Building Trees

by: Edward Spiegel & Eric Wenger

Methods for Exploration

One of ArtMatic Designer’s greatest strengths is the freedom allowed for many styles of rewarding exploration. It is possible to create beautiful art with either very limited or very detailed understanding of ArtMatic’s underlying principles. There are many different strategies that can be applied to creating works with ArtMatic. Below is a description of some strategies that have proved very productive. They are by no means mutually exclusive and are often used in combination.

The “Zen” Strategy

A system is mutated at random, zooming in and out to discover images. This requires no understanding of the underlying principles, but even the most knowledgeable artists find exploration liberating and a source of raw material that can be further manipulated and molded. ArtMatic’s imagespace is so large that there are many worlds that will only be discovered through this type of exploration.

The “Template” Strategy

The “Template” strategy is effective for both creating new works and for developing intuitions and deeper understanding. It’s a method where you choose an existing ArtMatic file as a departure point for exploration. The starting point could be one of the example files provided, or your own. If you are still developing your intuitions and understanding, you should spend some time exploring the effect that particular parameter changes have, and seeing how changing tile components affects the system. As you become more advanced, you will be able to use a system as a starting point and know what components to add or change to achieve a desired effect.

The Structure Presets folder provides many structural starting points for Tree building. You can add your own templates to it.

The “Modified Zen” Strategy

Finally, there is the “Modified Zen” technique where selective mutation and randomization are used to explore the richness of a particular system using any of the preceding strategies. Beginners will find this technique an interesting way to discover new systems and to develop their intuitions.

Start by locking all of a system’s functions and parameters by shift-clicking the lock icon of any unlocked parameter.
Next, selectively unlock parameters. Only unlocked parameters are affected by mutations and randomization.
Mutate the system to see how mutations affect it. Use the Mutations dialog or the randomizing dice.
The Mutations dialog is invaluable for such exploration. Locking all of a tile’s parameters makes the tile immune from change. You can turn on the Mutate Function Type option in the Mutations dialog, and see what effect mutations have on an individual or select group of function(s).

One of the best ways to deepen your understanding is to explore your favorite systems and examples. You can quickly develop a sense for the contribution made by the components in any system. As you explore more and more systems, you can develop a sense of how to manipulate a system to create different images and effects.

As you learn, you will find that you’re able to take elements from one system and use them in your own systems, aiding design. Ultimately, the key to creating ArtMatic art is to combine discovery, chance and design.

A Word from Eric Wenger

Before diving into techniques, a few words from ArtMatic’s creator, Eric Wenger, may prove enlightening.

Eric received a letter from a user who was frustrated by not being able to predict how ArtMatic systems behaved. Eric wrote:

If it can cheer you up, I don’t understand ArtMatic either! I mean that as soon as several operators are connected and interact on each other the mathematical equation quickly becomes way too complex to be of any use.

But it does not matter.

ArtMatic lets me interact and experiment with the system. I don’t have to fully understand why it does this or that. I can still play with it and learn how it behaves. So, even if I don’t quite understand the system mathematically, I can develop a feel for it. And now that I have some experience, I can more quickly find or create a structure and settings that yield interesting results.

The order of the system’s transform components is important. You get very different results by inverting things. Complex system behavior can be mastered empirically.

What seems most helpful to me is to have a broad but clear understanding of the different classes or categories of transformations, and how to interpret the inputs and outputs.

Designing a Tree

When designing or trying to understand ArtMatic trees, it is helpful to understand the ways that components can function in structure trees. Where a function falls in the tree will influence how it behaves. A couple of categorization methods can help you to understand the structure and functioning of a particular tree.

It is important to keep in mind that many components can play multiple roles – the precise function being determined by the components’ placement in the tree and its connections. You can use components in ways that resist categorization or defy their original intent, and discover astounding new images and effects in the process. In order to get oriented, though, it is useful to think of components as falling into a few simple categories (even though the categorization is a bit of a fiction).

When exploring a new system, it is a good idea to discover the schema – the tree’s general abstract structure. The schema is a coarse outline of the tree that breaks it into functional units. You will find that complex trees often have very simple schemas with many tiles working together in functional groups.

To a degree, all systems share the following “classic” schema:

Space transforms/space distortions/motion over time
Surface/texture/shape generation
Shading/color manipulation

Each of these schematic units might be quite complex, and, because tiles can be connected to multiple inputs and outputs, it can be difficult to isolate these abstract structural units. Even so, the attempt to identify them will help you gain insight into the tree’s deeper structure. It should be pointed out that in a very complex tree, you often find individual branches that are schematically complete (i.e. they have this basic structure) and mixed together to create a composite image (each complete branch being an independent image mixed by a mixing component).

The basic functional categories are:

Space transform/distortion functions:
These components transform or distort and remap a system’s geometry and generally have the same number of inputs and outputs. They take a 2D or 3D space and remap it.

The most basic space transform functions are the 22 Scale & Offset and 22 Rotate functions that scale and rotate the incoming space respectively.

The 22 Twirl component is a good example that distorts a normal Euclidean space by warping it into a whirlpool, and there are more complex remappings like 22 Complex inverse that sometimes defy description.

