The One Four Conjecture

I can’t remember when I first heard of the “Four Fours” puzzle.  The goal is to use four 4’s to create as many different integers as possible using basic arithmetic operations.  For example,

42=44-\dfrac4{\sqrt 4}.

Which numbers are possible to obtain depends on the range of operations or functions allowed.  Using a decimal point and a bar for a repeating decimal, for example, allows expressions such as

103=\dfrac{44}{.\overline 4}+4.

You can read more about this puzzle on this Wikipedia page, and look at the references there for even more information.

A few years ago, I wondered about what numbers are possible using just one 4.  Of course there aren’t many if you restrict yourself to the basic arithmetic operations.  But you can write 4!= 24, for example.

So the factorial was a way to make numbers larger, but then what about bringing those factorials back down?  The square root function does that, but then you need to convert to an integer.  So perhaps use the floor function, so that the second function you use is

f(n)=\lfloor\sqrt n\rfloor.

Now this is a bit trickier — even small numbers are not so easy to obtain.  For example,

5=\left\lfloor\sqrt{\left\lfloor\sqrt{\left\lfloor\sqrt{\left\lfloor\sqrt{\left\lfloor \sqrt{(4!)!}\right\rfloor}\right\rfloor}\right\rfloor}\right\rfloor}\right\rfloor.

That’s a lot of work just to get 5….  But as I kept on exploring using Mathematica, it seemed that eventually, you could get every positive integer this way!

At first, it was really hard to believe — but the more I worked, the more plausible it became.  It soon became obvious that other numbers other than 4 were possible to begin with.  You could start with 9, for example, since then you could get 4 by

4=\left\lfloor\sqrt{\left\lfloor\sqrt{\left\lfloor \sqrt{9!}\right\rfloor}\right\rfloor}\right\rfloor,

and then proceed from there.

Further, why use a square root?  Could other roots work as well?  More experimentation seemed to suggest that any root might also work.  This led me to the following:

The One n Conjecture:  Suppose n>2 and p>1 are given.  Then using the factorial function and the function

f(n)=\lfloor n^{1/p}\rfloor,

all positive integers may be obtained by some composition of these functions.

This seemed really difficult to prove.  Suppose, for example — using the factorial and square root functions — it is impossible to obtain some particular integer no matter what input you start with.  Of course it is always possible to obtain n from n^2, and n^2 from n^4, and so on, but you’ve got to get larger first, and that requires some use of the factorial.

It turns out, however — in a particular sense (which you’ll see in a moment) — it is always possible, for any p.

The Possibility Lemma:  Suppose p>1 is given.  Then for any positive integer k>1, there exist positive integers q,m such that

k=f(f(f\cdots f(m!)),

where the function f is composed q times.

Let’s focus on the square root for now — that is, p=2.  The Possibility Lemma is only a starting point, since it turns out that most of the time, the smallest m need to generate a particular k is actually greater than k — making an induction proof based on the Lemma impossible.

For example, for k=42, the smallest m is 218 with q=8, so that

42=f(f(f(f(f(f(f(f(218!)))))))).

But for k=43, we get

43=\left\lfloor\sqrt{\left\lfloor \sqrt{10!}\right\rfloor}\right\rfloor.

It seemed that the least m needed to generate a given k exhibited rather erratic behavior.  So my next step was to plot a graph of the least m given k.

I won’t go into all the details here — but it took a little work to optimize the algorithms.  As an example, the smallest m needed to generate k=48,\!500 is m=890,\!827, and computations with such large factorials take time.  It turns out that the trick was to compute in advance the first 1,000,000 factorials as floating point numbers.  A little accuracy is lost as result, but several checks suggested that even so, the correct value of m is found each time.

So here is the plot of values for the least possible m for k from 1 to 5,000.

OneFour5000

And here’s the plot for values of k from 1 to 40,000.

OneFour40000

Now here’s something interesting!  You can’t help but think “fractal behavior” here.  And why the thin bands?  Not sure yet, but they somehow correspond to square roots of factorials.  For example, the m for 49,998 is 470,324, and the m for 49,999 is 248,312, and the square root of 470,324! is not too far off from 248,312!.

But although it looks like there are thin bands, they are not uniformly generated.  Here’s a closeup of the previous graph in the range 49,900 to 50,000.

OneFour100.png

There doesn’t seem to be a predictable pattern as to when the jumps are made.

And it does appear that the thin bands have an upward trend.  Might it be possible that every positive integer is eventually the m for some k?  Not an easy question to answer.

And this is only a beginning!  I just generated the graph up to 50,000 yesterday, so I haven’t had time to analyze it in any more detail.  Using p=3 generates the following graph, so it seems that there may be similar behavior for various p.

