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I keep wanting to call it a "space" but that conflicts with the 2-space, 3-space, 4-space, n-space nomenclature.

e.g., in 3-space, you can hold one variable constant to get a 2-space "slice" (a plane) of the 3-space object. e.g. if I have $x^2 + y^2 + z^2 = 1$ (a sphere), I can take $z=0$ which gives me a plane slice (the x-y plane) on which is a circle of radius 1.

In 4-space, if I have $e_1^2 + e_2^2 + e_3^2 + e_4^2 = 1$ (a 4-sphere), I can take $e_4=0$ to give me a [XXX] slice (the x-y-z [XXX]) in which is a sphere of radius 1.

What is XXX?

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Technically speaking, $x^2 + y^2 + z^2 + w^2 = 1$ is usually (though sadly not always) referred to as the 3-sphere. It lives in 4 dimensions, yes, but the sphere itself is 3-dimensional. – Jesse Madnick Jan 30 '13 at 10:28
up vote 3 down vote accepted

Such a geometric object is called a hyperplane (here's the Wikipedia article), in particular an affine hyperplane if you want to consider the case when the hyperplane need not go through the origin.

Thus, one might speak of "the hyperplane in $\mathbb{R}^4$ determined by the equation $e_4=0$".

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