This uses an abrigded computation; with this new way of computation, it is impossible to distinguish between very large and very small exponents, but several common Ultra-Fractal coloration methods can be applied easiely.

You can download jan3lyap.zip, (3 KB, 11 KB unzipped, a very cheap download). This contains jan3.ufm, which must be copied to the formulas folder of Ultra Fractal, and jan3.upr, which must be copied to the parameters folder of Ultra Fractal. Please note: the formula "1. Sinus" (which was used to produce all the images on this page) adds an additional "B" to all periodicity numbers, except for 2 = AB, that means e.g. 5 = ABA+B = ABAB = AB = 2, 6 = AAB+B = AABB, 7 = AAA+B = AAAB &c.



Tilings


All tiles show the AB-plane from (0.6526, 0.6525) to (3.0, 3.0) and computations for a starting value x0 = 0. Below, you can see in the left column the periodicity sheme AB, in the middle column AAB and in the right column AABB. The value of b increases from to top to bottom, from 1.0 to 3.0 in steps of 0.25. Click on an image to see a larger version.



r = AB, b = 1.0r = AAB, b = 1.0r = AABB, b = 1.0
r = AB, b = 1.25r = AAB, b = 1.25r = AABB, b = 1.25
r = AB, b = 1.5r = AAB, b = 1.5r = AABB, b = 1.5
r = AB, b = 1.75r = AAB, b = 1.75r = AABB, b = 1.75
r = AB, b = 2.0r = AAB, b = 2.0r = AABB, b = 2.0
r = AB, b = 2.25r = AAB, b = 2.25r = AABB, b = 2.25
r = AB, b = 2.5r = AAB, b = 2.5r = AABB, b = 2.5
r = AB, b = 2.75r = AAB, b = 2.75r = AABB, b = 2.75
r = AB, b = 3.0r = AAB, b = 3.0r = AABB, b = 3.0





The value of b can be changed continuously. The periodicity sheme is a discreet value and therefor cannot be interpolated smoothly. But one can try a trick: assume we want to interpolate between AB and AABB. Then we could use a periodicity sheme that alternates between AB and AABB, like ABAABB. Further interpolations could be (AB)3(AABB) and (AB)(AABB)3. The next five pictures therefor show, for b = 2.5, the shemes AB, (AB)3AABB, ABAABB, AB(AABB)3 and AABB. Unfortunatly, this works not too well. There is an undenieable familarity between the three interpolating images, but they differ much from both AB and AABB.

Please note also that the bailout coloring algorithm has to be adjusted to the length of the periodicity sheme (since one iteration of ABABABAABB calls the function b*sin2(x+r) five times as much as one iteration of AB) and that the algorithm works less smooth for longer shemes.



r = AB, b = 2.5
r = ABABABAABB, b = 2.5
r = ABAABB, b = 2.5
r = ABAABBAABBAABB, b = 2.5
r = AABB, b = 2.5





Zooms


The following pictures show closer looks on the AB-plane. Of course, they are no longer suited for seemless tilings.


The first three show the same region (center=1.19861337910797101/1.20328450928910133 magn=7.41165881725675053 angle=314.999991690200648) for A6B6, with sligthly different b = 2.25, 2.30, 2.35:



Zoom 1
Zoom 2
Zoom 3



The picture depends also on the starting value x0, but in a rather different way: depending on the starting value, different branches of the ordered region become visible. The following three pictures are computed for AABB, b = 2.0, (center=2.07665/0.629025000000000002 magn=5.80888759802497823 angle=45.0000080824282519) and x0 = 0.1, 0.5, 1.1 (all other pictures on this page are computed for x0 = 0, expect for the next two):



Zoom 4
Zoom 5
Zoom 6



I computed some pictures for the values periodicity = 1975, x0 = 0.6 and b = 2.1. Two of them I found worth mentioning, (center=3.92702594155721382/3.41448984148751173 magn=5.01006329582636191 angle=192.680384164402714) & (center=4.35870348871219635/3.59804216515542159 magn=11.909569538894675 angle=120.391519800442004):



Zoom 7
Zoom 8



Comfort chair (head over feet, could also be an alien spacecraft), r = A4B2, b = 2.5, (center=1.9263276234976675/3.06898385736961257 magn=10.9467720081693971 angle=179.575604374688415):



Zoom 9



Design chair, same parameters (center=1.90669004327188293/2.72226844154556642 magn=9.37376846525635982 angle=359.575595723266402):



Zoom 10



Taken from here (center=1.834075/2.866975 magn=5.01158930025684396 angle=0):



Zoom 11



The same region for r = A3B2 lacks the chairs:



Zoom 12



The comfort chair/alien spacecraft (with slightly different zoom (center=1.90979297962913116/3.1066442762076972 magn=8.92887282044948013 angle=179.575604374688415)) in a bigger (800x800) picture:



Zoom 13



The next three pictures show subsequent zooms (for the same parameters as the two chairs): the third one seems to be anisotropic, while the first one clearly shows prefered directions.
(center=2.147917556875/2.5537410325 magn=596.617773840100472 angle=0)
(center=2.146526381425/2.5565233834 magn=13258.1727520022327 angle=0)
(center=2.1465520259845/2.55661389361 magn=220969.545866703878 angle=0)



Zoom 14
Zoom 15
Zoom 16



Again three subsequent zooms for r = A4B2 and b = 2.5.
(center=2.39309907253667371/3.65373734211842009 magn=5.65584660333022319 angle=269.575596406511724)
(center=2.225096610658204/3.64967740823151531 magn=15.0822576088805952 angle=269.575596406511724)
(center=2.21250919509244828/3.65109674574900924 magn=232.034732444316849 angle=269.575596406511724)



Zoom 17
Zoom 18
Zoom 19



Ant
r = A3B3 and b = 2.2
(center=1.2785/1.255025 magn=2.0285004310563416 angle=45.0000080824282519)



Zoom 20



Hurrican
r = A4B2 and b = 1.8
(center=0.542950000000000002/2.726125 magn=11.6177751960499564 angle=0)



Zoom 21



Light on
r = A3B3 and b = 3.0
(center=1.865375/3.01565 magn=5.01158930025684396 angle=45.0000080824282519)



Zoom 22



Light off
r = A4B4 and b = 2.3
(center=0.542950000000000002/1.8732 magn=6.5536167772589498 angle=224.483838620333791)



Zoom 23



First contact
r = A3B3 and b = 3.0
(center=2.428775/2.42095 magn=3.23532980143163344 angle=314.999991690200648)



Zoom 24



Penetration
r = A6B6 and b = 2.1
(center=2.585275/2.569625 magn=5.21614396557344984 angle=314.474367652433147)



Zoom 25



Symmetrie
r = A6B6 and b = 2.5
(center=1.10433476599854962/1.10619270961043764 magn=8.49139715325910438 angle=314.999991690200648)



Zoom 26







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Jan Thor
www.janthor.com
jan@janthor.de