| Department of Geometry |
| Bolyai Institute, Faculty of Science, University of Szeged |
| Department of Geometry |
| Bolyai Institute, Faculty of Science, University of Szeged |
Dice, fragments, polytopes and the complexity of shapes
The Department of Geometry is pleased to divulge that
Gábor Domokos
gives a lecture at the Kerékjártó Seminar with title
Date and place of the lecture is:
Abstract of the lecture:
Motivated by the experimental and numerical study of natural fragment
shapes we define the mechanical complexity $C(P)$ of a convex polyhedron $P$,
interpreted as a homogeneous solid, as the difference between the
total number of its faces, edges and vertices and the number of its
static equilibria, and the mechanical complexity $C(S,U)$ of primary
equilibrium classes $(S,U)^E$ with $S$ stable and $U$ unstable
equilibria as the infimum of the mechanical complexity of all
polyhedra in that class. We prove that the mechanical complexity of a class
$(S,U)^E$ with $S, U > 1$ is the minimum of $2(f+v-S-U)$ over all
polyhedral pairs $(f,v)$, where a pair of integers is called a
polyhedral pair if there is a convex polyhedron with $f$ faces and
$v$ vertices.
In particular, we prove that the mechanical complexity of a class
$(S,U)^E$ is zero if, and only if there exists a convex polyhedron
with $S$ faces and $U$ vertices. We also provide upper bounds for the
mechanical complexity of the monostatic classes $(1,U)^E$ and $(S,1)^E$
and offer a complexity-dependent prize for the complexity of the
Gömböc-class $(1,1)^E$.
We show that the mechanical complexity of random polyhedra may be the
missing key to the general description of natural fragment shapes.
We also indicate that the concept of mechanical complexity may be extended
to general convex bodies and it appears to be related to natural
abrasion processes.
Here are some snapshots of the event:
