12 1. Euclid Perhaps the most convincing confirmation that Euclid’s is not the only possible con- sistent theory of geometry came from Einstein’s general theory of relativity around the turn of the twentieth century. If we are to believe, like Euclid, that the postulates reflect self- evident truths about the geometry of the world we live in, then Euclid’s statements about “straight lines” should translate into true statements about the behavior of light rays in the real world. (After all, we commonly judge the “straightness” of something by sighting along it, so what physical phenomenon could possibly qualify as a better model of “straight lines” than light rays?) Thus the closest thing in the physical world to a geometric triangle would be a three-sided figure whose sides are formed by the paths of light rays. Yet Einstein’s theory tells us that in the presence of gravitational fields, space itself is “warped,” and this affects the paths along which light rays travel. One of the most dramatic confirmations of Einstein’s theory comes from the phenomenon known as gravitational lensing: this occurs when we observe a distant object but there is a massive galaxy cluster directly between us and the object. Einstein’s theory predicts that the light rays from the distant object should be able to follow two (or more) different paths to reach our eyes because of the distortion of space around the galaxy cluster. Fig. 1.4. A gravitational lens (photograph by W. N. Colley, E. Turner, J. A. Tyson, and NASA). This phenomenon has indeed been observed Fig. 1.4 shows a photograph taken by the Hubble Space Telescope, in which a distant loop-shaped galaxy (circled) appears twice in the same photographic image because its light rays have traveled around both sides of the large galaxy cluster in the middle of the photo. Fig. 1.5 shows a schematic view of the same

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