Projective geometry is an important mathematical discovery that goes back to the work of Desargues in the XVIIth century.

A space filling curve is a continuous map $\gamma \colon [0,1] \to \mathbb R^n$ whose range reaches every point in a higher dimensional region. Examples are the Peano curve or the Hilbert curve whose image is a unit square.

The above photo displays the four-step approximation of a three-dimensional space filling curve whose image is cube.

This “paradox” was explain to me by a colleague. Take a square whose side have length $1$ (on blue on the figure below) and subdivide it in four squares of side $1/2$. In the middle of each smaller square place a disc which is tangent to the four sides (on white on the picture below). In particular the radius of these disc is $1/4$. Now place a fifth disc (red on the picture below), centered at the center of the original (blue) square which is tangent to the four other discs.