Six Degrees of Ramification – on Fractals, Models, Randomization (#Discrete #Math Class)

27 May

… of Fractal-ization?

I mean, with a question like:

“Random graphs are a fascinating subject of applied and theoretical research. These can be generated with a fixed vertex set … and edges added to the edge set … based on some probability model, such as a coin flip.

[First] Speculate (did you just say introduce ‘fuzzyness’ on the answer?) on how many connected components a random graph might have if the likelihood of an edge (v1,v2) being in the set E is 50%.

[Then] Do you think the number of components would depend on the size of the vertex set V? Explain why or why not.”

Where does one end up – BUT in a random corner of the cloud

– Enter Fractals

Again, from the realm of Aesthetics, it’s now commonplace (some may argue, even BORING!) to get to see these CGI-facilitated illustrations all over the place; as with most anything out there, I can bet you that someone already set their Ruby-on-Rails atop these equations, and well, the “there’s an app for that” slogan could yield a couple of interesting iterations..

So as to the math of fractals, since they’re graphs to begin with, let’s jump in, shall we? (and careful where one ends up at!)

“… One of the most amazing facets of mathematics is the experience of starting with a problem in one area of mathematics and then following the trail through several other areas to the solution (or several versions of the solution).

We *** illustrate *** [See? Gotta satisfy the Right Hemisphere with these most complex ones!] this with a problem that starts out as a problem in rendering the attractor of an Iterated Function System (IFS), which leads to a solution that involves finding an *** Eulerian *** cycle in a certain graph and then to finding generators for the multiplicative group of a finite field.

We start with an introduction to IFS fractals and the problem of generating an image of the attractor of an IFS.

[…and what is an] ITERATED FUNCTION SYSTEMS. The basic idea behind Iterated Function System fractals is that we wish to formalize the concept of self-similarity. That is, given an image like…  we [then] wish to formalize our notion that the set is made up of three smaller copies of itself…” (Mendivil, 2003)

Of course, thoughts of ‘Replicants’ jumpin’ through the roofs of LA in 2030 come to mind – yet, as usual, are we not merely extrapolating and drawing inferences, sometimes conclusions on already-existing processes?

– Enter Instantiation – CGI-Style

As I’ve mentioned elsewhere, the Computer Graphics Industry is one of the most complex, yet gratifyingly simply iterations of what Computing has become – and as the excerpt above includes, the need to draw up an illustration, based off of very precisely and accurately calculated principles?Timeless – and profitable, for those venturing into the realms of the unknown, coming back after spending a quarter billion dollars to see their ROI (Return on Investment) quadruple – and all off of a fable of runaway mining and organic blue giants? on a place with 60% of our gravity field?

So EBSCO Host, when queried “acm siggraph fractal instantiation” spat out a SINGLE answer – yet, how lucky does one feel when it states…

“We present a dynamic tree modeling and representation technique that allows complex tree models to interact with their environment.Our method uses changes in the light distribution and proximity to solid obstacles and other trees as approximations of biologically motivated transformations on a skeletal representation of the tree’s main branches and its procedurally generated foliage.

Parts of the tree are transformed only when required, thus our approach is much faster than common algorithms such as Open L-Systems or space colonization methods.

Input is a skeleton-based tree geometry that can be computed from common tree production systems or from reconstructed laser scanning models.

Our approach enables content creators to directly interact with trees and to create visually convincing ecosystems interactively. We present different interaction types and evaluate our method by comparing our transformations to biologically based growth simulation techniques.”

(.pdf is taking its time downloading… so)

(whoa!… so many equations!… guess there’s a CONNECTION?)

“”…focus on phototropism and gravitropism. Phototropism is the tendency of a given branch to grow towards the light direction. We estimate the effects of phototropism for each branch at the time the branch was growing using our temporal light model described above. Gravitropism controls bending of the branches either away from or towards gravity. While we compute the strength of the tropisms directly from the input tree, we later expose it as a parameter that the user can modify to control the
transformation behavior of the trees…” (ibid)

Well, let’s just say that yes, we’re pretty good at deciphering and explaining what’s ALREADY there – and yes, all of the work above, again, serves to build near-perfect backgrounds to a story? one that only happens within arrays and CPU’s, rendering farms, mice and tablets – and in out heads, as everything’s put together within our synapses?

Wikipedia starts right off the bat on its article on “Computer Generated Imagery” with an illustration of a “rural landscape,. generated with fractals… that looks very real” and the link? Bingo!

“Real landscapes also have varying statistical behaviour from place to place, so for example sandy beaches don’t exhibit the same fractal properties as mountain ranges. A fractal function, however, is statistically stationary, meaning that its bulk statistical properties are the same everywhere. Thus, any real approach to modeling landscapes requires the ability to modulate fractal behaviour spatially. Additionally real landscapes have very few natural minima (most of these are lakes), whereas a fractal function has as many minima as maxima, on average. Real landscapes also have features originating with the flow of water and ice over their surface, which simple fractals cannot model” (Wiki, 2013)

So if one were to flip a coin?… what would it be? connection or no connection?
@Kankuchito on the Interwebs


Mendivil, F. (2003). Fractals, graphs, and fields. The American Mathematical Monthly, 110(6), 503-508,510-511,513-514. Retrieved from

WIkipedia (2013) Computer Generated Imagery Retrieved from:

Wikipedia (2013) Fractals Retrieved from:

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