The configuration space of a pentagon P is defined as the topological space of all possible realizations of P in the Euclidean plane, modulo the group of orientation-preserving isometries of the Euclidean plane.

By considering the pentagon as a mechanical linkage, the configuration spaces of all planar pentagons can be classified using techniques from basic topology.

There are 21 topologically distinct spaces: the orientable surfaces of genus 0, 1, 2, 3 and 4 with up to 4, 3, 2, 1 and 0 singularities respectively, the union of two tori identified at up to 3 points, as well as the empty set and the set of one point.

We introduce an alternative method for determining the configuration space using an extended version of the Euler characteristic which makes it possible to classify the configuration space of any given pentagon within a matter of seconds.

Authors

Dr. Robyn Curtis
Prof. Dr. Marcel Steiner, IGN ( http://personenseiten.fhnw.ch/marcel.steiner )

 
Publications

• Configuration Spaces of Planar Pentagons, R. Curtis and M. Steiner, American Mathematical Monthly, 114, No. 3, 2007, p.183-201.
• Ist das Öffnen eines Notenständers trivial? D. Jordan und M. Steiner, Elemente der Mathematik, 55, No. 4, 2000, p.163-172.
• Visualisation of Configuration Spaces of Polygonal Linkages, O. Mermoud  and  M. Steiner, Journal for Geometry and Graphics, 4, No. 2, 2000, p.147-157.