Corrupt Text file in the formula database

ea.txt (Ed Algra) is not displaying properly, looks like a bunch of gibberish.

My ea.txt is the same, Kathy.

I periodically archive all formulas and, looking back, the problem seems to be a long standing one. This happened sometime between September 2014 (when the file was fine) and January 2015 where it was unreadable.

I've contacted Ed to ask him if he can restore the original version of the file.

Thanks for the feedback, Chris. It's awesome that you take the time and effort to archive the data.

Thank you very much, Frederik, for the help and for reaching out to Ed.

Thanks for the feedback, Chris. It's awesome that you take the time and effort to archive the data.

I get my backup program to do that once a month now, before that it was a bit random and on a "whenever I thought about it" basis.

I can recommend the practice. More than once I have found formula changes that have 'broken' old images of mine so this is insurance that I can find a copy that works properly somewhere in my archives. It's not always possible to contact the author to try and get the problem fixed and it's heartbreaking to think your image is more or less gone because of a formula change that you may not discover for months or years.

Ed has restored the ea.txt file now.

Here's ea.txt from 1/4/2008; I originally got it in UF4.

Avariant

Avariant is provided in both the M(andebrot) and the J(ulia) version. All options described here can be applied to the M- and the J-version
but it's always wise to start in the M-version and then to switch to the J-version from the selected point in the M-plane.

The formulas, as given here, are not mathematically new. Indeed two of the features used in each of the ufm’s are not original. The basic
principles are:

1. Every iteration combines two different mathematical formulas. We can see this elsewhere in UF, e.g. in some files in jam2.ufm. The
   Japanese fractal software ‘Flight from Fractals’ and ‘Raimuraito’, also have two formulas working together.

2. A procedure to reduce the amount of inevitable spirals in Mandelbrot- and especially in Juliafractals, as described by Albert Smooks*).
   This has consequences: instead of spirals we get curves of rupture – one or more axes of symmetry with similar images on both sides. The
   curved nature of these axes may obscure the symmetries but you will often get surprising results.

The two formulas we use in every iteration are indicated as A and B in the layer properties. They each have four modules:

a. The Mandelbrot/Julia formula and related polynoms. This and the next two modules have four sub variants with respectively 1, 2, 3 and 4
z terms. These sub variants are accessed by checking the ‘Non-standard’ box where a choice is made within the Polynoms section. Additional
to the number of z terms there is always a term with c in it. Thus a 3 z term selection will give 3 z terms + 1 c term. The c term is
shown as a function (above the Polynoms heading), but functions are selectable for any of the z terms though the default is always ‘ident.”

b. The formula of the French mathematician Samuel Lattes as explored by Benoit Mandelbrot and published by him in 1980. Mandelbrot was able to
simplify the formula into the form known today by his name (the Mandelbrot formula**). I have used the Lattes formula in a somewhat more
general shape.

c. The Talis formula, taken from ‘Fractal Explorer’. This too has been given a more general shape*.

d. the fourth module (Combi z&c) is also based on a polynom. Selection of this module makes three terms available as a unit as shown in the
drop down section ‘z&c.” These are respectively: 1, z only; 2, z & c combined; 3, c only.

N.B. Explanations of ‘c’ and ‘z’ may be found by studying the fractals literature in Wikipedia or elsewhere on the Internet. You also can find
details in the UF help file under ‘type setting’.

When combining formulas A and B different modules can be used within each. You are also able to vary the way they interact with each other by
changing the ‘decider’ (the decider is found within the General section). Formula B is disabled when the decider = 0, in this case only
formula A is active..

If decider = 1, at each iteration the formulas calculate in series, provided ‘parallel’ is not enabled; with formula A being followed by
formula B.

Formulas A and B can be set to work in series or in parallel. When parallel is unchecked and the decider = 1, at each iteration formula A
calculates to be followed by formula B. The formulas work in series, but there are several options.

One such scheme for the iteration in series is:

  z1 --> by A --> z2 --> by B --> z3,

Here z3 is the initial value for the next iteration. When ‘series options’ are at ‘normal’, then z2 only is an intermediate station. If it is
at ‘var 1’ then the initial value for the next iteration is the real part of z2, combined with the imaginary part of z3. However at ‘var 2’,
the real part of z1 is combined with the imaginary part of end value z3.

