#### Milestone Year

### 1995

#### When circles touch

Consider a collection of circles of different sizes in a plane, packed so
that they do not intersect. Some pairs of circles may touch each other and
others will have no points of contact. Characterizing these relationships
(known as tangencies) constitutes an interesting problem of combinatorics,
related to other areas of mathematics and its applications.

A Weizmann Institute mathematician made major contributions to this
field of research. For example, in 1995, using circle packing as above,
he gave a basically combinatorial and constructive proof of Riemann
mapping theorem. This 19th century theorem lies in the basis of the
area of complex analysis and continues to be relevant even
today (and even in applied areas such as image processing).
It also lies at the basis for the contribution to SLE.
Informally it states that all "hole-free" domains in the plane,
except the whole plane, are conformally equivalent;
that is, they can be mapped one onto the other
by an invertible map that preserve infinitesimal angles.
In particular, the shape of small enough region
is approximately geometrically preserved,
except for possible scaling and rotation.