Archive for the ‘Fractals’ Category

Koch Snowflake

Tuesday, January 5th, 2016

Koch Snowflake by Nick Berry.

From the post:

We didn’t get a White Christmas in Seattle this year.

Let’s do the next best thing, let’s generate fractal snowflakes!

What is a fractal? A fractal is a self-similar shape.

Fractals are never-ending infinitely complex shapes. If you zoom into a fractal, you get see a shape similar to that seen at a higher level (albeit it at smaller scale). It’s possible to continuously zoom into a fractal and experience the same behavior.

Two of the most well-known fractal curves are Hilbert Curves and Koch Curves. I’ve written about the Hilbert Curve in a previous article, and today will talk about the Koch Curve.

There wasn’t any snow for Christmas in Atlanta, GA either but this is one of the clearest and most complete explanations of the Koch curve that I have seen.

Whether you get snow this year or not, take some time for a slow walk on Koch snowflakes.


Practical Fractals in Space

Friday, October 9th, 2015

Highly entertaining presentation on fractal curves and how to determine when things are both close in a search and close in terms of distance.

Updates from Complexity Explorer

Wednesday, October 7th, 2015

Updates from Complexity Explorer

Among other news:

Sid Redner on his Introduction to Random Walks tutorial [interview]

From the post:

Three tutorials coming soon: We are in the process of developing three new tutorials for you. Matrix and Vector Algebra, Information Theory, and Computation Theory. Stay tuned! And in the meantime, have you taken our latest tutorials, Maximum Entropy Methods and Random Walks?

Current courses: Fractals and Scaling and Nonlinear Dynamics are happening now! You can still join in these two fantastic courses if you haven’t already. Fractals and Scaling will end October 23rd, and Nonlinear Dynamics is set to end December 1st.

Agent-based Modeling: The Agent-based Modeling course has been delayed and will now be launched in 2016. We will let you know as soon as we have a clearer idea of the timeframe. You just can’t rush a good thing!

If you haven’t visited Complexity Explorer recently then it is time to catch up.

It is clear than none of the likely candidates for U.S. President in 2016 have ever heard of complexity! At least to judge from their overly simple and deterministic claims and policies.

Avoid their mistake, take a tutorial or course at the Complexity Explorer soon!

Complexity Updates!

Thursday, March 12th, 2015

The Complexity Explorer (Santa Fe Institute) has posted several updates to its homepage.

No news courses for Spring 2015. The break will be spent working on new mathematics modules, Vector and Matrix Algebra and Maximum Entropy Methods, due out later this year. Previous Santa Fe Complexity Courses are online.

If you need a complexity “fix” pushed at you, try the Twitter or Facebook.

If you are more than a passive consumer of news, volunteers are needed for:

Subtitling videos (something was said about a T-shirt, check the site for details), and

Other volunteer opportunities.


The Complexity of Sequences Generated by the Arc-Fractal System

Sunday, February 22nd, 2015

The Complexity of Sequences Generated by the Arc-Fractal System by Hoai Nguyen Huynh, Andri Pradana, Lock Yue Chew.


We study properties of the symbolic sequences extracted from the fractals generated by the arc-fractal system introduced earlier by Huynh and Chew. The sequences consist of only a few symbols yet possess several nontrivial properties. First using an operator approach, we show that the sequences are not periodic, even though they are constructed from very simple rules. Second by employing the ϵ-machine approach developed by Crutchfield and Young, we measure the complexity and randomness of the sequences and show that they are indeed complex, i.e. neither periodic nor random, with the value of complexity measure being significant as compared to the known example of logistic map at the edge of chaos. The complexity and randomness of the sequences are then discussed in relation with the properties of associated fractal objects, such as their fractal dimension, symmetry and orientations of the arcs.

Very heavy sledding but I suspect worth the effort. Recalling the unexpected influence of fractals on computer science.

In any event, the mental exercise will do you good.

I first saw this in a tweet by Stefano Bertolo

The golden ratio has spawned a beautiful new curve: the Harriss spiral

Friday, January 16th, 2015

The golden ratio has spawned a beautiful new curve: the Harriss spiral by Alex Bellos.


Yes, a new fractal!

See Alex’s post for the details.

The important lesson is this fractal has been patiently waiting to be discovered. What patterns are waiting to be discovered in your data?

I first saw this in a tweet by Lars Marius Garshol.

Math, Choices, Power and Fractals

Tuesday, January 6th, 2015

How to Fold a Julia Fractal by Steven Wittens.

From the post:

Mathematics has a dirty little secret. Okay, so maybe it’s not so dirty. But neither is it little. It goes as follows:

Everything in mathematics is a choice.

You’d think otherwise, going through the modern day mathematics curriculum. Each theorem and proof is provided, each formula bundled with convenient exercises to apply it to. A long ladder of subjects is set out before you, and you’re told to climb, climb, climb, with the promise of a payoff at the end. “You’ll need this stuff in real life!”, they say, oblivious to the enormity of this lie, to the fact that most of the educated population walks around with “vague memories of math class and clear memories of hating it.”

