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What is a Hypersurface?

  1. Jan 31, 2004 #1
    I'm new to the physics scene. I'm trying to get into it. I just read my first book the other day in fact. In the book it mentioned hypersurface. I've also heard it referred to as hyperplane. Hypersurface intrigued me a lot and I wanted to learn more about it. I did some research on the internet. The one thing I wanted to find I couldn't find... the formula. Does anyone know the hypersurface formula? Know where I can get it? Any help will be greatly appreciated.
  2. jcsd
  3. Jan 31, 2004 #2
    You'd do best to start your research with a search for "Flatland: A Romance of Many Dimensions." It's a copyright-expired work, so you'll find many copies of it available free of charge on the 'net. To be honest, I've never read the original work, which I understand is as concerned with political and social satire as it is with mathematical rigour; I've read many works that cite it and expand on its principles.

    The basic idea is this: to understand something in four dimensions, imagine yourself explaining the 3D version to a 2D person. Want to know what a 'hypercube' is like? Imagine explaining 'cube' to someone who has only seen squares.

    We use 'hyper' to refer to anything that exists in more than three dimensions, but often to four dimensions. A table of terminology:

    2D 3D 4D
    line plane hyperplane
    circle sphere hypersphere (or 'glome')
    curve surface hypersurface

    You can develop the functions for hyperplanes and glomes by analogy:

    line: ax + by = c
    plane: ax + by + cz = d
    hyperplane: ax + by + cz + dw = e

    circle: x^2 + y^2 = r^2
    sphere: x^2 + y^2 + z^2 = r^2
    glome: x^2 + y^2 + z^2 + w^2 = r^2

    The glome and the hyperplane are two examples of hypersurfaces. Just as you can create a three-dimensional surface by rotating, dragging, or otherwise mistreating a two-dimensional curve (like a parabola, circle, line, exponential curve...), there are any number of four-dimensional hypersurfaces that you can create by starting with three-dimensional surfaces.

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