This is the third in a series of posts arising from the Fibonacci cusps of Kleinian double-cusp groups.
As previously discussed, a series of fractions Fib(n)/Fib(n+1) will tend towards 1/ϕ over time. Correspondingly, the sequence of Fibonacci cusps on the Maskit curve will converge to the point corresponding to 1/ϕ, which generates what I suppose might be called the Golden Mean Group.
The μ parameter for this group for the Maskit slice where the second and fourth generators are parabolic ("tb=2") is provided in [1], specifically μ=1.2943265032+1.6168866453i (there is a mirror group with essentially the same properties at μ=0.7056734968+1.6168866453i). Curt McMullen and Troels Jorgensen were major proximate contributors to nailing this down.
Attached are Maskit and Jorgensen renders of the Golden Mean group. Interestingly, this set doesn't meet up like the Fib cusps that led up to it, even though the value of ε was set extremely low It looks like one of the intermediate frames from the walk animations, not a proper group. But what there still seems to be made up of tangent circles. What's going on? We'll start to explore that in the next post.
A completely different take on rendering this group is found in [1], fig 10.4 which is definitely worth a look if you're interested..
Traversal in R, rendering using Cairo, image editing in gimp.
[1] Mumford, D., Series, C., & Wright, D. (2002). Indra's Pearls: The Vision of Felix Klein. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107050051
When using a DFS to draw a Kleinian limit set, you partially traverse an infinite 3-ary tree. Each node corresponds both to a particular word in a 4-letter alphabet and an ordered set of points. Line segments can be drawn connecting those points in order; the deeper you go into the tree before drawing, the more detail you get. But when do you stop?
Many visually-appealing groups have the property that the node path length monotonically decreases as you descend any particular path, making "ε-termination" quite useful : when the distance along the a node's path is less than a provided value ε, draw and terminate.
The optimal ε depends on the size of your pixels in math space, your drawing technique (for example, in general a higher ε wants a higher line width), and how much quality you want.
Attached is an animation showing how the renders of three different Kleinian limit sets change as the {ε, line width} parameters are decreased from {500 pixels, 4.5 pixels} to {2 pixels, 0.5 pixels}. An animation with more limit sets included is available at https://www.youtube.com/watch?v=gH4kacpgw_A.
The groups shown are :
Riley group with parameter c=0.05+0.93i
Jorgensen projection of the 3/31 double-cusp group
Jorgensen group with parameters ta=1.87+0.1i, tb=1.87-0.1i
I wrote the rendering code in R, used Cairo to export to .png, and ffmpeg to convert to .mp4.
[1] Mumford, D., Series, C., & Wright, D. (2002). Indra's Pearls: The Vision of Felix Klein. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107050051