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4D space? 4D Cubes?

  1. Feb 13, 2015 #1
    I've read the descriptions and watched the videos over and over that describe these shapes, and my skepticism has simply not gone away. Can anybody explain what I might be missing?

    Here is my reasoning:
    Physics supercedes, or at the very least, is parallel to math. Math is not something that produces meaningful results on its own. Math is merely abstracted physics; it's never really led the pack in terms of revolutionizing scientific knowledge. Any great math discovery seems to be paired with a great physics advancement. Therefore, the math was simply the language of the physics discovery itself. It was not a separate thing.

    And with that said, the hypercube, as I understand it, is not a shape that exists. There may be some equations that point to more than 3 spatial parameters, such as the Kaluza-Klein, but I intuitively feel that this is misunderstood. This is an abstraction gone astray. It cannot mean 4 literal spatial dimensions.
     
  2. jcsd
  3. Feb 13, 2015 #2

    jedishrfu

    Staff: Mentor

    Math and Physics don't supercede one another. Math is the language of Physics. It allows physicists to analyze and predict things from their experiments. Its more accurate to say physics theory drives experiment and experiment drives physics theory and all this is done thru math and measurement.

    With respect to Math, it can be used as the language of choice for many other fields as well. Its a tool that provides us with a logical and self-consistent means to study problems. However, Math is even more than that and has been used to extend itself into other even more abstract areas that someday may find some practical use in our everyday lives.
     
  4. Feb 13, 2015 #3

    Mark44

    Staff: Mentor

    I disagree. There are many concepts in mathematics that did not arise out of physics, with just a couple being the Pythagorean theorem and the Quadratic formula, not to mention all of trigonometry.
    There are a number of concepts in mathematics that came about with no connection to any applications, let alone physics. One of these is the algebra developed by George Boole, that bears his name, that eventually became the basis for th logic used in computer circuits. Another example is number theory, that was at first considered "pure mathematics." Much of it has more recently found to be applicable in cryptography.
    We cannot perceive more than three spacial dimensions. However, mathematics has no such limits, and can deal with spaces with much higher dimensions. One of the applications of high-dimension vector spaces is error-correction algorithms in digital media such as CDs. In this case, the mathematics came first, and the application of it came much later.

    Your argument that physics supersedes (I think you mean "precedes") does not take into account the examples I have mentioned. There are many more.
     
  5. Feb 13, 2015 #4

    nsaspook

    User Avatar
    Science Advisor

    A classic example of 4D space math is in most computer games for 3D rotations using quaternions. Simple XYZ rotations will 'gimbal lock' when two planes overlap and is sequence order dependent. Using '4D' space is one solution this problem. Three.js is a very easy to use JavaScript API that using quaternions internally.

    http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/geometric/unit4dSphere/
    http://www.gamasutra.com/view/feature/131686/rotating_objects_using_quaternions.php
    http://threejs.org/
     
    Last edited: Feb 13, 2015
  6. Feb 13, 2015 #5
    Going through that first link, I'm not reading that there are 4 spatial dimensions, as the hypercube concept attempts to explain. It appears to just be an algorithm for moving within 3 spatial dimensions.
     
  7. Feb 14, 2015 #6

    Mark44

    Staff: Mentor

    The information that nsaspook linked to uses a four-dimensional space (which we cannot perceive, being creatures of strictly three spacial dimensions) to perform various transformations of objects in three dimensions.
     
  8. Feb 14, 2015 #7
    Thank you for the solid reply.
     
  9. Feb 18, 2015 #8
    I don't think those are very good examples because geometry can be considered as a kind of physics. It's just not traditionally called that, but it is physics because it makes physical predictions. Take a right triangle. Measure the sides other than the hypotenuse. Hypothesis: the length of the hypotenuse will agree with the value obtained from the Pythagorean theorem. Experiment: measure the hypotenuse. Result: agrees with hypothesis, within reasonable error bounds. And just as with Newtonian mechanics, if you extend the theory beyond its reach, it won't work. For example, if you tried this near a black hole.

