#kleinianlimitset — Public Fediverse posts

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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.

#kleinianlimitset #kleiniangroup #fractals #mobius #mobiustransforms #mathematicalart #mathart #mastoart #perfectloops

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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.

#kleinianlimitset #kleiniangroup #fractals #mobius #mobiustransforms #mathematicalart #mathart #mastoart #perfectloops