OneFour5000p3

I plan to keep working on this little puzzle — although I think a proof of or counterexample to the One n Conjecture is rather far off.  When I do make more progress, I’ll give you an update.  Despite the difficulty of the problem, this is a really fun puzzle to play with!  I hope you might give it a try, too….

Imagifractalous! 5: Fractal Binary Trees III

Last week I talked about working with binary trees whose branching ratio is 1 or greater.  The difficulty with having a branching ratio larger than one is that the tree keeps growing, getting larger and larger with each iteration.

But when you work with software like Mathematica, for example, and you create such a tree, you can specify the size of the displayed image in screen size.

So the trees above both have branching ratio 2 and branching angle of 70°.  The left image is drawn to a depth of 7, and the right image is drawn to a depth of 12.  I specified that both images be drawn the same size in Mathematica.

But even though they are visually the same size, if you start with a trunk 1 unit in length, the left image is about 200 units wide, while the second is 6000 units wide!

So this prompted us to look at scaling back trees with large branching ratios.  In other words, as trees kept getting larger, scale them back even more.  You saw why this was important last week:  if the scale isn’t right, when you overlap trees with r less than one on top of the reciprocal tree with branching ratio 1/r, the leaves of the trees won’t overlap.  The scale has to be just right.

2017-04-08ris2d.png

So what should these scale factors be?  This is such an interesting story about collaboration and creativity — and how new ideas are generated — that I want to share it with you.

For your usual binary tree with branching ratio less than one, you don’t have to scale at all.  The tree remains bounded, which is easy to prove using convergent geometric series.
2017-01-20ris1

What about the case when r is exactly 1, as shown in the above figure?  At depth n, if you start with a trunk of length 1, the path from the base of the trunk to the leaf is a path of exactly n + 1 segments of length 1, and so can’t be any longer than n + 1 in length.  As the branching angle gets closer to 0°, you do approach this bound of n + 1.  So we thought that scaling back by a factor of n + 1 would keep the tree bounded in the case when r is 1.

What about the case when r > 1?  Let’s consider the case when r = 2 as an example.  The segments in any path are of length 1, 2, 4, 8, 16, etc., getting longer each time by a power of 2.  Going to a depth of n, the total length is proportional to 2^n in this case.  In general, the total length is about 2\cdot r^n for arbitrary r, so scaling back by a factor of r^n would keep the trees bounded as well.

So we knew how to keep the trees bounded, and started including these scaling factors when drawing our images.  But there were two issues.  First, we still had to do some fudging when drawing trees together with their reciprocal trees.  We could still create very appealing images, but we couldn’t use the scale factor on its own.

And second — and perhaps more importantly — Nick had been doing extensive exploration on his computer generating binary trees.  Right now, we had three different cases for scaling factors, depending on whether r < 1, r = 1, or r > 1.  But in Nick’s experience, when he moved continuously through values of r less than 1 to values of r greater than one, the transition looked very smooth to him.  There didn’t seem to be any “jump” when passing through r = 1, as happened with the scale factors we had at the moment.

I wasn’t too bothered by it, though.  There are lots of instances in mathematics where 1 is some sort of boundary point.  Take geometric series, for example.  Or perhaps there is another boundary point which separates three fundamentally different types of solutions.  For example, consider the quadratic equation

x^2+c=0.

The three fundamentally different solution sets correspond to  c < 0, c = 0, and c > 0.  There is a common example from differential equations, too, though I won’t go into that here.  Suffice it to say, this type of trichotomy occurs rather frequently.

I tried explaining this to Nick, but he just wouldn’t budge.  He had looked at so many binary trees, his intuition led him to firmly believe there just had to be a way to unify these scale factors.

I can still remember the afternoon — the moment — when I saw it.  It was truly beautiful, and I’ll share it in just a moment.  But my point is this:  I was so used to seeing trichotomies in mathematics, I was just willing to live with these three scale factors.  But Nick wasn’t.  He was tenacious, and just insisted that there was further digging to do.

Don’t ask me to explain how I came up with it.  It was like that feeling when you just were holding on to some small thing, and now you couldn’t find it.  But you never left the room, so it just had to be there.  So you just kept looking, not giving up until you found it.

And there is was:  if the branching ratio was and you were iterating to a depth of n, you scaled back by a factor of

\displaystyle\sum_{k=0}^n r^k.

This took care of all three cases at once!  When r < 1, this sum is bounded (think geometric series), so the boundedness of the tree isn’t affected.  When r = 1, you just get n + 1 — the same scaling factor we looked at before!  And when r > 1, this sum is large enough to scale back your tree so it’s bounded.

Not only that, this scale factor made proving the Dual Tree Theorem so nice.  The scaling factors for a tree with r < 1 and its reciprocal tree with branching ratio 1/r matched perfectly.  No need to fudge!