    On some level, I'm not sure there's really anything outside of physics, math or not. That's what's so great about physics. It covers everything. That doesn't mean that mathematicians are always thinking about physics, though, at least in the sense of the formal discipline of physics, when they come up with things.

    It is helpful to think of things as being higher dimensional spaces. This isn't so clear to someone who hasn't developed their higher dimensional intuition, but it allows us to think of complicated things that have many degrees of freedom, using visual intuition and analogies with lower dimensional space. It's not unphysical. One of the big examples of a higher-dimensional space is a configuration space in classical mechanics.

    http://en.wikipedia.org/wiki/Configuration_space

    This doesn't mean that these spaces actually exist physically, only that they are a useful conceptual tool for reasoning about physics.
     
  10. Feb 18, 2015 #9
    This is exactly what I was arguing. You can't just make up a shape that could never exist in real life, in the three dimensional world. You can create an equation with more dimensions, but that's just an algorithm like you explained in the other part of your post. I don't know if higher dimensional intuition is really valuable. Maybe it's better to take higher dimensional algorithms and still imagine them as 3D shapes? Perhaps we are arguing semantics, but I would consider a 3D projection of a 4D object to merely be a 3D object. That doesn't count as actually visualizing 4D space, which is why I refer to that as seeing a 3D algorithm.

    Thanks for the link though. I'll check it out.
     
  11. Feb 18, 2015 #10
    Why not? If it helps you to think about things, then you should absolutely go ahead and make up shapes that could never exist in real life.


    Then, you haven't done enough math or physics. One reason why it's valuable is because visualization is always helpful as a memory aid, if nothing else. I'm pretty good at linear algebra. Why? Because I can imagine higher dimensional vector spaces and linear transformations. I understand it better, remember it better, and see more beauty in it because I can visualize it. The fact that I'm only visualizing it by analogy doesn't make much difference. If you prefer, I think about 4 dimensional spaces visually, rather than actually visualizing them. That's a subtle, but important distinction.


    Yeah, but it's a little more than that because you have to remember that what you are picturing isn't the real deal. It's just an analogy for your brain to grab onto that helps you think about it. Otherwise, you may risk getting confused and coming to false 3-dimensional conclusions about your 4-dimensional object. That's part of why I think mathematicians emphasize rigor and writing down formal proofs to make sure you are getting it right. But as far as the inspiration goes, it may well be visual. Not always, but it is a powerful way of thinking.
     
  12. Feb 18, 2015 #11
    Name a shape that you can imagine but can't exist in real life.
     
  13. Feb 18, 2015 #12
    That's what I've been doing when I said configuration spaces. Of course, you may say that you are only imagining a shadow of the actual shape to get a handle on it. So, maybe you're getting hung up on the semantics of what we mean by a shape. Maybe what we really mean is a mathematical object whose lower dimensional-analogues are things we can visualize.
     
  14. Feb 18, 2015 #13
    I think if we are to argue that a 3D projection of a hypercube (or some other higher dimensional shape) is a real object, then I would make a counter point. Hypercubes account for their 4th dimension by moving. Therefore, motion is a dimension. This means that a point moving for a period of time is a line segment, a line segment moving for a period of time is a polygon (assuming the line doesn't move precisely in the direction that its original point moved in), a polygon moving in a direction is a 3D Euclidian object (again, assuming it doesn't move in the direction of its previous two movements), a 3D object moving in space is a 4D object similar to a hypercube, a 4D object moving in space (essentially a 3D object with two orthogonal motions, a direction and a spin) is a 5D object.

    That's actually how I perceive derivatives/antiderivatives, so that perfectly matches my thoughts. In fact, it makes it clear that electrons are 5 dimensional objects, with a height, width, depth, linear velocity, and angular velocity. I can't speak with authority, but maybe that correlates to the Kaluza-Klein unification that gave space 5 dimensions.

