Wikipedia:Reference desk/Archives/Mathematics/2019 July 29

= July 29 =

Shouldn't the sin of 90 degrees be undefined?
There's no triangle with two degrees = 90 degrees. I'll understand if we could approximate arbitrarily close to this triangle, but the sin 90 = 1 would imply a triangle that's impossible, wouldn't it? C est moi anton (talk) 13:49, 29 July 2019 (UTC)


 * Luckily, this is explained quite well by the first two paragraphs of our article sine, corresponding to the more detailed discussion in the first two body sections of that article. --JBL (talk) 14:04, 29 July 2019 (UTC)


 * To add to this a little bit, there are multiple ways to define the Trigonometric functions, and not all of them require the right triangle to do so. The unit circle definitions are particularly elegant in that the don't have the problems you note; they also more fully capture the harmonic nature of the sine and other functions, with the right triangle version would miss (since the right triangle cannot have any angle greater than 90 as you note; but working on a unit circle allow any arbitrary number of rotations).  -- Jayron 32 16:07, 29 July 2019 (UTC)


 * It's not really covered in History of trigonometry but I suspect it wasn't until the 18th century that people started think of sin, cos etc. as functions of numbers rather than properties of a triangle. Because there is a series expansion you can talk about the sine of all sorts of unlikely things: complex numbers, matrices, differential operators, etc. Useful for complex numbers; I'm not sure about the others but it seems interesting. --RDBury (talk) 17:33, 29 July 2019 (UTC)
 * Indeed. People like Rene Descartes, Leonhard Euler, Carl Friedrich Gauss, and many others who gave us other mathematical bases for considering the trigonometric function that generalizes them from simple algebraic relationships of right triangles.  What can be taught in a semester-long mathematics class at the high school or college level in a matter of months took centuries for humanity to come up with from first principles.  We forget that from our perspective.  -- Jayron 32 18:14, 29 July 2019 (UTC)
 * We have an article on Trigonometric functions of matrices. Double sharp (talk) 05:27, 2 August 2019 (UTC)
 * In the case of sin 90°, you can imagine in the limit a sort of degenerate triangle with two coinciding sides and the other of zero length: then you have two right angles (sort of), the opposite and hypotenuse are identical, and the adjacent side is zero. So you get sin 90° = 1/1 = 1, cos 90° = 0/1 = 0, and tan 90° would have to be 1/0 which is undefined. A similar rationalisation works for 0°. But in general it's easier to think of the trigonometric functions of real angles from the unit circle rather than from right-angled triangles. Double sharp (talk) 10:29, 31 July 2019 (UTC)

