
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0006
M. Dunajski
{"title":"6. Other geometries","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0006","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0006","url":null,"abstract":"‘Other geometries’ offers an overview of some modern geometries and their links with other areas of mathematics, such as Gauss lemma in number theory and Gaussian distributions in statistics. The Atiyah–Singer index theorem, proved in 1963 by Michael Atiyah and Isadore Singer, is a celebrated result of geometry in the last 60 years. Felix Klein had the idea of defining geometry as a pair consisting of a space a two-dimensional plane and a group of all affine transformations of the plane. Grigori Perelman’s proof showed the Poincare conjecture, which was formulated in the early 20th century by French mathematician Henri Poincaré.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"110 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121121361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
4. Geometry of curved spaces 4. 弯曲空间几何
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0004
M. Dunajski
{"title":"4. Geometry of curved spaces","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0004","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0004","url":null,"abstract":"‘Geometry of curved spaces’ shows that the curvature of a curve is an extrinsic property that is visible to people when it is viewed sitting on the plane and can be defined as a curve on the Euclidean plane. German mathematician Carl Friedrich Gauss discovered that curvature is an intrinsic property of the surface in 1828. Gauss called it Theorema Egregium, which translates from Latin as Remarkable Theorem. The extrinsic curvature of curves, defines Gaussian curvature can be computed intrinsically. Higher-dimensional generalizations of surfaces are called manifolds.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127554991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
1. What is geometry? 1. 几何是什么?
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0001
M. Dunajski
{"title":"1. What is geometry?","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0001","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0001","url":null,"abstract":"‘What is geometry?’ mentions the Greek philosopher Pythagoras of Samos and his followers, the Pythagoreans, who spent their time unveiling the relationship between numbers and geometric forms. They were credited for what is now known as the Pythagorean theorem, wherein for any right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. Geometry stands out from most other branches of mathematics, as the proof of a theorem can be given in pictorial terms. The Pythagorean theorem is valid in Euclidean geometry and relies on concepts that play a central role in the theorem.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126593737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
3. Non-Euclidean geometry 3.非欧几里得的几何学
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0003
M. Dunajski
{"title":"3. Non-Euclidean geometry","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0003","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0003","url":null,"abstract":"‘Non-Euclidean geometry’ begins with a discussion on spherical geometry, which is the study of objects on the sphere and has lines that are defined as great circles. Spherical geometry is an example of a non-Euclidean geometry, as the lines do not satisfy Euclid’s parallel postulate. Hyperbolic geometry is another example of a non-Euclidean geometry, as it violates the parallel axiom and cannot be embedded in ordinary space. Hyperbolic geometry can be introduced as an abstract surface wherein lines are singled out and the distance which makes these lines the shortest are shown. With hyperbolic geometry, the apparent paradoxes of M. C. Escher’s angels and devils can be revealed.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123480167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
7. Geometry of the physical world 7. 物理世界的几何学
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0007
M. Dunajski
{"title":"7. Geometry of the physical world","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0007","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0007","url":null,"abstract":"‘Geometry of the physical world’ reviews Euclidean geometry, which provided a framework for interpreting empirical observations and making accurate predictions for over two millennia. However, as measuring apparatus improved, the geometry of the Universe is not that of Euclidean space. There were a number of 20th-century theories of physics which are based on geometries different to Euclidean geometry. These describe the Universe more accurately. The most basic of these is Minkowski geometry, which replaces Pythagorean distance by a space-time interval. The theory of gravitation is based on a curved version of the Minkowski distance and the electromagnetic and strong nuclear interactions are manifestations of curvature.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116977339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
2. Euclidean geometry 2. 欧几里德几何
Geometry: A Very Short Introduction Pub Date : 2022-01-27 DOI: 10.1093/actrade/9780199683680.003.0002
M. Dunajski
{"title":"2. Euclidean geometry","authors":"M. Dunajski","doi":"10.1093/actrade/9780199683680.003.0002","DOIUrl":"https://doi.org/10.1093/actrade/9780199683680.003.0002","url":null,"abstract":"‘Euclidean geometry’ talks about the cultures that grew up in the arid region of Mesopotamia in the fourth millennium BC that needed to find geometrical solutions to their problems. Dividing and surveying the land after periodic floods relied on measuring distances and computing areas. Euclidean geometry forms geometric intuition that gives an accurate description of the space of land. Euclid of Alexandria put geometry in a logical framework and used axioms to define complicated objects such as triangles. The theorems are logical consequences of axioms that take a long time to prove.","PeriodicalId":420147,"journal":{"name":"Geometry: A Very Short Introduction","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115613707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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