a cylinder has six plane surfaces.
how many plane surfaces does a cylinder have?
six
the process by which the master plane gages were produced dates back to antiquity but was refined to an unprecedented degree in the maudslay shop. the process begins with three square plates each given an identification (ex., 1,2 and 3). the first step is to rub plates 1 and 2 together with a marking medium (called bluing today) revealing the high spots which would be removed by hand scraping with a steel scraper, until no irregularities were visible. this would not produce true plane surfaces but a "ball and socket" concave-concave and convex-convex fit, as this mechanical fit, like two perfect planes, can slide over each other and reveal no high spots. the rubbing and marking are repeated after rotating 2 relative to 1 by 90 degrees to eliminate concave-convex "potato-chip" curvature. next, plate number 3 is compared and scraped to conform to plate number 1 in the same two trials. in this manner plates number 2 and 3 would be identical. next plates number 2 and 3 would be checked against each other to determine what condition existed, either both plates were "balls" or "sockets" or "chips" or a combination. these would then be scraped until no high spots existed and then compared to plate number 1. repeating this process of comparing and scraping the three plates could produce plane surfaces accurate to within millionths of an inch (the thickness of the marking medium).
machine tool
for surfaces, the generalization from the geometry of a plane – a flat surface – to that of a general curved surface had been described in the early 19th century by carl friedrich gauss. this description had in turn been generalized to higher-dimensional spaces in a mathematical formalism introduced by bernhard riemann in the 1850s. with the help of riemannian geometry, einstein formulated a geometric description of gravity in which minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces. embedding diagrams are used to illustrate curved spacetime in educational contexts.
introduction to general relativity