Fractal Fascination, Aesthetics, and Applied Science

Fractals are back? They were never gone!

What are fractals anyway? Let’s have a look:

It’s got something to do with geometry and nature, although most of Mandelbrot’s classic textbook is full of mathemetical formulas and scientifics plottings. Plus some complex colorful graphics that might have been done by a painter on drugs, albeit not with the same level of detail and geometrical precision. The kind of images commonly known as “fractals”.

Apart from some recent music releases with fractal titles, there is little data to back my pop culture claim that we’re on the verge of a new fractal hype. But whenever I check my websites’ insights, there’s a constant, albeit small, number of readers coming to one of my oldest blog posts that is still online: Fractals – Chaos Theory, Butterfly Effect, and Psychedelic Images. I did a quick web search and found that there is still active research and a continuous stream of papers and books about fractals, just that I never noticed in the past years. The scientific fractal theory assists our daily life by improving wi-fi connectivity, city traffic, medicine, image compression and computer vision.

Chaos Theory, Memetics, and Fractal Philosophy

Slightly overlapping with fractal science, memetics, and the buddhist worldview, the Chaos Theory, also called Butterfly Effect, describes the effect that a little trigger or information sometimes gets amplified to have a large impact, like the proverbial butterfly’s wing’s wind contributing to creating a storm on another continent. That’s science, philosophy, and an metaphor giving hope to everyone who might feel insignificant and overlooked in this world.

Likewise, simple iterative formulas create a complex form like the Mandelbrot set that contains microscopic repetitions of its macroscopic shape when zoomed into infitely.

Fractal Aesthetics in Art, Architecture, and Nature

We don’t need to spend more time staring at computer screens, as we can go outside and discover fractals in nature.

Fractals and similar mathematical geometry has also influenced architecture, like gardens being built inspired by the Fibonacci Sequence or the related golden spiral ratio.

Self-similar repetitive patterns can be found in plants, deep sea organisms, coast lines, clouds, as well as analytics and data science. Looking for fractal patterns in big data, human behavior or social media memetic, we might spot the next butterfly effect inspiring a breakthrough discovery.

Fractals in a Broader Sense

Linguistically, the word fractal derives from the Latin fractus, meaning broken or fragmented. In its mathematical sense, it is a general term for complex, often self-similar structures that can be described or generated by fractal formulas. In popular culture, the term fractal(s) is even broader and sometimes used figuratively. A text search for “fractal” in my digital photo album yields many colorful patterns, artwork and decoration of different origin and appearance, including mould on a wooden panel.

Musicians, tracks and artists were named related to fractals, inluding some new techno tracks released recently. Nice, but not groundbreakingly innovative.

Innovation beyond the Seen Unseen

As a creative developer, I wonder what a fractal aesthetic beyond trippy party pictures might look like today.

Early fractal literature featured formulas, monochromatic sketches, and occasional multi-color images often limited to a default color palette of scientific computers. I remember computer software back in the 1980s when it took computers several hours to draw a 300 x 200 pixel still image of a fractal. As a nerdy teenager, I sometimes entered a fractal formula prompt before leaving for school in the morning to discover an unseen artwork after coming back in the afternoon. Historic computers were far from zooming into fractal formula visualization to reveal self-similar patterns, so we used color palette animation to add a fake feeling of interactivity instead. When fractal image generation software advanced, so did the aesthetic aspect, and new books focused on “the beauty of fractals” rather than its mathematical background. Thanks to new graphics hardware and digital printing, the new wave of fractal visualization become excessively colorful and contrasting, creating a psychedelic experience inspired by and inspiring trips and party decoration.

