Ciro Santilli @cirosantilli  40

The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.

The counts per dimension are:

The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.

A convex regular polyhedron.

Their beauty is a classification type result as stated at classification of regular polytopes.

No good per-sub way as of 2022:

Just a circle.

Take ${\mathbb{R}}^2$ with a line at $x=0$. Identify all the points that an observer

For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.

To see why this is called a plane, move he center of the sphere to $z=1$, and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator $z=1$. Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.

Important 4D spaces: