Originally Posted by flashgordon2!
I mean if higher dimensions are 90 degrees from one another the common saying goes, then, as they say it is pretty hard to imagine(even Stephen Hawking has said this) what is 90 degrees after you get the third dimension?
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Yes, it is difficult to make visual representations of four-dimensional objects in 3- or 2-dimensional space (though it is certainly possible - see
tesseracts and, for that matter,
penteracts), and is quite perilous to try to visualize higher-dimensional objects; However, the arithmetic and theory of such spaces is straightforward, and is presented in any introductory linear algebra textbook (for vector spaces).
I basically came to think that the fourth and higher dimensions are overlapping on one another by symmetry. Each symmetry is a higher dimension. How many symmetries a given space defined by a given shape is how many higher dimensions it can at least possibly have. I mean a Cartesian plane turned 360 degrees on each other can have an infinity of dimensions of multiples of four.
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What is a "symmetry"? What does it mean for a symmetry to be a higher dimension? How do you define the number of symmetries belonging to any given space? What does it mean to 'turn a cartesian plane 360 degrees'?
It is important to realize that mathematics is not a subject of idle speculation. It does not mean anything to have an idea unless you can present it rigorously under some set of axioms and with specific definitions. The subject of vector spaces of all dimensionalities is very well-understood (including the theory of infinite-dimensional spaces, which are critical to quantum mechanics and to solutions techniques to the equations describing myriad physical systems). If you want to learn about the mathematical representations of these objects (which are the same ones used by physicists and everyone else working in science), then you need only to take a few textbooks out of a library and work through them.
I was a bit hesitant to post this because just stating the above doesn't give much to do or functionality; i mean, even if I'm right, so what? Well, just looking at the list of threads below, I noticed 'complex numbers', and I immediatelly see one way of making this insight fly . . . complex numbers are the arithmetic of higher dimensions according to my 'theory'!
Complex number at least in one explanation based on polar coordinates of moving the cartesian plane at 90 degrees successively.
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Complex numbers are not "the arithmetic of" anything. The set of complex numbers can be represented as a two dimensional real vector space, or (trivially) as a one dimensional complex vector space. They have nothing to do with the Cartesian plane (though the complex plane is certainly very similar to two-dimensional Euclidean space!). Polar coordinates simply provide a different way of representing a complex number; it is not difficult to show that for any real
and
, there are some reals
and
with
.
Unfortunately, the presentations of mathematical ideas in typical popular science books are uniformly reprehensible. Do not take such presentations too seriously; Almost without exception they bear only superficial resemblance to the actual mathematical descriptions of systems. If you really want to understand the structure and power of modern science, the only way to do it is to work through the math.
