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Hi Ken & Ali, On Wed, 13 Feb 2002 Kengrimes123 w aol.com wrote: >Please find attached as requested our latest draft of the article, which we>will be forwarding to our editor this Friday.Great! >We fear you may be somewhat disappointed with the finished product, since, as>you can see, we have not managed to cover the 'psychological crutch' aspect>of the 4th dimension. Although we found this discussion most interesting, we>simply found it impossible to fit into a popular account with a strict word>limit which had to cover so much other material. Our apologies.On the contrary, my general opinion of your article is that you've done a good job of making a simple as possible introduction to the subject and of covering the variety of research done in Europe/North America, given a non-expert background and a finite ;-) amount of time to learn the subject. (Including the several Brazilians and one Japanese researcher would have been fair, but would have required more time and reading and would have been difficult within the word limit.) I think there are several small changes that can be made in order to be correct about the "psychological crutch" aspect, which I've listed below as (1). These would only add a dozen or so words. Since I'm convinced that you've made an effort to try to understand the question - but probably without (yet) succeeding (it might happen sometime when you're in the shower or on summer holidays and daydreaming or whatever - and you'll suddenly say, of course!), I certainly won't hold it against you if you prefer not to make these changes. After all, only you two are signing it, not me! But I think it would be both in readers' and researchers' interests if if you did consider my suggested changes... There are also a few other corrections I've suggested. >We would, of course, still be most interested in any comments you might like>to offer (especially if we've made any howlers!) In the meantime, we would>like to take the opportunity to thank you for the time, interest and support>you have given us on this project.The most fun part of science is understanding the Universe, and helping others to understand the same things. Your help in transmitting this to others is definitely worth it. It's been a pleasure to work together. :-) All the best Boud ---------------------------------------------------------------------- (1) The "psychological crutch" aspect of the 4th dimension. (2) What are "photons"? (3) topological images of galaxies, not stars (4) Recent constraints show Universe flat like continents are flat (5) Infinitely many spherical spaces ---------------------------------------------------------------------- >This hard-to-visualise fourth dimension is particularly intriguing to>cosmologists because, if it exists, then the universe might be curved>within it. An immediate critique of this idea is that, to the best of(1) I would put: ... % cosmologists because it could be used to help imagine how the universe % might be closed. An immediate critique of this idea is that, % to the best of ---------------------------------------------------------------------- >joined to each other simultaneously [EXAMPLE AT ABOVE URL, PAGE 2]. If>this `cube' were the entire space of our three-dimensional universe,>then the bending operation, possible only in a fourth dimension, would(1) I would put: ... % then the bending operation, which can be imagined by using % a fourth dimension, would ---------------------------------------------------------------------- >began. The same would be true for our closed, curved,>higher-dimensional universe. Like an Asteroids-type starship, any(1) I would put: % began. The same would be true for our closed % universe. Like an Asteroids-type starship, any or % began. The same would be true for our closed % universe thought of inside a higher-dimensional artificial universe. Like an % Asteroids-type starship, any ---------------------------------------------------------------------- >The term any object includes, of course, photons.(2) I think that there are probably 10-100 times more people who have some notion of both "light" and "particles" but haven't the foggiest about "photons", so why not help enlighten them: % The term any object includes, of course, photons (particles of light). ---------------------------------------------------------------------- >In a closed, curved universe,(1) I would put: % In a closed universe (which could sort of be thought of as curved) ---------------------------------------------------------------------- >light itself would describe a curved path, eventually(1) I would put: % light itself could sort of be thought of describing a curved path, % eventually ---------------------------------------------------------------------- >The implications for contemporary cosmology are truly staggering. When>we look up at the night sky, it is possible that what we are seeing is>not a single, vast or infinite number of stars, but the same finite>set of stars repeated throughout the sky, like torchlights in a hall(3) You probably know that you're making an error here, with the goal of simplicity. We are certain that the size of the Universe is much bigger than our Galaxy - none of the stars we see with the naked eye are repeated, topological images of one another. (Unfortunately!) What solutions are possible to try to be correct without sacrificing understandability to the ordinary reader? Replacing "stars" by "astrophysical objects" would be correct but sound heavy and academic. In French you could use the word "astre" which is short and snappy and means an arbitrary object in the sky. But you're not writing in French. ;-) How about: % The implications for contemporary cosmology are truly staggering. When % we look up at distant galaxies in the night sky, % it is possible that what we are seeing is % not a single, vast or infinite number of galaxies, but the same finite % set of galaxies repeated throughout the sky, like torchlights in a hall ---------------------------------------------------------------------- >observations suggest that the universe is flat.(4) More accurate would be: % observations suggest that the universe is pretty close to flat, like % any continent on the Earth is pretty close to flat. ---------------------------------------------------------------------- >The number of contenders for shapes which could tile a flat or>spherical universe is limited, but a hyperbolic universe offers(5) A "howler", at least from the mathematical point of view! ;-) The number of shapes (in 3D) for both the spherical and hyperbolic cases is infinite [in mathematical jargon, these are both countable infinities, i.e. the same type of infinity as that of the ordinary counting numbers: 1,2,3,4,5, ...]. I made the *error* (published, I think) of repeating the claim by someone else (who I won't name) who said that the number of hyperbolic spaces is a bigger sort of infinity than the infinity of the spherical spaces. From Jeff Weeks' explanation, I accepted that this is wrong. It is true (as far as I understand) that the hyperbolic spaces are not yet completely classified by mathematicians, and their families are certainly more complicated in certain ways, while the spherical spaces are completely classified and form relatively simple families. But both numbers of spaces are infinite. I suggest: % The number of contenders for shapes which could tile a flat % universe is limited, but a spherical or hyperbolic universe would offer ----------------------------------------------------------------------

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