# What do you call the 4-dimensional analog of a plane?

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

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