#
Monday Lecture and Colloquium

**Monday, February 12, 2018 **

Freie Universität Berlin

Institut für Informatik

Takustr. 9

14195 Berlin

room 005

** Lecture - 14:15**

### Michaël Rao - ENS de Lyon

### Exhaustive search of all convex pentagons which tile the plane

*Abstract:*

In 1918, Karl Reinhardt asked which convex polygon can tile the plane, and the pentagons was the only opened case.
Fifteen types of such pentagons have been found between 1918 and 2015.

I present an exhaustive search of all families of convex pentagons which tile the plane.
This proof is in two parts. First we show that if a pentagon tiles the plane, then it is in one of 371 families.
Then a computer program tries, for each family, all possible tile arrangement, and shows that there are no more than
the already known fifteen types. In particular, this implies that there is
no convex polygon which allows only non-periodic tilings.

**Colloquium - 16:00**

### Andrei Asinowski - TU Wien

### Enumeration of triangulations of (some kinds of) subdivided convex polygons

*Abstract:*

We compute the number of triangulations of a convex k-gon with
subdivided sides for the following cases: (1) all the sides are
subdivided by the same number of points (r); (2) k=3. We find explicit
formulas and generating functions, and we determine the asymptotic
behaviour of these numbers as k and/or r tend to infinity. Finally, we
link these results to a well-known open problem from computational
geometry: finding a planar set of n points in general position that has
the minimum possible number of triangulations.

joint work with C. Krattenthaler and T. Mansour

Letzte Aktualisierung:
26.01.2018