In the parameter window we select the Formula tab (there are five tabs: Parameter, View, Light, Formula, Coloring - see image at the right side).

We then change the iteration formula currently used by clicking with the left mouse button onto the icon beneath the formula name - see image, labelled with number (2)

ChaosPro will then open a window in which all formulas currently available are presented in a tree view.
There is an item called AttractorConstructor.cfm, if collapsed, expand it to see the contents. There is a formula named "Sphere". That's the one we are looking for! We use it! Simply double click onto that formula.

ChaosPro has a preview mode in which the first few thousand pixels are drawn quickly in order to show you the basic shape of the object. Whenever ChaosPro needs to redraw the fractal it first draws the preview.

We now can rotate the object: We click into the fractal window with the left mouse button and move the mouse around. This will rotate us around the object. Although it looks like the object itself rotates, mathematically "we" are rotating around the object. But in principe this does not make any difference for us as long!

To be even more exact we do not rotate around the object, but around the view point: Let's switch to the View tab, there we can see the view point which is (0/0/0). And the object, i.e. our sphere, is centered around (0/0/0). And due to that it seems as if we are rotating exactly around the object.

If the object by chance would not be a sphere, but rather some other complex sophisticated object, which lies around anywhere in the space, then it would run out of round, if we rotate...

Let's practice this and change the view point: Either complicated, by entering some values, or easily and more intuitively: Hold down the Shift key and "grab" the fractal with the left mouse button (i.e. press left mouse button inside the window and move the mouse): By doing that we do not rotate around the sphere, but move around the view point.

Well, and now stop holding down the Shift key, grab the fractal again, move the mouse. Oh! It runs out of round!

Fig. View Tab

All these actions can also be done in the corresponding tab, i.e. in the View tab: The sliders and entry fields found there in principle do exactly the same: Rotate, zoom and such stuff. Whether you prefer to use them instead of directly grabbing the fractal is up to you.

If activated, ChaosPro will determine the middle of the object while doing the preview render and set the view point to that point every time the fractal changes. It thus "follows" the object and makes sure the object is halfway "centered" in the window.

This might be very reasonable if you are wildly searching for interesting fractals and you must constantly "find" the fractal as it may be outside the current window. As such you then the must constantly readjust the view point so the visual focus is at the center of the object.

On the other hand it can be very bothersome if by accident the checkbox is activated!
Suppose you've just repositioned the fractal so it is perfectly centered in the window, then you shortly adjust the light settings and some other parameter and suddenly ChaosPro discards your view point and sets it to what it thinks is the center of the object.

Let's go back to the main topic- our sphere. Basically very boring! What can we do with it to make it more interesting?

Well, ChaosPro constantly generates points which then are drawn. The lighting algorithm then detects points, tries to find a surface using neighboring points and "throws" a light ray on it to determine the light intensity. And the resulting object, the sphere, then is drawn.

Now what would be if we "displace" the points "by random" immediately before they are drawn?

Let's do that - let's add another formula after the sphere formula: See the image at the left side: You need to click the right button: The button immediately above the "desired" button lets you change the current formula (that's not what we want), and our button lets one add another formula.

Click onto the button.

The new formula which performs a random based displacement of a point in 3D space, is in the folder AttractorTrafo.cfm and is labelled "Perlin based FBM".

After the formula has been added we can see the effect almost immediately: Oops! Our sphere is distorted!

As a side note: We should make sure the checkbox "Final" is checked with our new formula "Perlin based FBM" so that the formula is applied after the input for the next iteration has been determined, immediately before the point is drawn. Otherwise the input for the next iteration is the result from the random displacement which could be nice but if not desired...

• Scale=1
• Scale Factor=1.5
• Scale Step=0.5
• Octaves=7
• In Coloring Tab Speed set to 0.1 (so that the colors do not change so much...)
• Zoom out a little bit, rotate

Finally it should not be too difficult to obtain an object which looks like a stone or a comet or a crinkled piece of paper (assumed you chose a proper color...)

To sum up: Originally there was a sphere. By the subsequent formula (Perlin based FBM) the sphere was distorted. And now imagine what happens if you apply the Perlin displacement formula to a more complex base shape...

1. Now lets check the checkbox "Hide" for the formula "Perlin based FBM", so we can see the base shape, our sphere.
2. Now we change the sphere formula: Press the button as shown in the image aside. In the formula window expand the folder "Attractors.cfm" and choose the formula labeled "Polynomial A/B".
3. Enter the following parameters manually:

   Displace:	0.42/0.57/0.65
   P1:	0.76
   P3:	0.44
   P5:	0.15

4. Move/rotate the object so it fits into the window.

Fig. Search Button

Great many combinations of parameters lead to just an endless cloud of points which cannot be drawn correctly.
Due to that ChaosPro (as well as Chaoscope, another very nice program! You should definitely try it out!) an automatic search function: This search function picks parameter values randomly, checks whether the resulting fractal can be drawn. If yes, it stops, if no, it continues. This task lasts until ChaosPro has found parameters, which seem to form something reasonable. Whether the resulting fractal then is nice or rather boring...well - ChaosPro cannot know that.

