As some of you may know, I’ve done a few puzzles based off of fractals now. Because of this, something has come to my attention as I’ve been playing around with the coding sequences in the player puzzle designer. It seems to me that it’s hypothetically possible to bind up extremely significant quantities of genetic data inside a relatively finite space using a model akin to the ‘Koch Curve RNA’ puzzle I posted not long ago.

For the intents of this article, ‘iterations’ shall refer to the number of passes of the function that are made on the RNA. The best way to exemplify this is to take the model I made, which is shown at iteration 2. It is based off of the Koch Snowflake, which is made by taking an equilateral triangle (that is, all internal angles are equal to 60 degrees each, for a sum of 180) and adding another triangle a third of the size of the original at the middle third of each leg of the original triangle. Done correctly, Iteration 0 will be an equilateral triangle, and iteration 1 will look something like a star of David. So, iteration 0 is 3 legs, and iteration 1 has 12. On iteration 2, you repeat the process and add triangles to the middle third of each leg, each new triangle being a third of the size of the origin triangle. Done correctly, iteration 2 will have 48 legs, if I recall. With each iteration, the curve gains 4/3rds in length without adding tremendously to the space it consumes. That is to say, you can draw a perfect circle around iteration 0 and run the system to iteration five thousand- while you may have grown the overall length of the curve by 4/3rds 5,000 times, you still will not have reached the edge of the circle. Infinite length within a finite space.

Of course, a line takes up no volume by existing, and may be as small as we imagine it to be, as is convenient. That and the fact that RNA takes up space means that for RNA, the idea of fractal folds taking place MUST be a matter of proportions. The higher the iteration of the fractal, the more bases will have to be present overall in order for the design to work. That also means that you’re binding up more bases in the design of the fractal, because the larger the number of bases, the higher iteration number you can achieve, and therefor the more space is saved.

As of the time of this writing, I’ve attempted fractal curves with two types of systems and a total of four different models from those systems. I’d like to talk primarily about the second kind of system, which is the fractal curve. I’ve made three models based off of the fractal curve, one of them unpublished because of problems I’ll explain shortly. The first model is the Koch Curve, which, as described above, follows a value system of 1/3. I’ve also made a curve on a value system of 1/2, and was nearly able to bring it out to iteration 3 before I hit the structure limit. The third and unpublished RNA fold is one based off a value system of 1/4, and remains unpublished due to the very high number of bases necessary in order to achieve anything stable past iteration 1.

What I’ve noticed in playing with the curve systems in particular is that the higher the fraction (for example, 1/2 > 1/3), the lower the amount of complexity. At iteration two, the curve with the value of 1/4 became intensely complex, and an exorbitant number of bases were necessary in order to make the fold work. At iteration 3, the curve with the value of 1/2 is still fairly simple, but more bases are necessary in order to bring the structure out past that level.

What the level of complexity implies for RNA seems to be how quickly and effectively the design will tie up bases in folds and pairs. For smaller RNA chains, in order to reduce the size of the strand, a curve with a value of 1/2 would probably be more than adequate. For very large RNA strands, a curve with a value of 1/4 or 1/5 may be quite beneficial in reducing the overall volume that the strand occupies.

I’ll also add that once they bond together correctly, the ‘arms’ of an RNA curve are very stable with respect to neighboring arms, and will very rarely break apart to form delinquent bonds when and where they should not.

I… I don’t remember where I was headed with this, but those are my observations. Perhaps more later if I think of more. I hope this wasn’t too confusing.