Think of these components as functions that map the points of the incoming space to new locations. Typically, these components are the first ones in a system. They establish the geometry in which the subsequent surface, texture, space or object is drawn.
Space translation/rotation/mirroring functions:
This category is a special kind of space transform function that doesn’t change the scale of the space. These functions are particularly useful for DF modeling as they always keep the Distance fields accurate.
See: Building 3D Objects: DFRM guide.
Surface/texture generators/shaders:
These components map a point in space to a particular elevation or color or both. They take multiple input values (usually spatial coordinates) and generate a single output value (elevation, Mask, DF field) or a vector value (RGB color, RGBA color + Alpha).

These components provide a surface or a texture for the space or objects which feed it. The 21 Ax+By+C (plane) and Gaussian Dot components are good examples of surface generators. Most components in this group may also be used as mixers. Context determines whether the component acts as a texture shader or a mixer.

There are many components that output both a color texture and a surface elevation, such as 24 MultiFractal noise # or 24 Color Regular tiles, for example.
Mixers:
These components mix the outputs of multiple branches/inputs into a single composite surface or image. Mixers can include mathematical operators like a simple addition, logical functions (Maximum / Minimum), or any operation that takes 2 or 3 inputs for a single output like a surface or a texture generator. Mixers can also work with packed vector data like the various packed logic or Maths tools. For example, a typical mixing component would be 21 interpolate or 21 Logic tools.
Filters:
These components have the same number of inputs and outputs and modify or remap the incoming values. All of the 1D (one input/one output) functions can be used as filters. For example, the 1D sine function will remap incoming values so that the output is restricted to values between -1 and 1; as a result, steadily increasing input values become cycling output values. If a sine filter is used at the output stage, the result is an undulating surface. By contrast, applying a random filter will result in the output value varying from the input value by a random amount.
Color functions:
These functions can have 3 inputs and 3 outputs 3:3 and transform and filter the input colors like Color Controls. They can also translate the incoming points/pixels between different color representations such as RGB to HLS (and the reverse). As with 3D space transform functions, they manipulate 3D vector data by assuming the data is an RGB color. Some components with 1 input and 3 outputs 1:3 can generate a color according to various algorithms, for instance: 13 RGB gamma or 13 Main Gradient.
Flow and data controllers:
This category is different in nature from most ArtMatic components. They are meta-functions that don’t generate anything, but are important for flow control and data formatting.
Pack/Unpack:
A pack component enables the 3 values of an RGB component or a 3D component to be packed into a single packed value. Special functions that have “packed” in their names can act on packed values (usually to mix them). This mechanism allows RGB/3D values to be passed through a single thread rather than three separate threads.

There’s also a 4-input Pack component. There are unpack components that take one input (a packed stream) and split it into its 3 or 4 constituent streams.

Iterations and loopers like the 23 Looper component control the evaluation flow, and can be used to create recursive and/or iterative calculations.

Universal Rules for Designing Good Trees

Analyse your needs and try to make a schema of the task at hand. Identify the input dimensions and output type. Is it 2D or 3D? Does the output need an alpha channel? Is this a surface? Is it textured? The answers to these questions will provide an overall Tree structure.
Divide the task into smaller chunks to simplify the problem. Test each chunk separately to ensure they work as intended.
Use Compiled Tree to make a function re-usable and name the CT meaningfully.
If a part requires several components and serves a limited purpose, use Compiled Tree to keep the overall structure of the Tree as clear as possible.

In ArtMatic Designer, a good knowledge of the components and their functions is extremely useful when learning how to implement a particular behaviour.

The Mysterious World of Complex Dynamical Systems

A comprehensive guide to image creation is beyond the scope of this manual. While many simple ArtMatic systems produce easily predictable results, it is easy to create systems with varying degrees of unpredictability.

You can easily create or discover systems that will surprise even ArtMatic masters because it is possible to create trees which implement complex mathematical or geometrical systems that have inherent unpredictability.

One of the joys and (sometimes) frustrations of complex dynamical systems (the technical term for many ArtMatic systems) is that a system may inherently defy predictability due to its extreme lack of linearity. With such systems, it may not be possible to precisely predict how a system will respond to different input values and parameter settings.

This unpredictability is often apparent when zooming far in and out of a system, or when modifying its settings. However, there’s a lot of wonder and beauty to be discovered by chance. The trick is knowing how to harness those discoveries and mold them. It isn’t necessary to understand the underlying mathematics to have great success with ArtMatic. But by applying some of these lessons, you will develop intuitions and understanding that are more important than the underlying mathematics.

To precisely design the (nearly infinite) range of possible images from scratch would (even if it were possible) require a fairly deep understanding of a pretty tricky (and fairly new) branch of mathematics.

This level of deterministic control is effectively impossible, because many systems defy predictability or credulity!

Consider this: when Benoit Mandelbrot first ran a program to plot the equation for which he is best known (and which has captured the imagination of so many), he believed that the fuzzy irregular graph being plotted was the result of a bug in the computer hardware. Entire books have been devoted to exploring the Mandelbrot Set and the images it can create, and the Mandelbrot Set is just a single ArtMatic component. Imagine how hard it would be to pre-design the kind of images generated by a combination of tiles, each of which was as unpredictable as the Mandelbrot Set!