# Graphics

### Subdifferentials

The limiting subdifferential of $f(x) = 2 \|x\|_1 - \|x\|_2$ for $\mathbb{R}^2$ and $\mathbb{R}^3$ at zero (notice that originally there was a mistake in the description of the function):

Please click on the thumbnails to see larger images.

#### Facially exposed and nice cones

Cross-sections of a four-dimensional cone that is facially exposed but not nice and of its dual, arXiv:1301.1000

### Billiard Balls Count π

An illustration to G. Galperin's article [G. Galperin, "Playing Pool with π (the Number π from a Billiard Point of View", Regular and Chaotic Dynamics, V. 8, No 4, 2003],
the simulation is for N=1, and it outputs two first digits of π.

#### A body invisible from a point

A POV-Ray animation of an object invisible from a point. The camera flies around the object, eventually getting into the point of invisibility, from which the body can't be seen.
The mathematical details can be found in [Alexander Plakhov, Vera Roshchina, "Bodies invisible from one point",
arXiv:1112.6167v1], also see Figure 7 in the paper. Animation by Vera Roshchina.

#### A body of zero resistance in one direction

A 3D model of a body or zero resistance [Alena Aleksenko and Alexander Plakhov,
"Bodies of zero resistance and bodies invisible in one direction" arXiv:0809.0108v3].
The building in the background is Palacio da Pena in Sintra, Portugal. Blender animation by Vera Roshchina.

#### A fractal body invisible in 3 directions

A fractal body invisible in 3 orthogonal directions. See [Alexander Plakhov, Vera Roshchina, "Fractal bodies invisible in 2 and 3 directions",
arXiv:1107.5667v1]