Gravity in a 4D universe
I suppose, in a 4D universe gravity might be at least as useful for living beings as here in our 3D one. But, while we have one vertical (towards <> away from the gravity center) and two horizontal dimensions, would there in a 4D space/universe be 1 vertical plus 3 horizontal D's or would it rather be 2 plus 2 (or maybe something totally different)? --85.76.32.219 (talk) 22:11, 29 July 2019 (UTC)
 * Assuming the 'vertical' is a direction of the gravitational force, that is a line joining the two gravitationaly acting point masses, I suppose the 'horizontal' should be a subspace normal to the vertical line, hence a 3D space. --CiaPan (talk) 16:36, 30 July 2019 (UTC)
 * I wonder if it's meaningful to inquire whether gravity is defined, in this hypothetical scenario, as a force field or as a potential energy field or as some other more general field?
 * It seems like this is an effort to reconstruct Newtonian gravity, but generalizing it to accommodate an additional spatial dimension. I think this is going to be an ill-fated endeavor, because Newtonian gravity is already ... less general than, say, the generalized theory of gravity, in which we describe the distributions of mass, energy, momentum, and stress, at all locations, and at all times, using a tensor formulation.
 * To put this in simpler terms - if you're using four spatial dimensions, what is the fourth coordinate of the object that has mass (and "exudes" a gravitational effect)? In what manner do you imagine the dependence, or change in the effect of gravity, as we change the value of that fourth coordinate?  ...I think you'll have to construct answers, and you'll probably end up with "math that doesn't work," because you're describing something "un-physical."
 * Nimur (talk) 18:20, 30 July 2019 (UTC)
 * I think one of the biggest things with 4D physics is there would be no cross product, or if there was a cross product it would live in 6 dimensions instead of 4, and physics uses cross products a lot. Objects would be able to rotate in two directions at different speeds at the same time. I'm pretty sure gravity would work on an inverse cube law as well; I couldn't find much on how that would affect orbits but I would think it means things are a lot more likely to either crash into each other or fly away from each other forever. Not good if you want a stable planet to evolve life on. --RDBury (talk) 21:22, 30 July 2019 (UTC)
 * I think you raise many good questions - especially noting the trouble with the generalization of the cross product - but the bit you mention about the inverse-cube laws worries me. The Euclidean distance, and the inverse square law, apply in any number of dimensions: why would you change the power-law just because you've added a geometric degree of freedom?
 * If you change the power law - for gravity or for any other field - as discussed in our article, it profoundly impacts important physical laws like the conservation of energy; because, of course, the force due to gravity is defined as the gradient of a conservative potential energy field. We don't want to meddle with that force law unless we know what we're doing!
 * One cannot casually speculate what consequences might follow from some weakly-specified arbitrary change to the geometry of the universe.
 * I found this very nice review article, Tests of the Gravitational Inverse-Square Law (2003), with lots of citations to further experimental and theoretical work exploring "alternative" formulations of gravity.
 * Nimur (talk) 22:30, 30 July 2019 (UTC)
 * Well, you get the inverse square law because the surface area of a sphere is proportional to the square of its radius. If you change it to a hypersphere, its surface area is proportional to the cube of its radius, so an inverse cube law is the obvious consequence. Double sharp (talk) 10:33, 31 July 2019 (UTC)
 * (ec) The supposed inverse-cube law follows from the Divergence theorem extended to a 4D space, where the ball's hypersurface is proportional to the radius cubed. --CiaPan (talk) 10:42, 31 July 2019 (UTC)
 * That is only true if you use the Euclidean distance! But if you're happy to change the number of dimensions, why wouldn't you feel comfortable changing the definition of your distance metric, or redefining the metric over which you integrate?  ... This is actually what we do with general relativity, and it's why we say that the actual shape of space time changes when we redistribute mass and energy.  Our article on the comoving and proper distances might give you some insight: during your surface integral, the radius of your hypersphere needs to be Lorentz-transformed!
 * The key difference is, when we work out the physical consequences in general relativity, they match our experimental observations, validating that this is a more useful mathematical model of the universe. I do not think we can say the same for a model that arbitrarily adds additional spatial dimensions; and I think the textbooks and the peer-reviewed literature are pretty much in agreement on these points.  This is exactly why we use general relativity as the framework for studying the geometry of gravitational interaction.
 * For more reading, here's a SIAM review article: Planetary Motion and the Duality of Force Laws (2000), which uses Hooke's law to explain why you kind of require an inverse-square law: "A fascinating footnote to this discussion is that Newton actually knew that his law of gravitation and Hooke’s law are dual..."  And there is even a nice discussion about "alternative universes" and their consequences!
 * Nimur (talk) 14:59, 31 July 2019 (UTC)
 * Because when the OP asks for a "4D universe", without further clarification, it seems fairly reasonable to guess that he or she means a Euclidean 4D space. The question is essentially based on a counterfactual, so the fact that visible higher dimensions are obviously something that do not exist in our universe is not relevant. Instead we would want to know: well, if you simply change dimensionality from three to four and try to work as analogously as you can until you get stuck, how does gravity work out?
 * For some more reading about the duality of power laws (and more besides): here on John Baez's blog. As Tamfang mentions, with an inverse cube law there are no stable orbits: the solution family for the motion of a particle moving under such a law is the Cotes's spirals. (The only closed bound orbits are circles.) Double sharp (talk) 15:44, 31 July 2019 (UTC)
 * A chunk of Greg Egan's novel Diaspora is set in a five-dimensional world; you might read that for some hints. One surprise is that with more than three spatial dimensions there are no stable orbits. —Tamfang (talk) 02:32, 31 July 2019 (UTC)