Smooth Koch curve using analytic tree fractals

smooth koch curve

Smooth Koch curve

The Koch curve, one of Mandelbrot’s “monsters” becomes a little more friendly when plotted using smooth (analytic) tree fractals.

For details on this curve, refer to this paper.

13 thoughts on “Smooth Koch curve using analytic tree fractals

      • Well, my math background is definitely not extensive. More than layman, far less than a professional, so quite a bit of it is over my head. Also, I tend to go straight for the applications, and take it on faith that everything was derived correctly, lol!

        Apart from some minor wording suggestions, there is one question I have (and again, the answer may be clear when I look at the paper in more detail). This basically makes fractals continuous (a fascinating idea), but it looks like the fractals are still disjoint. That is, the sub-trees are still treated as separate units; it isn’t that there’s a single mathematical formula that can generate this approximation, correct? Heck, it may not even be possible (I haven’t thought this through).

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      • There is a single analytic equation that describes the complete tree with all its branches (the equation in Theorem 2). The reason that works is due to the multivalued integral on Image 2. This is an integral of the unit function u(s) which is 1 and -1 between nodes and is 0 at a node. When it is 0, just after that it becomes 1 and -1 and the (multivalued) integral branches left and right. Then when u(s) becomes 0 again, it it splits into branches again. This formulation in Theorem 2 really is a single equation that genuinely described all the branches. Does that make sense to you?


      • No worries – the pseudo code is recursive for efficiency but it could have been written as a loop that plots every path from root to canopyand you wouldn’t see branch pointsanywhere in the code.They would result when the overlapping paths part ways. Thanks for the read!


      • My pleasure. Ironically, one reason I keep getting side-tracked with your paper is that it keeps making me think, which makes me go off on a tangent (ha!).

        For instance — and this may have been your point — your path theorem brought to mind a tree = a set of paths. An analytic tree could then be drawn by drawing each path as a curve that passes through each branch point in the path. So a tree with N paths becomes N curves.

        A tree could be drawn non-recursively in this way (just loop N times with the curve function for each set of paths), but this assumes the existence of the set of paths, which would have had to have been generated recursively, unless this process allows skipping that step.

        Again, I need to study your paper more 😛

        If this is what you were getting at, then kudos! If not, then kudos anyway for getting me to think!


      • The same happened to me when i wrote the paper. I kept thinking: oh, i should add this or that. I didn’t add the part about a tree being the set of all its paths from root to the branch tips. I wrote about that in an unfinished paper (in installments): . It was that realization and lemma 2.3 in the new paper (a tree is defined by any of its paths) that led me to develop an equation for the most simple path (the ultimate right hand branch) and then extend that to a full fractal tree.

        So yes, we could loop through all the paths, each of which is defined by the specific route through the u(s) function: 1,1,-1,1,-1,-1 etc. seen from that perspective, each path is fully defined by that sequence of 1 and -1. That is essentially a binary number whose number of bits corresponds to the number of node generations. I described this in the earlier paper and hesitated to include in in the new paper: if you multiply that binary number with the div phi(s) function, you get a specific path. That binary number then becomes an address for that specific path AND defines it mathematically. I didn’t include this in the new paper because it takes the brain into whole different (and cool) direction 🙂

        I’m glad the paper triggers further thoughts. It is partly why I put it out there: this a new direction for tree fractals and people should build on that further.

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