Ok, let's click onto the button which shows a magnifying glass. Does a new fractal appear?
In any case a counter will appear in the window where the fractal is drawn: It shows you how many parameters ChaosPro has already tested while searching for new reasonable parameters.

If the counter disappears, then ChaosPro thinks, the current parameters lead to something useful. If the counter does not stop increasing, then you can press onto the search button again: ChaosPro will stop the current search and restarts from scratch. Sometimes this helps, sometimes not.

If the counter disappears and the window is black, there are two reasons:

1. The fractal is outside the current view area (perhaps it is behind us, or too wide left/right/up/down). So we zoom out (mouse wheel is the most easiest thing) in order to check whether there is a fractal anywhere.
2. If all stays black, then bad luck, the search yielded no valid result. We then continue with the search and keep on clicking onto the search button. And at any time ChaosPro will find something. Keep on searching, you won't break anything! Some people spend hours clicking onto the search button and watching the results.

Not later than now you will become quite fast in mouse handling and using the mouse wheel: When using the search function you need to constantly adjust the position of the object: Sometimes it is far away then you need to zoom in, the next object is too near and you need to zoom out in order to see it completely. Sometimes the object is not centered and is somewhere near the edge of the window and you need to center it (you remember...holding down Shift key will let us "move" the object instead of rotate it). And not later than now - if you use the search function - you will recognize the benefit of the checkbox "Follow Object" in the view tab which automatically adjusts the viewpoint so that the object is halfway centered: It spares yourself some work with the mouse.

Fig. Search Settings

If you have found an interesting object then pause for a moment: The object looks nice, but how would slight variations of it look like? What if ChaosPro would change the parameters by random only slightly? ChaosPro can do that: The slider at the left side of the magnifying glass specifies how strong the parameters should be changed. Normally the slider stays at 100 (unit: percent), that means, the parameters will be chosen completely at random from the defined range of values.

Set the slider to 0.5%. That will instruct ChaosPro to change the parameters only by 0.5% of their defined range of values based on the current value. This is ideal in order to get only slight variations or mutations. If you want stronger variations, choose a higher value.

Now click again onto the magnifying glass. You will see: The fractal looks quite similar to before, but it has changed somewhat. Sometimes it has changed to be more interesting, sometimes it will be more boring. If you don't like the way the parameters are taking, then press Ctrl-Z or choose the menu item "Fractal/Undo", in order to go back to the previous settings.

Fig. Parameter Change

Using the search function in ChaosPro is one way of finding interesting objects. But of course you can change the parameters yourself by directly entering numbers. And regarding that I now want to describe another small new feature which ChaosPro offers:
ChaosPro allows you to change a number in an entry field not only by typing in a new number, but also using the mouse. This feature is "general" and not limited to the new fractal type "Attractor": Whenever you see an entry field with a number inside, you can change it with the mouse. Now how does it work:

• Choose a digit from the number using the mouse (by selecting a digit with the left mouse button).
• Keep the left mouse button pressed, do not release it.
• The digit which you've just selected defines how strong the number will change.
• After that move the mouse up/down while holding down the left mouse button.

If there is the number 0.77903204 and for example you choose the 0.77903204, then you can easily explore how the fractal looks like, if the parameter is 0.78903204, 0.79903204, 0.80903204. This is easy going with the mouse.
Round about 20 pixels on the screen correspond to a change of the selected digit. By selecting the proper digit you can determine how much the number actually changes. As ChaosPro restarts the rendering each time a parameter changes you can see quite fast, how the fractal responds to changes of its parameters.

Enjoy yourself exploring fractals!

Fig. Presets

Let's try this out in order to show something of the flexibility of this fractal type: Let's choose three formula sequences, each sequence consisting of a single formula - a linear transformation formula (which effectvely does a matrix multiplication plus a vector addition). ChaosPro has a little-known feature, the so called "Presets": These are different default settings for fractals, ment to be starting points for exploring. If you like you can create your own additional presets, starting points for fractal exploration. You obtain a list of presets by clicking with the left mouse button onto the small arrow within the formula button (at the right side of the formula button) - or by clicking with the right mouse button onto the formula button. A popup menu will appear which lists the currently defined presets.

The list might look different than shown in the image, that's because I am writing this tutorial before the final release of ChaosPro - but the idea behind that stays the same.

We now select the menu item "Select IFS": Now our fractal will be changed, the parameters and formulas assigned to this preset will be assigned to our fractal.

Fig. Formulas

A quite interesting fractal. As you can see, the formula with the name "Scale, Shear, Rotate, Displace" has been assigned three times to it. Of course the three formulas have different parameters. This is what makes it interesting.

You can also see that "weight" and "color" is different for each formula:
The "weight" defines how often ChaosPro will choose this formula compared to the other one. If formula #1 has weight=1 and formula #2 has weight=8, then formula #2 will be chosen 8 times as often as formula #1.
And "color" assigns a unique color to the formula which allows a coloring formula to reference that parameter and create interesting coloring effects.

