[seqfan] Polygon sequence

[Charles Greathouse]

Last month Ryan Lawrence Austin posted this suggestion for a new sequence
to our wiki. (I just came across it today.) Do we have any geometers
willing to give this a crack? I've pasted the message below, unchanged
except for breaking it into paragraphs for easier reading in email:

Algebra that pertains to this function I will explain here. I used a
program called Geogebra to build a function that starts with a line of 5
units from origin vertical on the y axis. the units were as small as i
could get, to the -13 power(the function gets large to the 13th power
quickly). I used the regular polygon tool to make n+ polygons
counterclockwise, starting from origin as the first point and the third
point, counterclockwise, in the polygon prior, as the base for the next n+
polygon. as the polygons become larger zooming out is required and
eventually comes to the point that the program won't zoom out anymore.
Scrolling to the next point takes awhile using the keyboard on the computer
(there is a point list on the left that allows clicking the origin point
first and then clicking a point to make a base, so only getting to the next
point is necessary on the screen).

I started looking for patterns in the numbers of two properties. First is
when the base crosses an axis/90degrees. Second is when a base changes
direction off of a virtual origin and changes direction going to the next
point(richochets). What I mean is when the line of the polygon created
travels towards a flat plane and rebounds off it, first side and second
side (counterclockwise). I obtained some numbers for crossing axis
directions, being 5,8,12,20,32,52,84, the first number being crossing the
vertical axis +y/180degrees. The second set was [3?4?],6,8,15,30,50,82. The
first number being confusing but probably 3.

The sequences for the rebounds seem to always be before the turns over the
axis and also do not share as apparent similarity to those numbers as well.
The numbers are the numbers of points/sides to the polygon. I am wondering
if there are functions to explain the spirals formed in this function and
what they are. I was originally looking for other spirals than the golden
spiral.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University