#fibonaccicusps — Public Fediverse posts

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

#kleinianlimitset #kleiniangroup #fractals #mobius #mobiustransforms #mathematicalart #generative #generativeart #mathart #fibonacci #fibonaccicusps