Rarely is it made obvious that all of these things are entirely optional—that mathematics is the art of making choices so you can discover what the consequences are. That algebra, calculus, geometry are just words we invented to group the most interesting choices together, to identify the most useful tools that came out of them. The act of mathematics is to play around, to put together ideas and see whether they go well together. Unfortunately that exploration is mostly absent from math class and we are fed pre-packaged, pre-digested math pulp instead.

Even if you are not interested in fractals or mathematics, this is a must read post! The graphics and design of the page have to be seen to be believed. Deeply impressive!

If you are interested in fractals or mathematics, you will be stunned by the presentation in this post.

I am going to study the techniques used on this page. I don’t know if they will work with WordPress but if they don’t, I will create HTML pages and link to them from here. Not all the time but for subjects that would benefit from it.

Steven Strogatz says in the tweet I followed to this page:

This is SO good:… You’ll understand imaginary & complex #’s, waves, Julia and Mandelbrot sets as never before

That is so true.

BTW, did you catch the “secret” of mathematics above?

Everything in mathematics is a choice.

Very important to remember when anyone says: “The data says/shows/proves….”

BS. The data doesn’t say anything. The data plus your choices in mathematics gives X result. Not quite the same thing as “The data says/shows/proves….”

In a data driven world, only the powerless will be unable to challenge both data and the choices applied to it.

Which do you want to be? Powerful or powerless?

Fractals in D3: Dragon Curves

Thursday, February 27th, 2014

Fractals in D3: Dragon Curves by Stephen Hall.

From the post:

Dragon fractal

This week I am continuing to experiment with rendering fractals in D3. In this post we’re looking at examples of generating some really cool fractals called dragon curves (also referred to as Heighway dragons). This post is a continuation of the previous one on fractal ferns. Take a look at that post if you want some basic info on fractals and some links I found useful. Fractals are a world unto themselves, so there are plenty of interesting things to be investigated in this area. We are just scratching the surface with these two posts.

Great images, complete with source code and explanation.

See the Fractal entry at Wikipeida for more links on fractals.

Fractal Ferns in D3

Saturday, February 22nd, 2014

Fractal Ferns in D3 by Steve Hall.

From the post:

This week I have been busy exploring the generation of fractals using D3. The image above is a “fractal fern” composed of 150,000 tiny SVG circles generated using some surprisingly simple JavaScript. Fractals are everywhere in the nature world and can be stunningly beautiful, but they are also useful for efficiently generating complex graphics in games and mapping applications. In my own work I like to cast a wide net and checkout new data visualization tools and techniques – you never know when it may come in handy. Some knowledge of fractals is definitely a good thing to have in your toolbox.

There are three parts to this post. The first part will be light introduction to fractals in general with a few links that I found useful. Next, I put together three examples that explore generating fractal ferns using JavaScript and provide some insight into how a simple algorithm repeated many times can produce such a stunning final result.

The last part deals with scaling an SVG to fit the browser window which often comes up in doing responsive design work with D3 visualizations. The solution presented here can really be applied to any data visualization project. If you look closely at the examples, they are being generated to an SVG element that is initially 2px high by 2px wide, yet scale to a large size in the browser window without the need to re-generate the graphic using code as the window size changes.

If you are interested in fractals after reading Steve’s post, Fractal over at Wikipedia has enough links to give you a good start.

Fractals are a reminder that observed smoothness is an artifact of the limitations of our measurements/observations.

The observed smoothness of subject identity in most ontologies is a self-imposed limitation.

Fractal Fest

Monday, December 2nd, 2013

Fractal Fest

From the tweet by IBM Research:

FRACTAL FEST on IBMblr It’s Tumblr like you’ve never seen it!. Turn your favorite blogs into fractal works of art with the IBMblr Fractalizer. And keep following along as we explore the contours,…

Some things are worth mentioning simply because they exist.

Fractals are one of the few things that fall into that category.


Design Fractal Art…

Monday, October 21st, 2013

Design Fractal Art on the Supercomputer in Your Pocket


From the post:

Fractals are deeply weird: They’re mathematical objects whose infinite “self-similarity” means that you can zoom into them forever and keep seeing the same features over and over again. Famous fractal patterns like the Mandelbrot set tend to get glossed over by the general public as neato screensavers and not much else, but now a new iOS app called Frax is attempting to bridge that gap.

Frax, to its credit, leans right into the “ooh, neat colors!” aspect of fractal math. The twist is that the formidable processing horsepower in current iPhones and iPads allows Frax to display and manipulate these visual patterns in dizzying detail–far beyond the superficial treatment of, say, a screensaver. “The iPhone was the first mobile device to have the horsepower to do realtime graphics like this, so we saw the opportunity to bring the visual excitement of fractals to a new medium, and in a new style,” says Ben Weiss, who created Frax with UI guru Kai Krause and Tom Beddard (a designer we’ve written about before). “As the hardware has improved, the complexity of the app has grown exponentially, as has its performance.” Frax lets you pan, zoom, and animate fractal art–plus play with elaborate 3-D and lighting effects.