Pictures of fractal illustrations in old textbooks, all rights reserved by the original authors and publishers:

• Algorithmen für Chaos und Fraktale by Dietmar Herrmann (1994, ISBN 9-783893-196333)
• The Fractal Geometry of Nature by Benoit B. Mandelbrot (1982, updated 2021, ISBN 978-1-64837-041-0)
• Ein Weg zur fraktalen Geometrie by Reinhart Behr (1995, ISBN 3-12-722410-9)
• Chaos and Fractals: New Frontiers of Science by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe (1992, updated 2004, ISBN 978-0-387-20229-7)

These are classic fractal textbooks. I’ll list more recent publications further below.

Abundance of Nerdiness and Colorful Cliché

After fractal software had become commonplace and everybody got bored about their colorful coffee table books, popular cultured seemed to forget about fractals – apart from my local library still providing the books that you can see in the previous paragraph. Media coverage about fractal theory and practice is practically zero now. People seem to have more important things to do than discovering inherent sense and beauty in science and nature. Fractal philosophy, spirituality, art, and applied science are always there, there’s just not much talk about it, unlike there was when Benoît Mandelbrot and Julia dug into the topic, when home computers first popularized colorful fractal visualization, or when stereograms, also known as squinting pictures or “the magic eye” allowed to creative three-dimensional artworks using two-dimensional book printing techniques. But perhaps we are on the verge of a new wave of fractals in popular culture.

We can still relive the explorative astonishment with a classic textbook, ready-made app, or a tutorial website like Jörg Kantel’s blog posts about the strange attractor, Mira, (2025, in German), or his 2017 Python script to draw a dragon curve fractal. Did I mention that fractals were never gone?

There is still an infinity of possible pictures that nobody has seen before!

• The Useful JS fractals generator feels quite classic and not artistically pleasant or innovative, but with its quick rendering you can experience self-similarity on the fly.
• Nico’s Fractal Machine courtesy of Science versus Magic is another classic fractal plotter, enhanced with animation and modern rendering speed.
• Zazov’s Online Mandelbrot Creator allows registered users to go beyond its classic algorithmic and coloring presets.
• Jakub Rak’s Fractal Generator 2021 showcases artistic images that sometimes resemble deep sea corals and creatures.
• GIMP comes with at least 4 fractal filters: Fractal Trace (Map), Fractal Flame (Render), Fractal IFS (Render), and the Fractal Explorer Render filter, none of which seems easy and beginner-friendly though.

The MandelBrowser app, and various other freemium online services, including AI-prompt-based fractal image generators, promise a colorful exploration with the surreal psychedelic neon aesthetics everyone might have expected.

So far, so good, but if you’re my age or you’re not into science or techno music videos, you might find it a boring cliché, disappointing the expectation of unseen discoveries.

Beyond the Cliché?

Using ready-made software and tweaking it with uncommon settings and less contrasting color palettes, inspired by architecture and nature, might be an alternative approach to seek a new aesthetic encounter with fractal art. A first glimpse at all the books and software that I mentioned before didn’t take me far, at least not yet, as you can see on my screenshot collages.

Maybe it’s not the point to invent a new fractal aesthetic, but rather try to understand why fractal science is important today.

Applied Fractal Science

Which notable scientific achievements have been made in recent years?

I already mentioned some areas of applied science powered by fractal theory like fractal antennae. There are many more:

• Fractal data compression as an alternative to statistical, dictionary-based, text compression, more suitable to compress images by detecting and describing their complex structures in fractal formulas. This is especially useful for images of complex natural photography with organic shapes that don’t fit the traditional square raster approach, including geology, geography, and medicine.
• Multifractal approaches are used to improve image recognition (computer vision) using Convolutional Neuronal Networks.
• Fractal patterns are also used to model and predict complex natural and societal processes and ecosystems, city planning, traffic, weather, medicine and finance.

As I’m no academic, I haven’t read any of the books listed below. Maybe you should, depending on your intellect and ambitions. However, the variety and consistency of scientific publication proves that there is a lot going on right now involving research and applied science related to fractals.

Scientific Sources and Further Reading

• Fractal and Fractional Analysis in Biomedical Sciences and Engineering by Andjelija Ž. Ilić (2025, ISBN 978-3-7258-4847-8)
• Fractional and Fractal Derivative Models for Anomalous Sediment Transport by Sun, HongGuangLi, ZhipengNie, Shiqian (2025, ISBN 978-3-11-134876-6)
• Optical and X-Ray Polarized Scattering for Porous Nanoparticles and Aggregates such as Combustion Soot by di Stasio, Stefano (2025, 978-1-0716-4618-2)
• Fractal Geometry in Human Geography and Planning by Cécile Tannier (2024, ISBN 978-1-394-30655-8)
• Statistics of natural images using hash fractal image compression by Kostadin Koroutchev and José R. Dorronsoro (2023, DOI:10.1145/973620.973661)
• Fractalopolis — A Fractal Concept for the Sustainable Development of Metropolitean Areas by Pierre Frankhauser (2021, in Complex Systems, Smart Territories and Mobility, ISBN 978-3-030-59301-8)
• Reconfigurable Antennas for Wireless Communication: Design Mechanism, State of the Art, Challenges, and Future Perspectives by Mingyue Zhang, Guangyu Xu, Renjing Gao (2024, International Journal of Antennas and Propagation)
• An Integrated Modeling Approach Combining Multifractal Urban Planning with a Space Syntax Perspective by Claudia van der Laag-Yamu, Akkelies Van Nes (2017, DOI:10.3390/urbansci1040037)
• Fractal Physiology and Chaos in Medicine by Bruce J. West (2011, updated 2013, ISBN: 978-981-4417-79-2)
• Fractional Derivates for Physicists and Engineers by Vladimir V. Uchaikin (2013, ISBN 978-3-642-33910-3)
• Fractal Antenna Design using Bio-inspired Computing Algorithms by Balwinder S. Dhaliwal, Suman Pattnaik, Shyam Sundar Pattnaik (2009, ISBN 9789815136357)
• Fractal Image Compression — Theory and Application by Yuval Fisher (1995, ISBN 978-1-4612-7552-7)
• Fractal Concepts in Surface Growth by Albert-László Barabási and Harry Eugene Stanley (1991, ISBN 978-0521483186)
• Fractals and Scaling in Finance: Discontinuity, Concentration, Risk by Benoît B. Mandelbrot (1997, 978-0387983639)

Different Meanings of the Term “Fractal”

What else does “fractal” mean and why did they choose the wording?

• Fractal as a software is both an instant messaging client and and a programming framework for previewing and building web design systems collaboratively, similar to Storybook or Histoire.
  • The web development framework got its name for its modular, sometimes also called atomic, approach to build websites from small, reusable components which can be assembled together in various ways to create larger structures, mirroring the mathematical fractal concept where detailed patterns repeat across different scales and form complex structures, reusing the same basic elements over and over.
  • The messaging client, also known as Gnome Matrix) used to have a controversial name that developers sought to replace with a more positive alternative. The name “Fractal” was proposed by Regina Bíró and chosen as a mathematical term that reflects the modular, scalable nature of the software, and because it evokes a sense of intricate structures emerging from simple parts, and an analogy for open, federated messaging networks built from many interconnected rooms and users, much like the “fediverse” of alternative social media communities.
• “Fractal” hardware includes the
  • American guitar amplifier brand Fractal Systems
  • and Fractal Design, a Swedish hardware company.
• “Fractal” culture includes computer games, and a variety of music tracks and artists.
• Wikipedia’s disambiguation page about fractal meanings is a good starting point for further explorations, and so are
• search engines specialized in scientific papers,
• and maybe even your local library!

Conclusion

Fractals and related effects are relevant for several quite different areas of science and culture. Fractals can help us find hidden effects and unlikely insights and inspiration, not only as scientists and artists but even as common people, we can enjoy their ease and beauty and let knowledge about the butterfly effect increase our hope and optimism knowing that small things aren’t necessarily insignificant.