When parallel is enabled the ‘series options’ is renamed as ‘parallel options’ and the drop-down box makes additional choices available.

The scheme is as follows:

  z1 --> by A --> z2,  and  z1 ––> by B --> z3.  

The actions of both formulas are in fact parallel and mutually independent. The result of both are z2 and z3 and both have to be combined in
one or another way to give a value z as a starting point for the next iteration. For this joining there are many possibilities. We give here
11 possibilities: ‘sum’, ‘product’, ‘difference 1 en 2’, ’quotient 1 en 2’, arithmetic, geometric and harmonic mean, together with the two
options previously available when ‘in series’. Of course not all these possibilities provide usable images in all cases.

It’s also possible to adjust the ‘decider’ at other values than 0 or 1. For instance at decider = 3, then iterations #1 and #2 are only
carried out by formula A. At iteration #3 both A and B are carried out, but iterations #4 and #5 use formula A, then #6 with both, and so on.

So when the 'decider' is not zero the rule is:
If the number of the current iteration, divided by the decider as specified is an integer, then both formulas A and B are executed. If this
quotient is not an integer, then only formula A is executed, and in this current iteration formula B is ignored.

As stated previously ‘decider’ is found under the ‘General’ heading, where you will also find the ‘bail-out’ value and the different adjustable
bail-out types.

Modules ‘Mandelbrot’, ‘Lattes’, and ‘Talis’ are default in their most simple ‘normal ’shape, we only can change the power of z (not for
Lattes), and at Talis we also can adjust the Talis constant.

When ‘non-standard’ is chosen for any of the first three modules, the Smooks option also becomes available. However, module 4 is inherently
non-standard, so here the choice of Smooks options is always present.

Smooks operations.

A short survey of Smook’s method may be of assistance and particularly to those with some familiarity with the mathematical terms used in the
construction of fractals. It only uses functions already available in UF, particularly abs(). During every iteration the position of z is
controlled by the parameters set. In the simplest case z remains in the first quadrant of the complex plane (the north east part of the image)
when both the real part and the imaginary part are positive. When the orbit of z comes outside the NE part, the function abs() will act and
place z back in the NE part. In that case z will be reflected in relation to the real or/and imaginary axis, so the natural building up of
the spiral(?) will be disturbed. In this basic example the NE quadrant is passive and the others active. Of course it’s also possible to
divide the complex plane in other ways. In Avariant this idea has been worked out in the following six modes.

First a remark on the complex plane of our image. The centre point is 0/0, the half line to the east at 0 degrees, to the north at 90 degrees,
to the west at 180 degrees, to the south at 270 degrees. The maximum angle of 360 degrees coincides with 0 degrees on the half-line to the
east. (In math, the positive direction is anti clockwise.)

A brief description of the six modes follows, but it isn’t necessary to know the how they function to produce effective results. The modes can
be applied intuitively.

If modus ‘none’, then no Smooks operations will take place. Only partial scaling is possible as we'll describe later on.

Modus ‘original smooks’, the basic situation according to Smook.

Modus ‘3 segments’. First we have to adjust an angle alpha between 90 degrees and 360 degrees (default 360 degrees). This angle divides
the originally active area (NW, SW, and SE) into two zones, the first one from 90 degrees to alpha. The second from alpha to 360 degrees.
We distinguish the NO quadrant segment I, segment II the zone from 90 degrees to alpha, and segment III from alpha to 360 degrees. There are
two submodi: ‘disabled’: segments I and III passive, and II active. The second one, 'enabled': I and II are passive and III active.

Modus ‘4 segments’. Two angles have to be adjusted: angle alpha > 90 degrees and smaller than beta. So this second one, beta must be higher
than alpha and its maximum is 360 degrees. Default: alpha 90 degrees, beta 270 degrees. To simplify the adjustment with the ‘fractal
explorer tool’, both angles are combined into one complex number, gamma. Angle alpha is the real part of gamma, beta is the imaginary part.
Segment I again is the NO quadrant, segment II the zone between 90 degrees and alpha, segment III between alpha and beta, and segment IV from
beta to 360 degrees. Here too, there are two sub modes: 'disabled': segments I and III passive, II and IV active. In sub mode 'enabled' segments
I, II and IV are passive and segment III active.

In mode ‘S to N ’, all values of the orbit in the southern hemisphere are reflected to the north; in mode ‘W to E refl.’, the same with orbit
values in the west, they are reflected to the east.

