Parameter Ray Atlas

This page illuminates various interesting points of reference on the M-set, giving then the corresponding ray arc angle. This is a visual map; the corresponding tables of angles and formulas are on the Tables page.

Visually, the stuff here is pretty boring. Don't expect any hot graphics. The interesting part on this page is the relationship between arc-angles and features. The Parameter Rays page describes how the rays are computed and numbered, while a mathematical intro is given on the Douady-Hubbard Potential page.

Bud Vases

One of the main results of Douady-Hubbard theory is that two rays always pinch off the base of buds. Here are some buds coming off the main cardiod.

The 1/15 and 2/15 rays (in red) surrounding the m(1/4) bud.

The 1/31 and 2/31 rays (in red) surrounding the m(1/5) bud.

The 17/127 and 18/127 rays (in red) surrounding the m(2/7) bud. Note that the 2/7'ths bud is in the middle (via Farey addition) between 1/3 and 1/4.

The 273/2047 and 274/2047 rays (in red) surrounding the m(3/11) bud. Note that the 3/11 bud is in the middle (via Farey addition) between 2/7 and 1/4.

Here's a picture of the 4/15 bud, with the rays [4369/32767, 4370/32767] pointing at its base highlighted in red. Its quite small. Recall 4/15'ths position on the Farey tree: [1/3 2/7 3/11 4/15 1/4] i.e. to the right(4/15) to the right(3/11) of the one in the middle(2/7) between 1/3 and 1/4.

Tenna Tips

In general, rays that lie at multiples of inverse powers of two point at tips of antenna.

Rays at inverse powers of two touch the tips of the prominent antennas of the m(1/3) bud.

The 1/4 ray (in red) landing at the tip of the antenna of the m(1/3) bud.

The 1/8 ray (in red) landing at the tip of the antenna of the m(1/4) bud.

The 1/16 ray (in red) landing at the tip of the antenna of the m(1/5) bud.

A closeup of the 1/4 ray touching the tip of the antenna of the m(1/3) bud.

Furcations

The antennas mounted on top of a bud indicate its cycle count. A three-pronged antenna is mounted on the n=3 bud, a four-pronged antenna is mounted on the n=4 bud, and so on. The antennas can be split into their component rays, as shown below. For bud n, the splitting rays are located at (2ⁿ+2^m-1) / 2ⁿ(2ⁿ-1) for m=1,2,...,n.

A theoretical treatment is given by Devaney, Moreno-Rocha, Geometry of the Antennas in the Mandelbrot Set (or mirror here)

The bifurcation of the antenna at [9, 11, 15] / 56 on the m(1/3) bud.

The trifurcation of the antenna at [17, 19, 23, 31] / 240 on the m(1/4) bud. Note the artifacts, these are discussed on the ray page.

How can we be sure of the above fractions? Here, we split the 17/240 and 31/240 rays down the middle. We can see clearly bow the red edge points at the quadrifurcation.

The quadri-furcation of the antenna at [33, 35, 39, 47,63] / 992 on the m(1/5) bud. Note the artifacts, these are discussed on the ray page.

The mono-furcation of the antenna at the end of the m(1/2) bud isn't at all obvious, because, well, there's no obvious visual signpost. However, the formula tells us its at [5,7]/12. We've colored here so that the sharp red edge points exactly at it.

Friends

The buds that sit on top of buds, buddies or friends. These are always split off by the two rays (2ⁿ+2)/(2ⁿ-1)(2ⁿ+1) and (2ⁿ⁺¹+1)/(2ⁿ-1)(2ⁿ+1). The periods of the bulbs are 2n, and, of course, the first ray angle serves as the generator of the period-doubling cycle. (And, quite prettily, the generator doubles n times before wrapping around, doubles n-1 more times to wrap, then 1 to get back to the start. So its indeed 2n. )

The primary bud due west of the m(1/2) bud is separated by [2,3]/5 and thus has a period q=4 with cycle (2,4,3,1).

The primary buddy of m(1/3) is split off by [10,17]/63, and has a period q=6 with cycle (10,20,40,17,34,5)

The primary buddy of m(1/4) is split off by [18,33]/255, and has a period q=8 with cycle (18,36,72,144,33,66,132,9)

The primary buddy of m(1/5) is split off by [34,65]/1023, and has a period q=10 with cycle (34,68,136,272,544,65,130,260,520,17)

Double Troubling

Due west of the main cardiod are a set of smaller and smaller bulbs. If we count these, assigning n=1 to the period-2 bulb off the main cardiod, then we have that the period of each bulb is 2ⁿ. The smaller of the two rays that pinch these off are given by s^-_n = a_n / (2^{2^n-1}+1) where a_n = a_n-1 * (2^{2^n-2}-1) + 1 and a₁=1. The other ray is of course s⁺_n= 1-s^-_n.

In the limit, this appears to converge to 0.412454033640107 which is the Thue Morse codeword. Its not obvious to me why it appears here; but I haven't thought about it either. (See, for example, Thue Morse L-systems for its relevance to fractals. See also A Fresh Look at Number for the occurrence of complimentary (gray-code) Thue-Morse in the symbolic dynamics of the logistic equation.) The codeword is the Farey number of 0.418979789366342 but this number seems to be unknown.

The primary bud due west of the m(1/2) bud is separated by [2,3]/5 and thus has a period q=4 with cycle (2,4,3,1).

The bud due west of the bud due west is rayed by [7,10]/17 and thus has a period q=8 with cycle (7,14,11,5,10,3,6,12).

The bud due west ... is rayed by [106,151]/257 and thus has a period q=16 with cycle (106,212,167,77,154,51,102,204,151,45,90,180,103,206,155,53)

The bud due west ... is rayed by [27031,38506]/65537

Mini Me

The largest mini-M-set on the real axis is at c=(re,im)=(-1.75,0). The main M-set can be mapped into, but not onto, using a simple binary-codeword expansion algorithm, as defined on the Tables page. For any feature on the main bulb, the algorithm provides the ray angle for the corresponding feature on the mini-bulb.

The two rays entering its tail are 3/7 and 4/7. It has period q=3.

The two rays splitting off the primary bud are 4/9 and 5/9. The bud has period q=6, that is, twice the period of the mini-cardiod.

In general, the rays that go into the tails of mini-me's seem to be of the for p/(2ⁿ-1) for some integers p, n. The following pictures show some of these. Unfortunately, they are quite messy, because the ray algorithm breaks down in this area. As mentioned elsewhere, there doesn't seem to be any way f fixing this algorithm, and I don't know of others.

This one shows the rays 3/31 and 4/31. The 3/31 ray is the red-blue discontinuity coming in from the right side, and heading straight into the tail of the bud. The 4/31 ray comes in from the left, gets interfered with by the busted algorithm but if you look at it just right, you'll see that it goes into the tail as well. This mini-me is the largest one on the longest antenna spoke of the 1/4 bud.

The [5,6]/31 rays.

The [5,6]/31 rays from a distance.

The same view, using ordinary coloration. So which mini-me was that? Why, the largest one on the smaller antenna of the 1/3 bud.

There seems to be a pattern, but its hard to describe & intuit. For example, the [7,8]/31 rays enter the tail of the largest mini-me on the longest antenna of the largest mini-me of the 1/3 bud. The [9,10]/31 rays pinch off the 2/5 bulb, while the [11,12]/31 rays go to the tail of the mini-off-the-mini-off-the 1/2 bud.