Number of combinations of 4 dimensions choosing 3 at a time is 4.
Someone please give a description of a most elementary 4 Dimensional Hyper
surface which has its four 3D intersections with coordinate planes as:
( a plane, a cone, a cylinder, and a sphere ) from which hopefully one gains
an understanding/feel/reality of the 4D.
EDIT1:
The way I imagine a particular case: Three mutually perpendicular planes can cut a cone along an ellipse,hyperbola and pair of straight lines. The 3 planar space curves can be re-joined together with a zero Gauss curvature surface, re-constructing the cone.
Three small circles on a sphere are made by coordinate planes cutting a sphere whose center is not the origin.They can be re-joined together to get back the sphere, or even we can employ a minimal surface run through these 3 circles using a soap film or even a surface of constant mean curvature.
Possessing certain analogous higher curvature properties ( do not know what they are), by the same token arbitrary surfaces should be capable of being integrated to form hyper-surfaces, or so it appears to me.