After every iteration, when a Smooks operation takes place it will be finished by the ‘flipper question’. Parameter ‘flipper’ can be adjusted
at ‘none’, ‘only north west’, and ‘all’. At ‘north east’, for all points of the orbit in this quadrant the real and imaginary components will
be changed. If ‘flipper’ = ‘all’, the same occurs for every point of the orbit. Flipping, the change of the real and imaginary component of
z, means a reflection to the angle of 45 degrees to the real axis.

These modes greatly enhance the original suggestion of Smooks, although he had already indicated that more variations are possible. In
Avariant there also is another way to apply his idea: Smooks uses the function abs() for the total value of the orbit point z, but it is also
possible to apply it only on a part of z. In module a (Mandelbrot polynoms) and d (blended polynoms) z consists of a number of terms which
can be split up and combined in several ways. So when we have a function

z = a1(z) + a2(z) + a3(z) + c,

then it can be split up in a part u and v, for instance u = a1(z) + a2(z) en v = a3(z) + c. We use u to carry out the Smooks operations but
we don’t do that on part v. Afterwards we combine both u and v into one complex value z as a starting point for the next iteration. For both
other modules the practice can be similar. Lattes has a fraction and Talis has two terms, one of them a fraction. It’s possible to apply
Smooks on the total fraction, or only on the numerator or denominator separately.

The last variations are indicated in Avariant as ‘allocation’. When it is ‘all’, then we use the whole z value. The other options, ‘part 1’,
‘part 2’, etc. only work on the split part u. The remaining part, v, isn't changed.

Even when no Smooks operations are applied it is useful to have the facility to split z into u and v. In that case of course it’s not
necessary to have the Smooks option enabled. When we choose under ‘allocation’ another option than ‘all’, then a parameter ‘partial scale’
appears. This is a (complex) scaling factor that only works on the allocated part u, and not on part v of the total z.

The effect of Smooks operations is at first sight somewhat disappointing. The beautiful Mandelbrot set for instance is seriously deformed,
just as Picasso in certain periods distorted the human form. At least half of the coastline changes from a beautiful Norwegian fjord coast
into a dull, plain, sand beach. At other places we see amounts of pixels smeared out as unusable smudge, and possibly only some devotees of
‘modern art’ can use them. Other parts look as landscapes, seen from a plane, with strange perspectives. Using built in UF-functions as
‘stretch’ and ‘skew’, it will be possible to use them. But at some spots of the new coast line we’ll see elaborate structures with curved
axes of symmetry, mentioned earlier. Spirals are absent or rarely present there. It’s easy to find these useful areas. They provide a
wonderful basis for creating fractal art, using Ultra Fractal ucl-files and gradients.

In all cases it’s possible to work in the normal way to find Julia fractals, using the Mandelbrot set as a map. It’s a remarkable thing that
even the dullest parts of the 'deteriorated’ Mandelbrot set often will give useful Julia’s.

A final word

In Avariant we have 4 modules and with this limited amount innumerable variations will be possible. Nevertheless it’s easy to add other
modules with new mathematical formulas. It isn’t sensible to make them too complicated for the calculation times would become too long and
unduly extend screen rendering times as parameters are changed – including those caused by changing ucl’s and gradients.

Other effects are possible: for instance it is possible to change the scale of formula A in relation to B by ‘overall scale A’. Such a
parameter is lacking for formula B, as it isn’t necessary there.

Also we didn’t mention details of the individual modules, most of them are obvious and it will be easy for you to make effective use of the
options available..

) http://orbittrap.blogspot.com/2007_08_01_archive.html (Aug. 4, 2007)
) B. Mandelbrot describes this interesting story in The Beauty of Fractals’ of H-.O. Peitgen and P.H. Richter (1986) p.151-160.
) Also in the Talis formula more variations are possible. They have been extensively worked out by Tony Marshall in his ‘Talis and Friends’
(tma2.ufm). Maybe it's possible to implement some of these variations in a later update of Avariant.

Acknowledgement

Thanks to Cliff Tolputt for his kind positive critical remarks and his corrections on my broken English.

It has been updated since then, and is now fixed as Frederik noted. Just update your public formulas, and you'll have the latest version.

Thanks so much for your trouble in posting the full text here. Meanwhile, it has been restored in the formula database.