Bridges 2017 is next week in Waterloo, Ontario, Canada! We’re all looking forward to it…Nick and his parents are also going, and I’ll get to see a lot of new friends I met last year in Finland.
In preparation, I’ve been creating a portfolio of binary and ternary trees. I’ve been coloring and highlighting the branches to make them easier to understand, so I thought I’d take the opportunity to share several of them. Mathematics just as it’s happening….
Here is an example of what I’m talking about.
Recall that we’re generalizing the idea of “left” and “right” branches of a binary tree — rather that just rotating left or right (and possibly scaling), we’re allowing any affine transformation to correspond to “left” and “right.” In order to avoid any confusion, let’s refer to the transformations as “0” and “1.”
In the tree above, the 0 branches are shown in black, and the 1 branches in red. Notice that sometimes, the 0 branches seem to go to the right, other times, to the left. This is a common occurrence when you allow the branching to be determined by more general transformations.
Here, you’ll notice that 0 and 11 take you to the same place — beginning with the trunk (the vertical black line at the right), if you follow one black branch, you’ll end up at the same blue node as you would if you followed two red branches. In fact, no matter what blue node you start at, following one black branch will always take you to the same place as following two red branches.
Here is not the place to go into too many details, but let me give you the gist of what we’re studying. If you denote the transformation which generates the black branches by and that which generate the red branches by you will observe behavior like that in the above tree if
Right now, most of the work I’m doing revolves around saying that I want to generate a tree with a certain property (like 0 and 11 going to the same node), and then solving the corresponding matrix equation to find interesting examples of trees with this property. Usually the linear algebra involved is fairly complicated — feel free to comment if you’d like to know more. As I said, this blog is not the place for all the details….
Below is another interesting example.
In this tree, 0 and 00 go to the same node. What this means in terms of branching is that there can never be two consecutive black branches (again, 0 is black and 1 is red) — taking the additional 0 branch leaves you in exactly the same place.
Note that for this property to hold, there is no restriction on what the 1 transformation can be. Above, the transformation is a scaled rotation by 60°. This is what produces the spirals of red branches emanating from the nodes of the tree.
In this example, 0 and 01 go to the same node (0 is black, 1 is red). Implications for the tree are that the red branches zigzag back and forth, getting shorter with each iteration. Moreover, there aren’t any red branches drawn after a black branch (except the first time, off the end of the trunk of the tree). This is because 01 — taking a red branch after a black branch — takes you to the same place as 0, which is taking a black branch. In other words, the red branch doesn’t take you any further.
Here, 01 and 10 take you to the same node (again, 0 is black and 1 is red). This is easy to see — there are many quadrilaterals in the tree formed by alternating red and black branches. In addition, the iterations of this tree move progressively higher each time — so, for example, all the nodes at the top are at a depth of 4.
If you look at the number of new nodes at each level, you get a sequence which begins
2, 3, 6, 12, 24, 48, 96, 192, 384, 768, ….
After the second level, it seems that the number of new nodes doubles with each iteration. I say “seems” since it is generally very difficult to prove that such a relationship continues up arbitrarily many levels. Although it is easy to find the sequence of numbers of nodes for any tree using Mathematica, I have rigorous proofs of correctness for only three cases so far.
What makes the proofs so difficult? The following tree is another example of a case where 01 and 10 go to the same node.
But the sequence of new nodes at each level is just 2, 3, 4, 5, 6, …. (This is one case I’ve rigorously proved.) The transformations and used to make this tree, in addition to forcing 01 and 10 to go to the same node, also have additional properties which force any two strings of the same length with the same number of 0’s (and therefore the same number of 1’s as well) to go to the same node. So to prove the sequence of nodes in the previous case is 2, 3, 6, 12, 24, 48, …, you need to say which strings take you to the same nodes, and prove that this pattern continues no matter how far you go up — and that there are no “surprise” coincidences of other nodes as you proceed.
It is also possible to look at infinite strings going to the same node, as shown below. The linear algebra gets quite a bit more involved in these cases.
Here, 0 and 1 do in fact correspond to going left and right, respectively. The thicker black path corresponds to the string 011111…, that is, going left once, and then going to the right infinitely often, creating the spiral you see above. The thicker green path corresponds to alternating 10101010… infinitely often, creating a zigzag path which approaches exactly the same point as the black spiral does.
Here is another example, where the spiral 011111… (in black) approaches the same point as the infinite zigzag 100100100100… (in purple).
These are some of my favorite examples. But I should remark that once you know how to mathematically find transformations which produce trees with a desired property, it takes a lot of fiddling around to find aesthetically pleasing examples.
I hope these examples give you a better idea of the nature of our research. I’ll update you on Bridges when I get back, and let you know about any interesting comments other participants make about our trees. I leave on Wednesday, and will post pictures daily on my Twitter @cre8math if you’d like a daily dose of mathematical art!