Does not require straight line cuts.

Diffeomorphic to $SU(2)$.

A unique projective space can be defined for any vector space.

The projective space associated with a given vector space $V$ is denoted $\mathbf{P}(V)$.

The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e. $\vec{v}=\lambda \vec{w}$

The most important initial example to study is the real projective plane.

In those cases at least, it is possible to add a metric to the spaces, leading to elliptic geometry.

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.

Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ${\mathbb{R}}^3$".

Take a open half sphere e.g. a sphere but only the points with $z>0$.

Each point in the half sphere identifies a unique line through the origin.

Then, the only lines missing are the lines in the x-y plane itself.

For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.

Ciro likes this model because then all the magic is confined just to the $z=0$ part of the model, and everything else looks exactly like the sphere.

It is useful to contrast this with the sphere itself. In the sphere, all points in the circle $z=0$ are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.

A polygon is a 2-dimensional polytope, polyhedra is a 3-dimensional polytope.

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.

square, cube. 4D case known as tesseract.

Convex hull of all $\{-1,1{\}}^D$ (Cartesian product power) D-tuples, e.g. in 3D:

From this we see that there are $2^D$ vertices.

Two vertices are linked iff they differ by a single number. So each vertex has D neighbors.

The non-regular version of the hypercube.

Examples: square, octahedron.

Take $(0,0,0,\dots ,0)$ and flip one of 0's to $\pm 1$. Therefore has $2\times D$ vertices.

Each edge E is linked to every other edge, except it's opposite -E.

3D parallelogram.