Here's an example of the extension to 3 dimensions, along the lines that vortico proposed:

    x^2/2.0 + y^2/5.0 + z^2/3.0 = 10.0

or in the form closer to what vortico used:

    (x/sqrt(2))^2 + *y/sqrt(5.0))^2 + (z^2/sqrt(3.0))^2 = 10.0

Here's a plot: http://www.wolframalpha.com/input/?i=x^2%2F2.0+%2B+y^2%2F5.0... .

Vortico used a common text notation where "x_1" means "first dimension", "x_2" means "second dimension", and so on up to "x_12" which means the 12th dimension.

The "x_1" is an alternate way to type x₁, "x_2" denotes x₂, and "x_12" denotes x₁₂. In practice it's easier in a text-only forum like HN to style the notation as "x_2" than reach for the Unicode "x₂", and there aren't enough Unicode characters for all of the things that might be in subscript.

My example used the first 3 dimensions, which are often named 'x', 'y', and 'z'. You could say that 'x' by itself is an alias for 'x_1' (which is a notation for for x₁), 'y' is an alias for 'x_2' (that is, x₂), and 'z' is an alias for 'x_3' (that is, x₃).

(Sometimes people will use 'u', 'v', and 'w' for the next three dimensions, but you can see that eventually you'll run out of letters. That's one reason why people switch to the 'x_123' notation.)

The 'a_1', ... 'a_12' are similar. The 'a_1' (that is, a₁) is the constant for the first dimension, so 'sqrt(2.0)' in my example. 'a_2' as the constant for the second dimension, which is 'sqrt(5.0)', and 'a_3' for the constant 'sqrt(3.0)'.

You'll notice that my equation ends with '10.0' instead of the '1' that vortico used. This is a scaling factor. I could divide both sides by 10 and get the right-hand side to be 1.0, in which case the full equation would be:

    x^2/20 + y^2/50 + z^2/30 = 1

or in the form closer to what vortico used:

    (x_1/sqrt(20))^2 + (x_2/sqrt(50))^2 + (x_3/sqrt(30))^2 = 1

where a_1 = sqrt(20), a_2 = sqrt(50), and a_3 = sqrt(30).

Mathematicians prefer this form because it removes the otherwise arbitrary and meaningless scaling factor from the analysis, and because the a_1, a_2, etc. value give the minimum and maximum values along the x_1, x_2, etc. axes.

Mathematicians also prefer to write integers without the ".0" as I did, but as a long-time programmer I tend to write things like "3.0" to emphasize that it's a floating point number, where "3" would mean integer.