This isn’t the place to go into all the mathematics, but I’d be happy to share a copy of our paper if you’re interested.  We go into a lot more detail than I ever could in a blog post.

This is how mathematics happens, incidentally.  It isn’t just a matter of finding a right answer, or just solving an equation.  It’s a give-and-take, an exploration, a discovery here and there, tenacity, persistence.  A living, breathing endeavor.

But the saga isn’t over yet….  There is a lot more to say about binary trees.  I’ll do just that in my next installment of Imagifractalous!

Imagifractalous! 4: Fractal Binary Trees II

Now that the paper Nick and I wrote on binary trees was accepted for Bridges 2017 (yay!), I’d like to say a little more about what we discovered.  I’ll presume you’ve already read the first Imagifractalous! post on binary trees (see Day077 for a refresher if you need it).

Recall that in that post, I discussed creating binary trees with branching ratios which were 1 or larger.  Below are three examples of binary trees, with branching ratios less that 1, equal to 1, and larger than 1, respectively.

2016-12-18threetrees.png

It was Nick’s insight to consider the following question:  how are trees with branching ratio r related to those with branching ratio 1/r?  He had done a lot of exploring with graphics in Python, and observed that there was definitely some relationship.

Let’s look at an example.  The red tree is a binary tree with branching ratio r less than one, and the gray tree has a branching ratio which is the reciprocal r.  Both are drawn to the same depth.

2016-12-03-doubletree1.png

Of course you notice what’s happening — the leaves of the trees are overlapping!  This was happening so frequently, it just couldn’t be coincidence.  Here is another example.

tree1

Notice how three copies of the trees with branching ratio less than one are covering some of the leaves of a tree with the reciprocal ratio.

Now if you’ve ever created your own binary trees, you’ll likely have noticed that I left out a particularly important piece of information:  the size of the trunks of the trees.  You can imagine that if the sizes of the trunks of the r trees and the 1/r trees were not precisely related, you wouldn’t have the nice overlap.

Here is a figure taken from our paper which explains just how to find the correct relationship between the trunk sizes.  It illustrates the main idea which we used to rigorously prove just about everything we observed about these reciprocal trees.

tree2

Let’s take a look at what’s happening.  The thick, black tree has a branching ratio of 5/8, and a branching angle of 25°.  The thick, black path going from O to P is created by following the sequence of instructions RRRLL (and so the tree is rendered to a depth of 5).

Now make a symmetric path (thick, gray, dashed) starting at P and going to O.  If we start at P with the same trunk length we started with at O, and follow the exact same instructions, we have to end up back at O.

The trick is to now look at this gray path backwards, starting from O.  The branches now get larger each time, by a factor of 8/5 (since they were getting smaller by a factor of 5/8 when going in the opposite direction).  The size of the trunk, you can readily see, is the length of the last branch drawn in following the black path from O to P.  This must be (5/8)5 times the length of the trunk, since the tree is of depth 5.

The sequence of instructions needed to follow this gray path is RRLLL.  It turns out this is easy to predict from the geometry.  Recall that beginning at P, we followed the instructions RRRLL along the gray path to get to O.  When we reverse this path and go from O to P, we follow the instructions in reverse — except that in going in the reverse direction, what was previously a left turn becomes a right turn, and vice versa.

So all we need to do to get the reverse instructions is to reverse the string RRRLL to get LLRRR, and then change the L‘s to R‘s and the R‘s to L‘s, yielding RRLLL.

There’s one important detail to address:  the fact that the black tree with branching ratio 5/8 is rotated by 25° to make everything work out.  Again, this is easy to see from the geometry of the figure.  Look at the thick gray path for a moment.  Since following the instructions RRLLL means that in total, you make one more left turn than you do right turns, the last branch of the path must be oriented 25° to the left of your starting orientation (which was vertical).  This tells you precisely how much you need to rotate the black tree to make the two paths have the same starting and ending points.

Of course one example does not make a proof — but in fact all the important ideas are contained in this one illustration.  It is not difficult to make the argument more general, and we have successfully accomplished that (though this blog is not the place for it!).

If you look carefully at the diagram, you’ll count that there are exactly 10 leaves in common with these two trees with reciprocal branching ratios.  There is some nice combinatorics going on here, which is again easy to explain from the geometry.

You can see that these common leaves (illustrated with small, black dots) are at the ends of gray branches which are oriented 25° from the vertical.  Recall that this specific angle came from the fact that there was one more L than there were R‘s in the string RRLLL.

Now if you have a sequence of 5 instructions, the only way to have exactly one more L than R‘s is to have precisely three L‘s (and hence two R‘s).  And the number of ways to have three L‘s in a string of length 5 is just

\displaystyle{5\choose3}=10.

Again, these observations are easy to generalize and prove rigorously.

