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Everybody knows that when a plane intersects a cone at different angles and positions, we get conic sections. But, I wanted to know that if the same was possible in higher dimensions. If we take the 4 dimensional equivalent of a cone, and make it intersect with the 3 dimensional equivalent of a plane (cube or a cuboid) in 4 dimensional space, will we get 3 dimensional shapes like spheres or ellipsoids?

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Yes, you get ellipses and spheres. There are also 3D equivalents to a parabola and hyperbola. There are actually two types of 4D conic surfaces: one is a single connected surface, and the other has two separate surfaces. Here are some animations I just made on the topic:

3D View of Conic Sections from $x^2+y^2=z^2$

2D View

4D Conic Sections of $x^2+y^2=z^2+w^2$

4D Conic Sections of $x^2+y^2+z^2=w^2$

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