We show how an equilateral pentagon with fixed side lengths but variable angles needs to change its shape while keeping angular momentum equal to zero such that the overall amount of rotation is maximised.
Three different sets of equilateral pentagons are considered:
1) all possible pentagons including self-intersecting and degenerate pentagons,
2) the set of all simple (i.e. non-intersecting sides) pentagons and
3) the set of all convex pentagons
In each of these cases a contractible loop in shape space is found that is optimal in the sense that it yields the maximal overall rotation of the equilateral pentagon.
These optimal shape changes are shown in animations, available as movies or as Mathematica CDF files. Another CDF file illustrates the discrete symmetry reduced fundamental region of shape space in which every equilateral pentagon appears exactly once.
The CDF files can be opened by downloading Wolfram CDF player (free) at http://www.wolfram.com/cdf-player/.
| Author Institutional Affiliation | University of Sydney |
| Author Email | |
| Author Postal Mail | School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia |
| Keywords | geometric phase, equilateral pentagon, symmetry reduction, zero angular momentum, shape change |