Have just been reading Roger Penrose's big fat ouvre The Road to Reality, and thought I should post about something I have learnt.
Here is a picture called "Circle Limit IV" by M.C. Escher which very beautifully describes a certain kind of geometry that is alien to us regular folk, but one that appears completely naturally in the known universe - the so-called hyperbolic geometry. This "representation" by Escher explains how this universe behaves - if it were compressed inside a circle.The idea is, as an object tries to get closer and closer to the boundary circle, it gets smaller and smaller in the usual sense of the word in such a way that its size at any point in time completely prohibits it from ever reaching the circle (Think: Zeno's paradox, but this time it's for real!). This was exactly the idea that Einstein used to describe the behaviour of bodies that are going really fast - the speed of light behaves like an "infinity" - it something concievable but not actually attainable.
Of course, for one of the angels/devils shown above, the universe looks just like ours does to us - ie. completely flat and the far reaches of the universe (ie. the boundary circle) just as unattainable.
Penrose, of course, just like any other popular science author, goes to great lengths to explain to the reader that this is not just a figment of his imagination but something that is just as real as you and me. However, I have always wondered at the rationale behind such an effort to convince - it seems to me that those people who think this is "dumb math" are probably not going to be reading a 4000-page tome on science to begin with - and for those who are actually interested, what you are doing is going to great lengths to be pedantic - something almost every thinking individual utterly detests! Solution: say it like it is, and those who want to understand will do the necessary self-convincing.
Of course, for one of the angels/devils shown above, the universe looks just like ours does to us - ie. completely flat and the far reaches of the universe (ie. the boundary circle) just as unattainable.
Penrose, of course, just like any other popular science author, goes to great lengths to explain to the reader that this is not just a figment of his imagination but something that is just as real as you and me. However, I have always wondered at the rationale behind such an effort to convince - it seems to me that those people who think this is "dumb math" are probably not going to be reading a 4000-page tome on science to begin with - and for those who are actually interested, what you are doing is going to great lengths to be pedantic - something almost every thinking individual utterly detests! Solution: say it like it is, and those who want to understand will do the necessary self-convincing.
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