    Going beyond 5 dimensions is a bit harder, but maybe we can do it. A 6D object would be like a 5D object that oscillates in one dimension orthogonal to its linear velocity (like up and down). A 7D object would be like a 6D object except it is now oscillating in two directions orthogonal to its linear velocity (now it's moving in circles towards its linear vector while also spinning about its own axis). I'm not sure what to do next with this model of dimensions, but I will think...

    **PS I feel quite certain now that this is a superior way to make space and motion intuitive. Therefore, 6D objects DO exist in our world. We can see them everyday in the form of photons, which have a wavelength, a spin, and a velocity. It's argued they are massless and therefore width/height/depth-less, but that's besides the point. A 5D object exists in our world. It's an electron. Also, adding more dimensions to our typical conception of 3 spatial dimensions DOES bring more accurate predictions, as per string theory, because there can be abstracted layers of motion within motion, and each of those motions are a dimension.

    Try modeling an electron in a carbon atom, on Earth which is spinning around itself, on Earth which is elliptically moving around the sun, in the solar system which is moving around the center of the milky way, in the milky way which is moving away from the center of the universe. Each layer of abstraction must be an actual dimension according to the seemingly logical marriage of dimension and motion.
     
    Last edited: Feb 18, 2015
  15. Feb 18, 2015 #14
    Yes, I think this is semantics, like I said either. My quabble, which is seemingly insignificant to you but is the entire point I am making, is that "lower dimensional-analogues" are all that exist. That is all it could be. To say that there exists a world which has 4 orthogonal spatial dimensions is pure fantasy. However, look at the post that I just made. Motion may be the solution to this contradiction if we take motion to be a literal dimension, like we do with the hypercube "projection".
     
  16. Feb 19, 2015 #15
    There are 4-d objects in space-time is what you could say.

    Well, electrons are either 4-d if you're thinking of them as things that exist in space time, or if you're thinking more along the lines of configuration space, that's going to be an infinite-dimensional Hilbert space.

    That's something like what configuration spaces do. One of the examples is just n point particles. Each one has 3 directions it can wiggle in, so if you have n particles, they can wiggle in 3n different directions, so you get a 3n-dimensional configuration space and 6n-dimensional phase space if you want to describe velocities as well as positions. If you have particles that aren't just points, you would get more degrees of freedom.
     
  17. Feb 19, 2015 #16
    In some sense, these configuration spaces do exist in real life, by the way, provided you realize that the mathematical configuration space is only a model for the real life thing.
     
  18. Feb 19, 2015 #17


    Why isn't an electron 5 dimensional? Yes, it moves linearly in 4D space, having x, y, z in displacement over t time, but does it not also have an angular velocity accounted for as its charge? That means 5 pieces of information or 5 dimensions.

    Awesome. Do you have any suggestions on reading material?
     
  19. Feb 19, 2015 #18
    Well, they are still sort of imaginary. It's like the set of positions that your object COULD find itself in. So, we actually only observe one point in the configuration space at a time. And over time, we just see things moving along paths through the configuration space. So, it's kind of Platonic scenario, like how real circles are only reflections of some idealized circle that we can imagine. I don't know if I'm a Platonist or not. What it really means is that there's distinction between the actual things and the mathematical model of them.
     
  20. Feb 19, 2015 #19
    I don't agree; that's my fundamental disagreement. Something in math has to exist in the world (or it could exist if we created it). See my point about going from the electron to the universe. To model that motion, you would need a **** ton of dimensions. If you want to say "but they're still moving in the same spatial dimensions if you average it out", then I would point you to the very first abstraction of spatial size to movement, space-time.
     
  21. Feb 19, 2015 #20
    If that object itself is motion (which correlates to my point-->line-->2D-->3D model) (which, thanks to modern physics, we now know is true: most of materials are empty space with occasional moving chemicals, which are full of empty space but occasionally moving particles), then it's configuration space is not just an imaginary model; it's the real thing.
     
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