7 ms·
Any smooth cubic surface contains 27 lines
- stephengillie 11y agoThe Wikipedia article might be more accessible: https://en.m.wikipedia.org/wiki/Cubic_surface https://en.m.wikipedia.org/wiki/Cubic_surface
- johnhattan 11y agoAhh, thank you. My first reaction was "A surface of a cube is a square". I can define a square with way fewer than 27 lines.
- manofcode 11y agoHaha, my thoughts exactly
- Sniffnoy 11y agoNon-mobile link: https://en.wikipedia.org/wiki/Cubic_surface https://en.wikipedia.org/wiki/Cubic_surface
- ternbot 11y ago27 * 4 = 108
- jsprogrammer 11y ago>No matter how one goes about it, the proof of this theorem is nontrivial. Is there a proof of this? Also, AMS: adding additional text to the text I selected to copy is not cool.
- omaranto 11y agoAre you asking for a proof of the "fact" that all proofs of the theorem about cubic surfaces having 27 lines are non-trivial? If so, then no, there is not even a definition of non-trivial proof, let alone a proof that any specific proof is non-trivial or a proof that there exists a fact all of whose proofs are non-trivial.
- jsprogrammer 11y agoYes, that is what I was asking for. What does the sentence mean then? One particular proof (of which we do not know if it is exclusive) is non-trivial (whatever that means)? Or, perhaps, all known (to the author) proofs are non-trivial?
- omaranto 11y agoBaez didn't mean that as a mathematical statement rather as a statement in the field of history and human psychology: "So far nobody has been able to devise a proof that is widely regarded by people who study it as being trivial (where triviality of a proof is a subjective judgment for which there is (1) no formal criterion, but also (2) a surprising amount of agreement)."
- amelius 11y agoMy definition of a trivial proof would be a proof that is mechanically verifiable. I.e., it is verifiable by even something as "dumb" as a computer. Of course, such proofs tend to be longer than most proofs that are considered non-trivial.
- omaranto 11y agoMaybe years ago that would have been a plausible definition, but by now many proofs have been verified by computer, including several which probably no-one would feel comfortable calling trivial.
- flubert 11y ago>No matter how one goes about it, the proof of this theorem is nontrivial. ...I'm going to guess he really means: "No matter how one goes about it, this proof of the theorem is nontrivial." ...and not... "No matter how one goes about it, all proofs of this theorem are nontrivial."
- heinrichhartman 11y agoMore like: "I am aware of several proofs of this theorem. None of them is trivial."
- pklausler 11y agoIs the converse statement also true -- i.e., do 27 distinct lines define a cubic surface? Will have to think about this.
- enedil 11y agoTake 27 lines that intersect in exactly one point (all coincide in the same). Would you tell that they define a smooth surface?
- pklausler 11y agoGood point.
- duskwuff 11y agoCertainly. Those lines could all lie in a plane, for instance -- that's very much a smooth surface.
- pklausler 11y agoI don't know why you're getting downvotes. Thanks for the example.
- bcook 11y agoI would like to know the reason for the downvotes as well.
- impendia 11y agoIt would not be cubic though (a plane is defined by a linear rather than a cubic equation). (p.s. I was not the downvoter)
- thaumasiotes 11y agoIt's a little odd to say a polynomial "isn't cubic" just because the degree-3 coefficient is zero. Ordinarily what matters is that every coefficient of degree higher than three is zero. Think of the result that you can fit a polynomial of degree n to go through n+1 points. It's easy to show that the polynomial indicated by that theorem which goes through a set of collinear points must be a line (degree-1 polynomial), no matter how many points are given[1]. But we don't say "as long as the points aren't collinear, this theorem holds"; we just accept that a line is a special case of cubic polynomial. [1] Proof: the polynomial of degree n going through n+1 points is unique. The line going through n+1 collinear points goes through all of those points, and therefore must be the unique polynomial of degree n to do so. I do see in the wikipedia article, though, that "cubic surface" appears to be defined to exclude polynomials which have any nonzero term of degree other than three. It's a weird world. :/
- Cogito 11y agoIs anyone aware of which Greg Egan is referenced in the article? I couldn't see any links to the original work, but I might be blind. I suspect that it is the Australian scifi author, as it is the sort of thing he has done before, but I haven't dug into it yet.
- ttctciyf 11y agoYeah, it's the SF author - he's collaborated previously with John Baez, as for example at https://golem.ph.utexas.edu/category/2009/10/girih_by_egan.html https://golem.ph.utexas.edu/category/2009/10/girih_by_egan.h... where you can see the link to the applet goes to Egan's site at http://www.gregegan.net/ http://www.gregegan.net/ or on the paper http://arxiv.org/abs/gr-qc/0208010 http://arxiv.org/abs/gr-qc/0208010 co-authored by Egan, Baez and J. Daniel Christensen (referenced in Egan's wikipedia entry.)
- ourcat 11y agoSomeone needs to build a threeJS/WebGL scene of this. So we can drag it around.
- user2994cb 11y agohttp://matthewarcus.github.io/polyjs/clebsch.html http://matthewarcus.github.io/polyjs/clebsch.html
- ClayFerguson 11y agoIf you look up "-1/12" on Wikipedia page about; 1+2+3+4...=-1/12 it has a mention that: "Bosonic string theory not consistent in dimensions other than 26". But if you add one dimension for time you get 27. I wonder if there is a relation between all these pointing to the number 27.
- evanpw 11y agoThat 26 already includes a dimension for time.
- DigitalJack 11y agoCan we get one in the shape of a kitten?
- IngoBlechschmid 11y agoIncidentally, this topic gave rise to a much larger theory, namely mirror symmetry (unrelated to reflection at everyday mirrors). Please take the following account of the history with some grain of salt, I'm not an expert on this topic. After counting lines on a smooth cubic surface, you could also count lines on other manifolds, for instance three-dimensional ones given by a quintic polynomial (these are examples for "Calabi–Yau manifolds"). Also you could count curves of degree two, three, and so on instead of lines, which are curves of degree one. The calculations get increasingly harder: The case of degree two curves was only settled in 1986. It therefore came as a surprise when a group of physicists (Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes) announced in 1991 a formula for calculating the result for curves of any degree. They did so by inventing a new technique, mirror symmetry, in which one relates the "complex geometry" (as in "complex numbers") of the manifold on which you're counting curves to the "symplectic geometry" of a certain other manifold, dubbed mirror of the original one. Many aspects of mirror symmetry are still widely non-understood and many conjectures are motivated and made plausible by physical arguments. https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory) https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory... https://en.wikipedia.org/wiki/Homological_mirror_symmetry https://en.wikipedia.org/wiki/Homological_mirror_symmetry
- amelius 11y agoAny polynomial of degree n has n roots, but some of them may be degenerate, and some of them may be complex. So, how would that translate to the cubic-surface situation? Are there cubic-surfaces that are "degenerate", and thus have actually less than 27 lines? Are all the lines expressible in real numbers? Or do we need complex numbers, like it is the case for polynomial roots? Just some questions that popped into my mind.
- GFK_of_xmaspast 11y ago> Any polynomial of degree n has n roots, but some of them may be degenerate, and some of them may be complex This is not true with multiple variables, consider for example "x2 + y2 - 1"