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Thread: A tesseract in N-dimensional space

  1. #1
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    A tesseract in N-dimensional space

    In my previous post we've found out that there are two possible painting schemes for a 4D hypercube (its cubic cells must be painted with red and blue only and its red cells must be opposite its blue cells). Those variations can't be converted into each other by rotation of any kind inside 4D.

    What about N-dimensional space, including N=infinity? Is there any dimensionality that allows such a conversion?
    Last edited by Fox333; Nov 18th 2017 at 12:48 AM. Reason: grammar
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  2. #2
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    Re: A tesseract in N-dimensional space

    Suppose you have an n-cube where n is at least 4. Orient the cube as we did in the other post. If you alternate red/blue, you will get a configuration that cannot be reached by just assigning one color as you go from one adjacent face to the next.
    For the infinite-dimensional cube, number the faces on the edges by the integers (without zero) such that z is opposite -z. Do the same thing. Assign reds to positive odds and blue to positive evens. That cannot be reached when you assign reds to all positives.
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    Re: A tesseract in N-dimensional space

    Thanks. You mean N-dimensional hypercube in N-dimensional space, right?

    I mean only 4-dimensional hypercube. As an example we can consider 1D space. As it was written in my previous post, there are two possible configurations for segments (1D hypercubes) which can't be converted into each other inside 1D, but can be converted into each other in 2D. As far as I know more dimensions are available, more kinds of rotation are available (0 in 1D, 1 in 2- or 3D, 2 in 4D). So, is there a kind of rotation in N-dimensional space, which allows "to swap" colours of faces by the exact way to make the two configuration of 4D hypercube interchangeable?
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  4. #4
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    Re: A tesseract in N-dimensional space

    Maybe in non-orientable spaces. Try projective planes.
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