Recursive Fractal Art
What is Recursion
All around us are complex structures that began from a single point and branch outward; trees, rivers and their tributaries, snowflakes, ferns, and lightning. This complexity is not chaos; they are not created along randomly selected pathways. Instead, they are the same instruction performed again and again: branch to branch again to branch again, such patterns can be replicated mathematically.
Our eyes recognize this structure and find order in it; it is, after all, the shape of the universe. A branch looks like the tree it came from. A tributary looks like the river it feeds. This is recursion, and it is one of nature’s most reliable building tools.
The most interesting concept to see when we look at nature’s design is the level of shape repetition, from the source giant river, to the tributaries, to the side rivers, to the creeks, ending at the little trickles on the edge, they have the same shape. We call this self-similarity.
Structures that show this characteristic are called fractals, and can be studied mathematically both theoretically and in nature.
How to use the recursive fractal art tool
The Recursive Fractal Art tool lets you explore recursion starting from a preset — Spiral Tree, Snowflake, Flower, Diamond Web, Coral, Spiral, Stars, or Fern. Each preset uses one of six starting shapes: circle, square, triangle, line, diamond, or star. Select a color palette, then use the child rule sliders to set how each copy of the shape is transformed relative to its parent, how far it moves, how much it rotates, and how much it shrinks. In mathematical terms, these are called affine transformations: translation, rotation, and scaling applied recursively at every level.
Finally, set the depth, how many times the rule repeats. You can think of the fractal dimension as a number that describes how densely a shape fills a plane. A line is one-dimensional. A filled square is two-dimensional. A fractal sits between them: it meanders, so it is never truly one, but it also never fills the plane completely, so it never reaches two.
This tool indeed can generate beautiful images, images that look like they came from nature, but underneath it all, they are all equations. In our world, math and nature have the same underlying rules. The show rule button unmasks the math behind, manipulates the controls, and see how the calculations change.
What is affline transformation?
Fractal growth that varies by reflection, rotation, scaling, and translation, though not necessarily by angle or distance. The most common example is that of a fern; the parts of a fern frond vary by reflection, rotation, size, and translation; each part of the frond is an image of the others, though the angle and distances of parts vary.
What is recursive fractal art?
Fractal art is both mesmerizing and alien. The endless sweep of iterations looks like something that exists in our world, but on the other hand, does not.
What is fractal dimension?
A number with a value between one and two. A line is one, but a fractal will never be that, because it meanders and is length-wise unmeasurable. The same thought holds for surfaces; a fractal will never fill a surface because it endlessly iterates.
What is self-similarity in mathematics?
Self-similarity is a trait in fractals where the part is identical or in natural systems almost identical to the whole. It is endlessly similar no matter how far you divide.
What is an iterated function system?
An iterated function system is a generative math function that creates self-similar fractals varying by reflection, rotation, size, and translation.
Who first studied fractals mathematically?
Benoit Mandelbrot is considered the father of fractals. Working at IBM through the 1970s, he studied the Julia Set and developed what became known as the Mandelbrot Set, publishing his landmark work The Fractal Geometry of Nature in 1982
Where do fractals appear in nature?
Fractals appear in almost everything in nature, from galaxies to clouds to rivers. It would be easier, in all actuality, to list the things that are not fractals, and let everything else be considered a fractal.