Animation Window

• Please open the animation window and add the current fractal: Simply click onto the button "Add". A copy of the current fractal will be added to the animation.
• Then set Scale=0 on all three formulas in order to remove the random displacement.
• Add the fractal with the new settings to the animation by clicking the button "Add" again. A copy of the current fractal will be added.
• How much time do you have? How much time did it need to render the fractal? My computer needed 1 minute for the fractal, I have 2 hours spare time, thus I can now render 120 frames. This will give a smooth animation I think...ok, why not? I drag the second key frame in the animation window and pull it down until there are 120 frames between the first one and the second one.
• Ok, finished, rendering the animation can start, please click onto the button "Start" in the animation window.
• After choosing a file name ChaosPro will ask for the compression to use. I always use "no compression", as I always will use another animation editing tool which allows me to experiment with different compression/quality ratios, depending on my needs: If I want to upload an animation to my homepage, then the size should not be too big. If I want to show a friend such an animation, then the size does not matter, quality is what matters! This is of course best done with a perfect animation which then can be downsized to your needs.

It's very easy to create an animation. And by viewing the animation you can see the effect of the "Scale" parameter. It seems to be only a multiplicator for the displacement: From 0.4 (displacement) slowly moving to 0 (no displacement).

If you like you can rotate the fractal during, zoom in/out, you can add dozends or hundreds of keyframes, just as you like. I only show you a few basic possibilities, just let your imagination run wild.

Render Mode

But there are also other render modes: My favorit is the "Flame" mode, well-known due to flame fracals (you might google for that). The main trick done by this algorithm is that it counts how often a pixel on the screen gets hit and then the logarithm for that will be calculated - just to get to the point. As it operates on screen pixels the 3D effect basically is thrown away - it's not possible to render a light and light is very important for the perception of 3D. but on the other hand the object looks transparent and unveils its inner structure. Due to the logarithm applied to pixels the object still looks like 3D (dense areas seem to be in front, sparse areas seem to be far away), but it does not look realistic, it looks unnaturally beautiful.

Interestingly the flame coloring algorithm not only works for flame fractals, but quite well also for many other 3D fractals (although not ment for them!). Just choose the flame mode for our current fractal: Choose the "Parameter" tab, choose the "Flame" render mode.
And suddenly our fractal is rendered - well - transparently?

One thing to mention: The flame coloring algorithm needs a proper coloring formula which makes use of the additional information and creates correct colors from the density. Thus you need the coloring formula "Weighted Sequence" or "FlameSequence". The IFS by default has "Weighted Sequence", and due to that we do not need to change something here. But if you explore your own objects and the object looks boring it might be due to the wrong coloring formula.
Using a suitable coloring formula creates astonishing effects.

Fig. Presets

Ok, enough talk: How do we get a Flame? Using Presets! Simply click onto the formula selecton button using the right mouse button. And from the list of presets choose "Select Flame 2D".

The next step is the same as in Apophysis or UltraFractal: You start a random search for new useful parameters. So you do in ChaosPro - the magnifying glass instructs ChaosPro to perform a new search.

But beware, a short note: Flames are somewhat special: If you look at the formula of the flame, then the overall structure is that the flame master formula contains round about 50 completelty independent small formulas.
You can see that in the parameters listed: There are many of them! And normally a flame fractal only uses only a few of those 50 formulas, the others not. Otherwise the flame fractal would be too mixed up - and in order to be "nice" for a human being, there must be some kind of order in the chaos. Additionally too many active sub formulas just let rendering time increase.

Fig. Flame Search

So we need to tell ChaosPro to search, but choose only some of the 50 sub formulas. How do we do that?

Well, the formula for the flame fractal uses a special feature of ChaosPro made for such issues: It tells ChaosPro that there are many sub formulas inside the main formula. And ChaosPro then offers the possibility to choose some of them at random and then randomly chooses parameters for these formulas.
This is all done if you right click with the mouse onto the search button. See image at the left side: A small popup menu will appear which offers us the possibility to choose 5, 10, all or just keep the current ones and then to choose the values of the remaining parameters at random.

So choose "5 Vars" from the section "All Formulas": This will activate exactly 5 sub formulas from the first flame formula, then choose values for the remaining parameters at random, then choose another 5 sub formulas from the second flame formula, for the third, fourth, etc., until all formulas have been randomized.

Ok, now search can start: If you've chosen the menu item, ChaosPro starts the search. By the way, ChaosPro keeps in mind what you have chosen. So the next time you want to search you only need to click onto the search button with the left mouse button and do not have to right click, choose item from popup menu.

After some time you for sure have found an interesting flame fractal. Then you would like to render a few mutations, perhaps it gets more interesting: As before we just set the slider aside to 5%. And then we right click (!!) onto the search button and choose the item "Keep" in the section "All Formulas".
This instructs ChaosPro to keep the current sub formulas and only slightly changes the parameters. The sub formulas itself define the overall structure of the flame, if another sub formula would be chosen, then the flame would change completely. Using "Keep" prevents that. The result then are mutations.

So much for flame fracals. The 3D-Flame function is labelled "3D Superfunction". If you like you can replace the "normal" flame functions with that one and then continue your search. Have fun!