Since we can portray 3-D figures on a 2-D surface (ex. a cube drawn on piece of paper), is it possible to portray a 4-D figure in our 3-D world?

  1. Yes it is.

    Source: http://en.wikipedia.org/wiki/Tesseract

    Just as a drawing of a cube is a projection of a three dimensional object onto 2D space, you can create three dimensional objects that represent a projection of a hypothetical four dimensional object into 3D space.

    Of course, just as a flat drawing of a cube is not the same thing as an actual three dimensional cube, a three dimensional projection of a tesseract is not the same thing as an ‘actual’ tesseract.

  2. Personally, I prefer to think in terms of “slices”.

    Think of a cone. Put its base on the floor. If you slice the thing horizontally, you get circles with linearly increasing radii. If you slice it vertically, you get half hyperboles. If you slice it parallel to its edge, you get a succession of parabolas.

    Actually, forget I said that. Let’s use a really simple example: a sphere with its center at the origin. When you slice it horizontally, you get circles. For a sphere,
    rho^2 = x^2 + y^2 + z^2
    where rho is the three-dimensional radius.
    x^2 + y^2 = r^2 gives the radius in the x-y plane (which we slice parallel to)

    so rho^2 = r^2 + z^2, meaning that the radius of the circle that you slice depends on the height that you slice at. The corresponding radii look like a circle.

    For a hypersphere, 4-dimensional radius^2 = x^2 + y^2 + z^2 + a^2

    a is the fourth spatial dimension. If you slice the hypersphere parallel to a = constant, you get spheres with

    4-d radius ^2 = 3-d radius ^2 + a^2

    So, put a bunch of spheres next to each other, but with their radii increasing/decreasing in the shape of a circle. That’s my portrayal of a hypersphere.

    I find this method easier than imagining projections. Just like you can think of a cube as a bunch of squares stacked on top of each other in 3-space, a hypercube is a bunch of cubes stacked on top of each other in 4-space.

    Edit: I haven’t figured out how to do rotations yet. I would appreciate help on how to portray 3-d rotations in two dimensions. (I’m guessing you need to use polar coordinates of some sort, but holy crap describing a square in polar terms is masochistic.) I haven’t thought this through yet.

  3. Think of a cube drawn on a piece of paper as the ‘shadow’ cast by a 3 dimensional object into the 2nd dimensions. If you held up a see-through cube and shine a light from above, the shadow you would see would be the 2D projection of a 3D object.

    Now, bump that up one dimension with a 4th dimensional object projection in the 3rd dimensions. It’s hard to wrap your head around, but the shadow of a 4D object into the third dimension is called a tesseract or hypercube: (http://www.daviddarling.info/images/tesseract.jpg).

    The tesseract is the shadow/projection of a 4D object in 3D. The added fourth dimension, most perceive as time, and thus the 3D projection moves with time, such as in this youtube clip (https://www.youtube.com/watch?v=5xN4DxdiFrs

    TL;DR:
    – 3D object in 2D is the ‘shadow’ of the 3D object
    -the ‘shadow’ of a 4D object in 3D is called a tesseract or hypercube
    -example of tesseract: https://www.youtube.com/watch?v=5xN4DxdiFrs

  4. One thing you have to remember, is that we can’t portray a 3 dimensional figure on a two dimensional surface, in order for it to be seen or portrayed, the paper must be visible in the three dimensional plane. We must be outside the 2 dimensional plane to mark the paper (adding in depth) and outside it to view it or interact with it. We are not the outside the three dimensional plane, so it stands to reason that it could be impossible to properly portray a 4 dimensional object (many consider the tesseract to be flawed.)

    Another thing you have to remember is that we do live with four dimensions, three space dimensions, and one time dimension. So the four measurements that correspond are Length, width, depth, and longevity, these four things have to be present in order for an object to be present to us. Longevity is most often displayed with numerous pictures, or an infinite mirror effect (showing it in multiple “times” per say).