I was afraid of this day.

The day when I would see an iPhone or iPad app that I just could not live without. 😉

If you think fractals are just pretty, remember Fractal Tree Indexing? And TukoDB?

From later in the post:

Frax offers a paid upgrade which unlocks hundreds of visual parameters to play with, as well as access to Frax’s own cloud-based render farm (for outputting your mathematical masterpieces at 50-megapixel resolution).

The top image in this post is also from the original post.

I first saw this in a tweet by IBMResearch.

Introduction to Complexity course is now enrolling!

Tuesday, February 5th, 2013

Santa Fe Institute’s Introduction to Complexity course is now enrolling!

From the webpage:

This free online course is open to anyone, and has no prerequisites. Watch the Intro Video to learn what this course is about and how to take it. Enroll to sign up, and you can start the course immediately. See the Syllabus and the Frequently Asked Questions to learn more.

I am waiting for the confirmation email now.

Definitely worth your attention.

Not that I think subject identity is fractal in nature.

Fractals as you know have a self-similarity property and at least in my view, subject identity does not.

As you explore a subject identity, you encounter other subjects identities, which isn’t the same thing as being self-similar.

Or should I say you will encounter complexities of subject identities?

Like all social constructs, identification of a subject is simple because we have chosen to view it that way.

Are you ready to look beyond our usual assumptions?

Update: Introduction to Complexity [Santa Fe Institute]

Wednesday, December 5th, 2012

The Santa Fe Institute has released the FAQ and syllabus for its “Introduction to Complexity” course in 2013.

The course starts January 28, 2013 and will last for eleven (11) weeks.

Lecture units:

  1. What is Complexity?
  2. Dynamics, Chaos, and Fractals
  3. Information, Order, and Randomness
  4. Cellular Automata
  5. Genetic Algorithms
  6. Self-Organization in Nature
  7. Modeling Social Systems
  8. Networks
  9. Scaling
  10. Cities as Complex Systems
  11. Course Field Trip; Final Exam

Funding permitting there may be a Complexity part II in the summer of 2013.

Your interest and participation in this course may help drive the appearance of the second course.

An earlier post on the course: Introduction to Complexity [Santa Fe Institute].

MongoDB Index Shootout: Covered Indexes vs. Clustered Fractal Tree Indexes

Friday, September 7th, 2012

MongoDB Index Shootout: Covered Indexes vs. Clustered Fractal Tree Indexes by Tim Callaghan.

From the post:

In my two previous blogs I wrote about our implementation of Fractal Tree Indexes on MongoDB, showing a 10x insertion performance increase and a 268x query performance increase. MongoDB’s covered indexes can provide some performance benefits over a regular MongoDB index, as they reduce the amount of IO required to satisfy certain queries. In essence, when all of the fields you are requesting are present in the index key, then MongoDB does not have to go back to the main storage heap to retrieve anything. My benchmark results are further down in this write-up, but first I’d like to compare MongoDB’s Covered Indexes with Tokutek’s Clustered Fractal Tree Indexes.

MongoDB Covered Indexes Tokutek Clustered Fractal Tree Indexes
Query Efficiency Improved when all requested fields are part of index key Always improved, all non-keyed fields are stored in the index
Index Size Data is not compressed Generally 10x to 20x compression, user selects zlib, quicklz, or lzma. Note that non-clustered indexes are compressed as well.
Planning/Maintenance Index “covers” a fixed set of fields, adding a new field to an existing covered index requires a drop and recreate of the index. None, all fields in the document are always available in the index.

When putting my ideas together for the above table it struck me that covered indexes are really about a well defined schema, yet NoSQL is often thought of as “schema-less”. If you have a very large MongoDB collection and add a new field that you want covered by an existing index, the drop and recreate process will take a long time. On the other hand, a clustered Fractal Tree Index will automatically include this new field so there is no need to drop/recreate unless you need the field to be part of a .find() operation itself.

If you have some time to experiment this weekend, more MongoDB benchmarks/improvements to consider.

Fractals in Science, Engineering and Finance (Roughness and Beauty)

Saturday, January 7th, 2012

Fractals in Science, Engineering and Finance (Roughness and Beauty) by Benoit B. Mandelbrot.

About the lecture:

Roughness is ubiquitous and a major sensory input of Man. The first step to measure and simulate it was provided by fractal geometry. Illustrative examples will be drawn from the sciences, engineering (the internet) and (more extensively) the variation of financial prices. The beauty of fractals, an unanticipated “premium,” helps in teaching and bridges some chasms between different aspects of knowing and feeling.

Mandelbrot summaries his career as the pursuit of a theory of roughness.

Discusses the use of the eye as well as the ear in discovery (which I would call identification) of phenomena.

Have you listened to one of your subject identifications lately?

Are subject identifications rough? Or are they the smoothing of roughness?

Do your subjects have self-similarity?

Definitely worth your time.

First seen at: Benoît B. Mandelbrot: Fractals in Science, Engineering and Finance (Roughness and Beauty) over at Computational Legal Studies.