# Rotating Pentagon at Zero Angular Momentum

### Patterns and Simulations

By William Tong and Holger R. Dullin
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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.

 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

• Animations for optimized shape changing pentagons(.cdf, 510.57 KB) - 2970 download(s) Animations of the optimized shape change for: 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. Individual animations for each of the three categories are also available in mp4 format.
• Interactive CDF file(.cdf, 55.35 KB) - 2959 download(s) Interactive CDF File: Drag the mouse cursor around the fundamental region surrounded by the four borders to see the different pentagonal shapes on shape space. *Note: If the mouse cursor is outside the fundamental region it will be pulled to the boundary automatically.
• Optimized shape change for all pentagons(.mp4, 256.64 KB) - 2964 download(s) Optimized Shape Change for all Pentagons: An animation in mp4 format showing the optimal shape change of all possible pentagons including self-intersecting and degenerate pentagons.
• Optimized shape change for simple pentagons(.mp4, 212.34 KB) - 2974 download(s) Optimized Shape Change for Simple Pentagons: An animation in mp4 format showing the optimal shape change for the set of all simple equilateral pentagons.
• Optimized shape change for all convex pentagons(.mp4, 142.74 KB) - 2966 download(s) Optimized Shape Change for All Convex Pentagons: An animation in mp4 format showing the optimal shape change for the set of all convex equilateral pentagons.