#kleinianlimitset — Public Fediverse posts

6/

In general, the smoothly-connecting double cusp groups with numerator k are clustered into groups where the denominator is of the form \(kn + m, m < k \); m, k co-prime.

So, for example:

- if k=4, m ϵ {1,3}
- if k=5, m ϵ {1,2,3,4}
- if k=6, m ϵ {1,5}
- if k=7, m ϵ {1,2,3,4,5,6}
- if k=8, m ϵ {1,3,5,7}

There are always an even number of such sub-streams, and they come in pairs (where m=j, k-j) where the structure is similar, but complementary. For example, when k=7, the paired streams are {7n+1, 7n+6}, {7n+2, 7n+5}, {7n+3, 7n+4}.

Attached are several movies that show this effect when the numerator is 7.

- the first one, titled "7n-Smoosh", shows what a concatenation looks like that includes all denominators in order, with maximum denominator 150

- the other three are titled "7n+1", "7n+2", and "7n+3" with maximum denominator 250

As was the case for the case where the numerator is 3, the relative beauty of these videos is in the eye of the beholder, but the ones that are constrained to constant offsets of a multiple of the numerator are much more consistent to one another.

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5/

Attached are three different traces of the Maskit projection through the double-cusp groups where the numerator is 3, with a cap on the denominator at 188.

The first attached video (whose caption begins with "Smoosh") shows the result of concatenating every available cusp, sorted by increasing value of the corresponding fraction; that is, the cusp list starts with {3/188, 3/187, 3/185, 3/184 ...}. Note that there is no double-cusp group for 3/189 or 3/186 because those fractions reduce.

Unlike the movies for numerators 1 and 2, in this case the animation looks quite choppy and, although visual appeal is a matter of taste, certainly does not minimize differences between frames.

The second and third videos show the technique to generate smooth animations. Since the frames corresponding to cusps expressed as 3/(3n + 1) share an orientation, as do the cusps 3/(3n + 2), we must render two separate movies, one for each pattern, to get a smooth animation.

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3/

The previous post showed an effect where color gradients for 1/n cusps are highly concentrated when n is a multiple of the palette size, and widely distributed when n is a half-multiple of the palette size. This effect appeared to be conserved across projections.

Attached are 4 images demonstrating that this effect appears to also be conserved under resizing the desired output image. I found this far more counter intuitive than conservation across projections -- after all, the value of ε is half as much as it was for the smaller renders, you'd expect that to affect the depth needed to get to a terminal node. Honestly, I don't have a great explanation for it; any actual topologists who want to weigh in with your two cents, please consider yourselves invited.

The attached still images are 1920x1080 and 960x540 renders of the Unit Circle projections of the 1/31 and 1/63 cusps; the larger ones have more detail, but still show the same color gradient distribution pattern.

The 1920x1080 animations are too large to host on Mastodon; the YouTube links follow; the only parameter changing between these and the animations posted earlier in this thread is the image size.

Jorgensen: https://youtu.be/lEY6yWwDT74

Unit Circle: https://youtu.be/Y1rczGsU9Hk

Maskit: https://youtu.be/CMb7OdDTH1o

Some of you (my favorites) are skeptically thinking "she's asserting that this happens on multiples and half-multiples of palette size, but she's only showing one of each."

Unit Circle Looping up to 1/189 cusp: https://youtu.be/mNRp74Pe8b4

Traversal in R, rendering using Cairo, movie conversion with ffmpeg.

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2/

Attached see the same animations as in the first post in this thread, with the important exception that they are in different projections (the unit circle reference projection and the Maskit projection).

Just like the Jorgensen projection presented last time, these show the double cusp groups 1/n, where n is { 2, 3, 4, ... 63, 62, 61 ... 4, 3, 2 }. The palette (length 60) and color assignment rules similarly are the same.

At first it seems like quite a coincidence that the colors behave the same way with respect to n (that is, showing the most variation when n is a half-multiple of the palette size, the least when n is a multiple of the palette size), but it made sense (at least to me) after some reflection.

The underlying structure and topology of these projections are the same, so the node depth needed to draw what our minds perceive as the preserved elements from frame to frame proceeds the same way.

Interestingly, this phenomenon is conserved across resolutions as well, so renders of these groups at, say, 1920x1080, show the same high-level color patterning behavior (I'll post a demo later on YouTube).

Traversal in R, rendering using Cairo, movie conversion with ffmpeg.

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1/

The next few posts are going to explore how to arrange limit sets in time to make movies that appeal to human eyes. So far the animations on this channel have varied granularity ε or the μ parameter that defines the topology of the limit set.

This movie, and the ones following in this thread, consists simply of concatenations of double cusp limit sets, so there is no continuous movement to model.

However, to our visual systems there is a "preferred" way to sequence double-cusp groups that is appealing, and conserved across projections.

The art tax for this post is a simple example of a pleasing path: a looped animation of the Jorgensen tb=2 projection for the cusps 1/n, where n is { 2, 3, 4, ... 63, 62, 61 ... 4, 3, 2 }.

The palette and the rules for choosing line colors (by the depth of the terminal node in the tree associated with those line segments) do not change at all from frame to frame. In this case the palette size was 60.

The apparent color-cycling behavior comes about because the shapes that our minds map as "the same" from one frame to the next have a consistently increasing tree depth associated with them.

To look at this a little more closely, see the attached still image, which shows how the structure of the limit set changes as n varies. Each step along the path is the transformation of the previous circle by a group generator. For the 1/n cusp, for walking this path, we apply the loxodromic/spiral generator a (or A) n times, before applying the parabolic generator B (or b).

Traversal in R, rendering using Cairo, movie conversion with ffmpeg.

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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

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As scaffolding for a larger project, I had to figure out how to uniquely enumerate the generalized circles that make up any double-cusp Kleinian limit set, and determine their representative matrices.

Once I had my hands on that, it seemed natural to use those open interior spaces to recursively nest other limit sets.

The enclosing limit set is a 0/1 cusp displayed in the unit-circle reference projection described in _Indra's Pearls_. The nested limit sets were picked by starting at a random location in a large Farey sequence and simply using sequential elements.

The coloration comes from drawing the line segments based on how long the word associated with their corresponding terminal node is; this render used a palette of length 50, and flips the order of traversal at each nesting layer.

I wrote the rendering code in R, and rendered to .png using Cairo.

[1] Mumford, D., Series, C., & Wright, D. (2002). Indra's Pearls: The Vision of Felix Klein. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107050051

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