![]() It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. The underlying emphasis throughout this unique, challenging textbook is on how mathematicians think, and how they apply an apparently theoretical system to the solution of real-world problems. Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. As Professor Krause points out, While Euclidean geometry appears to be a good model of the 'natural' world, taxicab geometry is a better model of the artificial urban world that man has built.Īs a result, the book is replete with practical applications of this non-Euclidean system to urban geometry and urban planning - from deciding the optimum location for a factory or a phone booth, to determining the most efficient routes for a mass transit system. However, taxicab geometry has important practical applications. On the way to this spurious demonstration, Saccheri established several theorems of non-Euclidean geometryfor example, that according to whether the right, obtuse, or acute hypothesis is true, the sum of the angles of a triangle respectively equals, exceeds, or falls short of 180. Because the surface is curved, there are no straight lines in the traditional sense, but these distance minimizing curves known as geodesics will play the role of straight lines in these new geometries. Because of this non-Euclidean method of measuring distance, some familiar geometric figures are transmitted: for example, circles become squares. In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180. Non-Euclidean geometry is the study of geometry on surfaces which are not flat. Distance is not measured as the crow flies, but as a taxicab travels the grid of the city street, from block to block, vertically and horizontally, until the destination is reached. ![]() In taxicab geometry, the shortest distance between two points is not a straight line. This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry - undergraduate math students, advanced high school students, and puzzle fans of any age - an opportunity to explore taxicab geometry, a simple, non-Euclidean system that helps put Euclidean geometry in sharper perspective. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |