This is the second part of a previous post.
The figure was generated with our Chaos Game Explorer tool, using the algorithm described at
and as presented in Barnsley’s books (Fractals Everywhere, 1988; The Desktop Fractal Design HandBook, 1989).
The game was played N = 100,000 times by randomly placing a point within an n-gon (polygon with n vertices), using different combinations of vertices (n) and scale ratios (r), and by coloring in white the emerging patterns. Some combinations produce patterns somehow resembling ancient calendars, medallions, rings,… from different ancient cultures.
For the above figure, I used n = 12 and r = 0.30.
Running the algorithm by coding the pixels in different colors reveals that the patterns are just the result of partially overlapping the same n-gon across many scales of observations. Did ancient cultures know about this technique?
Just for fun, you may want to try with other values, then run searches in Google Images for ancient calendars, medallions, rings, etc and compare results. Share your images and let me know if you found something interesting. I’m documenting results.
Fractals Miner: Fractal Patterns and Growth Phenomena – Theory, Experiments, & more. Available now at
Research the fractal geometry literature. Use the images tool below a result to view beautiful patterns or recrawl search results to build your own curated collection.
Note: Image below was created with Manglar, an experimental tool under development.
Closing a gap in the fractal geometry theory: Definition of fractal topography to essential understanding of scale-invariance.
One year old, but very relevant these days. Very important and enlightening research article.
A better understanding of scale-invariance by means of defining fractal topography opens the door to many practical applications. This was something loosely suggested in the literature, but not fully addressed. Great job!
The Chaos Game Explorer is our most recent tool. It was developed to help users replicate many of the patterns found in the fractal geometry literature. The tool is available at
Again, for those at the intersection of IR/Data Mining/Chaos/Fractals, check this out
Google matrix analysis of directed networks, by Leonardo Ermann, Klaus M. Frahm, and Dima L. Shepelyansky
A great article.
My research career is now in its first full circle.
We have recently launched the Bifurcation Diagrams Explorer. This is a tool for examining the behavior of low dimensional nonlinear dynamical systems.
Well, what does that have anything to do with information retrieval (IR)?
If you are an IR guy at the intersection of nonlinear dynamics, you already probably know that bifurcation diagrams are relevant to:
- Google search inefficiencies and that popular sites are like attractors on the Web.
- The growth and evolution of spatial networks can be modeled with bifurcation diagrams.
So the implications to social media, search, and data mining are there, if you can grasp the relevant research out there.
I wonder how long it will take for pseudo-scientific marketers/seos to prey on that, as they tried in the past with LSI/LSA, LDA, Vector Theory, and few other IR topics.
Bifurcation diagrams are used in the study of dynamical systems and are applicable to a wide range of fields: from the modeling of biological populations and financial systems to the modeling of chemical reactions and nonlinear circuits, to mention a few.
We have developed a new tool for students and researchers interested in Nonlinear Dynamical Systems, called the Bifurcation Diagrams Explorer. It is available at
This tool lets you explore many of the bifurcation diagrams found in the literature, providing a visualization of the underlying behavior of a dynamical system as a parameter c is changed. We assume that you have a basic knowledge of bifurcation diagrams and dynamical systems.
The tool is powered by our Minerazzi Grapher, a lightweight PHP class that generates all kind of graphs through a web browser. No additional libraries or software needed. A great resource for introducing users to Chaos Theory.