SJf_Wettbewerbs_Broschüre_2007 - Die Goldene Sonne am Calanda

Lucas Brönnimann
5415 Nussbaumen AG
1988
Physik / Technik
Martin Lanter
5400 Baden
1987
Aargauische Kantonsschule Baden
Würdigung
Lucas Brönnimann und Martin Lanter haben
folgende mathematische Frage experimentell
untersucht: Wie können
Punkte mit maximalen Abständen auf der
Oberfläche einer Kugel platziert werden?
Mit zwei verschiedenen Methoden, unterschiedlichen
physikalischen Interpretationen
sowie der dazu notwendigen mit
MATLAB selbst entwickelten Software
sind sie diese Frage systematisch und
eigenständig angegangen. Ihr grosser
Einsatz hat zu sehr guten Resultaten geführt,
die mit (einzeln vorhandenen) theoretischen
Werten übereinstimmen.
Prädikat
Hervorragend
Sonderpreis
„European Union Contest
for young Scientists“ in Valencia
Sonderanerkennung
Metrohm Stiftung Herisau
Experte
Dr. Stefan Niggli
Mathematiklehrer, Kantonsschule Baden
46
Simulation on the sphere: Distribution of points and calculation of energy
The following two related properties for a set of n points on a sphere are being investigated:
1) The set has the maximum property if the minimal distance occurring between the points is
maximal.
2) Given a law of repelling forces, the configuration of points can reach an equilibrium with
(globally or locally) extremal energy.
While it is difficult to verify in general that the maximum property is satisfied, the energy criterion
may be approximately checked by numerical simulation. The present investigation uses deterministic
and stochastic simulations to find suitable approximations to extremal energy configurations
as well as arrangements of points on the sphere enjoying the maximum property.
The stochastic simulation mimics Darwinian Evolution. The deterministic one is inspired by
Euler’s method for the solution of ODE’s.
With the help of two independent methods it was possible to check our own results and find
out, e.g., that only the tetrahedron, octahedron and icosahedron possess the maximum property
among the platonic solids inscribed in a sphere. Using Darwinian Evolution or deterministic
simulation, the cube is transformed into an antiprism, and the dodecahedron evolves into a triangulation
of the sphere involving 20 vertices and 56 triangles.
In the case of Hooke’s law, applied to repelling springs between n points, we found out that the
energy can be kept constant while the points move in such a way that the center of gravity remains
fixed in the center of the sphere.
All the other simulations are based on Coulomb’s law.
The simulations involve programming of geometric algorithms such as finding the convex hull,
distance matrices, various invariants of point configurations on the sphere, as well as 3-D plots.
We discuss examples involving Fullerenes with 24, 60 and 120 points, and a discussion of equilibrium
configurations involving only triangles and quadrilaterals.