1||0|794|0|
0|0|0|||||Possible mistake in %22What Is Geometry%22|David|1|07:00:02|09/25/2008|I was just reading through the %22What Is Geometry%3F%22 article %28http%3A%2F%2Fwww.cut-the-knot.org%2FWhatIs%2FWhatIsGeometry.shtml%29. The final paragraph of that article says%3A%0D%0A%0D%0A%22There are many geometries. All of these share some basic elements and properties. Even finite geometries deal with points and lines and universally just a single line may pass through two given points. Thus I believe that a frequently used term %22Taxicab Geometry%22 is a misnomer...%22%0D%0A%0D%0AIf I understand correctly%2C this criterion is violated by spherical geometry%2C where some pairs of points are joined by an infinite number of lines %28as with the meridians all passing through the poles on the earth%27s surface%29.%0D%0A%0D%0AThe comment about %27taxicab geometry%27 might be right%2C but maybe for a different reason.%0D%0A%0D%0A%2A%2A%2A%2A%2A%0D%0AI vaguely remember reading somewhere that a geometry must satisfy%3A%0D%0Adr%5E2 %3D dx%5E2 %2B dy%5E2 %2B dz%5E2 ...and so on for higher dimensions%0D%0A%0D%0AThat definition of a %27geometry%27 would include spherical geometry and exclude taxicab%2C but I don%27t know what other interesting %27geometries%27 it might exclude.%0D%0A%0D%0A%2A%2A%2A%2A%2A%0D%0ABy the way%2C that was an excellent article. I keep forgetting about this site and then stumbling on it again in web searches. Keep up the good work. %3A%29
1|1|0|||||RE%3A Possible mistake in %22What Is Geometry%22|alexb||09:05:19|09/25/2008|%3EIf I understand correctly%2C this criterion is violated by %0D%0A%3Espherical geometry%2C where some pairs of points are joined by %0D%0A%3Ean infinite number of lines %28as with the meridians all %0D%0A%3Epassing through the poles on the earth%27s surface%29. %0D%0A%0D%0AIt%27s not just that there is an infinity of lines through two antipodes. Any line passing through one also passes through the other. In this sense%2C the points are indistinguishable. When it comes to defining elliptic geometry%2C the antipodes are identified and are justly considered as a single point.%0D%0A%0D%0AAs opposed to elliptic geometry%2C modelled on a sphere%2C  in the commonplace spherical geometry%2C considerations are usually carried out %22in the small%2C%22 i.e.%2C in the regions that do not include antipodes.%0D%0A%0D%0AA remark to this effect is certainly in order.%0D%0A%0D%0A%3EI vaguely remember reading somewhere that a geometry must %0D%0A%3Esatisfy%3A %0D%0A%3Edr%5E2 %3D dx%5E2 %2B dy%5E2 %2B dz%5E2 ...and so on for higher dimensions %0D%0A%0D%0AThere of course no unique and universally accepted definition of geometry. It%27s rather a field of study that admits a branching%2C multi-level classification. Yours is probably more relevant to the idea of manifolds and differential geometry.%0D%0A%0D%0AI would rather admit finite geometries under the same umbrella than make differential geometry into the root of classification.%0D%0A%0D%0A%3EBy the way%2C that was an excellent article. I keep forgetting %0D%0A%3Eabout this site and then stumbling on it again in web %0D%0A%3Esearches. %0D%0A%0D%0AThank you for the kind words.%0D%0A