Outer Billiards Dynamics for a Regular Polygon with 17 Sides

This is a CDF Demo written in Mathematica 8.04. This is the latest release of Mathematica 8 with enhanced CDF capability.The controls should be active in Mathematica or any CDF reader. This is a self-contained notebook. The code is embedded because “SaveDefinitions” is set to True in the Manipulate command shown below. There is fairly large amount of data involved so it may take a while to initialize. To download or watch the companion video at GeneticsOfPolygons, select Christmas2011.flv from the Animation menu.

This demo is based on a regular polygon with 17 sides. See Projections.pdf and N17Summary.pdf at GeneticsOfPolygons.org for details. With N = 17 there are 8 possible projections including the ‘null’ projection which is the orbit itself. These are numbered P1-P8 below. The Christmas animation is based on the P2 projection, but the Demo below allows the user to choose any of the 8 projections.

The projections all use the same initial point which is the center of the canonical Step-1 'bud' which we call S[1]. This bud is also a regular polygon with 17 sides. It is shown below on an edge of the generating polygon which we call ' Mom'. By convention, Mom is centered at the origin with radius 1 and the coordinates of the S[1] center are cS[1] = {-0.367499035633140663,-0.948624361851015331}

By the symmetry of the outer billirds map (Tangent Map) on a regular polygon, Mom is surrounded by endless rings of congruent Moms with their 'surrogate' S[1]’s. The surrogate S[1] in Ring k has coordinates cS[1] + {-2k*cS[8][[1]],0} where cS[8][[1]] is the first coordinate of the center of the canonical Step-8 region (also known as 'Dad'). Dad’s center is at cS[8] = {-10.791718657261583420, 1.0} so the ring spacing is about 22 units. Below we can see the P2 projections in the first few rings. The region around Mom we call Ring 0.

Under the Outer Billiards Return map, the period of the S[1] center is 17. All the Ring 0 projections have the same period as this ‘null’ projection because each projection is a simply a re-mapping of the vertices of Mom. These periods increase as we move away from the origin and for k > 0, the surrogate S[1] in Ring k has period 17*(30k -1). Since the Demo only goes out to ring 3, the maximum number of iterations is 1513

The P2 projection for Ring 0 is not shown in the graphic above, but it can be seen in the Demo by selecting Ring 0 and Projection 2. Note that all the P2 projections are centrally symmetric and their centers are aligned horizontally but the spacing between centers is not the ring spacing. In the Demo below, click on the small ‘+‘ boxes to get better control over the variables.

Note: The 'Pinwheel' map also generates projections of orbits, but the process is different.The Pinwheel map is designed to filter out rotations around Mom by focusing on just a single strip. For a regular polygon, this filtering process is most effective beyond Ring 1. On the left below is the P2 Pinwheel Projection in Ring 2. The red dot is the local S[1] center. This Projection has period 15 while the P2 Orbit Projection in Ring 3 has period 1003. This is because the Orbit Projections are not filtered, so they are ‘embellishments’ of this primitive period 15 orbit. This can be seen clearly when we overlay the two plots below. Not only do these plots have the same initial point, but every point in the Pinwheel Projection is also a key point in the Orbit Projection.

As we pass to the outer rings, the Pinwheel Projections remain the same, but the Orbit Projections grow in period and in size. Eventually the structure is dominated by simple rotations about Mom. Below is the P2 Orbit projection of the S[1] center in Ring 41 along with close ups of the central region. This is period 17*(30*41-1) = 20893 and the initial point (red dot) is the local S[1] center at cS[1] + {-82*cS[8][[1]],0} = {884.553430859816,-0.948624361851015}.

By Ring 100 the dynamics are almost pure rotations and the central region is mostly rotational ‘noise’. If we looked at the real-world orbit of a small satellite around a large planet it would also be dominated by simple rotations which could drown out the small-scale perturbations in the orbit.

As projections go, these plots are very small. Even the Ring 100 plot above has radius less than 80 units For complex orbits, the projections tend to be very large and the Orbit Projections and Pinwheel Projections often appear to be identical because the high-frequency embellishments are not discernable.Below is a P7 Orbit Projection of a point which may be nonperiodic. This shows 500,000 points and the scale is thousands of units. At this scale we cannot distinguish the Pinwheel Projection and the Orbit Projection.

Below is a smaller P7 Orbit Projection in blue and the corresponding Pinwheel Projection in pink. These plots have the exact same scale and appear to be identical - except for the offset in position. But the blue plot has almost twice as many points, because of the small-scale embellishments.

Below is a close-up of two matching regions. Note that even the pink Pinwheel Projection has its share of rotational embellishments which are not filtered out.