Fractals 4 - Deterministic IFS and Common Mistakes in recognising Fractals

Summary

This lecture delves into deterministic Iterated Function Systems (IFS), explaining how to identify parameters like reflection, scaling, rotation, and translation for fractal generation. Examples include the Sierpinski gasket and a fractal tree, showcasing how initial objects transform into the same attractor based on defined rules. The lecture also addresses common misconceptions about fractals, emphasizing the importance of self-similarity across multiple levels, not just repetition or symmetry. It distinguishes between true fractals and patterns like the Cantor set or nested dolls, which do not exhibit infinite scaling. The session concludes with a nod to the significance of education, quoting Einstein on the essence of learning and setting the stage for future discussions on random IFS algorithms and chaos games.

Takeaways

• 🔍 Deterministic IFS (Iterated Function System) involves parameters like r, theta, phi, e, and f that correspond to reflection, scaling, rotation, and translation in both horizontal and vertical directions.
• 🌟 The initial object in an IFS can be anything, as the final product, or attractor, is determined by the rules of the IFS, not the starting object.
• 📏 Standard rotations like 90 or 180 degrees are easily identifiable, but non-standard angles require physical measurement.
• 🌲 An example of creating a fractal tree using IFS involves breaking down the tree into parts, scaling, rotating, and translating to form the trunk and branches.
• 🎨 Salvador Dali's 'Visage of War' painting is discussed as an example of fractals in art, where the self-similarity of skulls within skulls creates a recursive effect.
• ❌ Common misconceptions about fractals include thinking that symmetry or simple repetition creates a fractal, but true fractals require scaling and self-similarity across multiple levels.
• 🔢 The IFS table is a systematic way to record the transformations used to create a fractal, including scaling factors, rotational angles, and translation values for each color or part.
• 🌱 The concept of fractals extends beyond mathematics into nature and art, with Dali's painting and tree structures being examples of natural fractals.
• 📚 Education, as quoted from Einstein, is not just about what is learned but what remains after forgetting, emphasizing the importance of understanding and applying knowledge.
• 🔄 The next lecture will cover the random IFS algorithm, exploring randomness, chaos games, and the differences between random and deterministic IFS in generating fractal images.

Q & A

What are deterministic IFS (Iterated Function Systems)?

Deterministic IFS are a set of mathematical rules that define transformations such as scaling, rotation, and translation applied to an initial object to generate a fractal pattern. These transformations are defined by parameters r, theta, phi, e, and f, which correspond to reflection, scaling, rotation, and translation in both horizontal and vertical directions.

How does the initial object in an IFS transformation affect the final fractal?

The initial object in an IFS transformation does not affect the final fractal. Once the rules are defined, any initial object will eventually converge to the same fractal, which is the attractor of that particular IFS.

What are the standard rotations discussed in the script?

The standard rotations discussed are 90 degrees and 180 degrees, both in a counter-clockwise direction, which are considered positive angles. These rotations are easily identifiable and are used in the transformations within IFS.

Why is it necessary to measure non-standard scalings and rotations?

Non-standard scalings and rotations, which are not obvious amounts like half or one by four, require physical measurement of distances and angles because they cannot be easily identified. This measurement is necessary to accurately define the transformations in the IFS rules.

What is the significance of the origin of the coordinate system in IFS transformations?

The origin of the coordinate system in IFS transformations is significant as it provides a reference point for translations. It can be set to any point, often the lower left corner, but any symmetry available can be used. The rules will change accordingly.

How are the translational values determined in IFS?

Translational values in IFS are determined by measuring the distance the different parts of the fractal are moved in the horizontal or vertical direction. These values are crucial for accurately reconstructing the fractal pattern.

What is the role of the 'trunk' in the IFS tree example?

In the IFS tree example, the 'trunk' is created by using highly shrunken and rotated copies of the tree. This technique adds realism to the fractal by giving it a distinct trunk, which is essential for the overall structure of the tree.

What is the importance of experimenting with IFS rules according to the script?

Experimenting with IFS rules allows for the creation of different tree designs, simulating the variety found in nature. It encourages understanding of how slight changes in the rules can lead to diverse fractal patterns.

What does Einstein's quote about education imply in the context of the script?

Einstein's quote suggests that true education is not just about memorizing facts but about retaining the knowledge and understanding that remains after formal learning is forgotten. In the context of the script, it implies that the understanding of IFS and fractals is more important than memorizing specific examples.

Why is it a misconception to assume that symmetric objects are fractals?

Symmetry alone does not define a fractal. A fractal requires self-similarity across different scales, not just symmetry. The script clarifies that even though physical fractals may exhibit a limited level of scaling, true fractals must show a pattern that repeats at multiple levels of magnification.

What are common mistakes people make when identifying fractals?

Common mistakes include assuming that any symmetric object is a fractal, that repetition of a pattern is enough to define a fractal, and that objects like Russian dolls or nested bottles are fractals. The script emphasizes that fractals require self-similarity across scales and not just a single point of symmetry or a single level of repetition.