And where does this take us?

canopies.png

On the right are 12 copies of a tree with a braching ratio of r less than one and a branching angle of 30°, and on the left are 12 copies of a tree with a reciprocal branching ratio of 1/r, also with a branching angle of 30°.  All are drawn to depth 4, and the trunks are appropriately scaled as previously discussed.

These sets of trees produce exactly the same leaves!  We called this the Dual Tree Theorem, which was the culmination of all these observations.  Here is an illustration with both sets of trees on top of each other.

2016-12-14gtree.png

As intriguing as this discovery was, it was only the beginning of a much broader and deeper exploration into the fractal world of binary trees.  I’ll continue a discussion of our adventures in the next installment of Imagifractalous!

A New Cantor Set?

Although we’ll do a little review, this post assumes some familiarity with the Cantor set as well as working with fractions in bases other than 10.

So what is the Cantor set?

2017-02-11cantor1

The usual geometrical definition is to begin with a line segment of length one (as shown at the top of the figure).  Next, remove the middle third of that segment, so that two segments of length 1/3 are formed.  Then remove the middle thirds from those segments, and continue recursively.

Even though — after this process is repeated infinitely, of course — you remove segments whose lengths sum to 1, there are still infinitely many points left!  (Actually, uncountably infinitely many, although this observation is not necessary for our discussion.)

There are many ways to describe the remaining points, but one common way is to say that the Cantor set consists of all those numbers between 0 and 1 (inclusive) which, when written in base 3, may be written only with 0’s and 2’s.

Again, we’ll briefly review.  We have .23 = 2/310, for example — since the places after the ternary point (no “deci”mals here) represent powers of 1/3.

So if we wanted to find 3/4 in base 3, we note that we’d need 2/3 = 0.23, but there would be 1/12 left over. This is smaller than 1/9, so we’re at 0.203 so far. Next, we need 2/27, giving 0.2023, with 1/108 left over. And so on. The result is that 3/4 is equal to 0.202020…3, where the “20” repeats. This can also be shown using infinite geometric series, if desired:

\dfrac34=\dfrac{2/3}{1-1/9}.

Surprisingly, the idea for a possibly new type of “Cantor set” came from studying binary trees!  I say possibly new, since I couldn’t find any reference to such a set online, but of course that doesn’t mean someone else hasn’t come across it.  And I call it a type of Cantor set since it may also be formed by taking out thirds of segments, but in a slightly different way than described above.

2017-01-02binarytree

Now I’ve talked a bit about binary trees before, so I won’t go into great detail.  But here is the important idea:  when you make a branch, you’re pointing in some particular direction, and then turn either left or right, but you can’t just keep going in the same direction.

So what if you looked at ternary expansions, and as you added digits, you had the option of adding 1 to the previous digit (like turning left), or subtracting 1 (like turning right), but you couldn’t use the same digit twice consecutively.  So 0.21021201 would be OK (we’ll drop the 3 subscripts since we’ll be working exclusively in base 3 from now on), but 0.12002120 would not be allowed since there are consecutive 0’s.

Note that adding 1 to 2 gives 0 in base 3, and subtracting 1 from 0 gives 2.  So essentially, starting with 0., you build ternary expansions with the property that each digit is different from the previous one.  And, of course, the expansions must be infinite….

What do iterations of this scheme look like?

2017-02-11cantor2.png

We start with a segment of length 1.  Recall we begin with 0., so that means the ternary expansions may begin with 0.1 or 0.2.  Expansions beginning with 0.0 are not allowed, so this precludes the first third of the segment.

Now here comes the interesting part!  On the second iteration, (the third line from the top), we remove different thirds from the first two segments.  Since the 0.1 may continued onto 0.10 or 0.12, but not 0.11, we remove the middle third from the 0.1 segment.  Further, 0.2 may be continued as 0.20 or 0.21, but not 0.22, so we remove the last third of the 0.2 segment.  The iteration process is not symmetrical.

We continue on….  Since 0.10 may be continued as 0.101 or 0.102, but not 0.100, we remove the first third of the 0.10 segment.  You get the idea.  Seven iterations of this procedure are shown in the figure above.

Note that since the process for creating the original Cantor set is symmetrical, this imposes a self-similarity on the set itself.  The Cantor set is exactly the union of two duplicate copies of the original, scaled by a factor of 1/3.

In other words, the Cantor set may also be created using an iterated function system with the following two transformations:

y=\dfrac13x,\qquad y=\dfrac13x+\dfrac23.

What about the self-similarity of the new Cantor set?  To help in seeing this, here’s a slightly more detailed version of the iteration scheme.

2017-02-11cantor3.png

Is there any self-similarity here?  Yes, but the fewest number of transformations I’ve found to describe this self-similarity is five.  The curious reader is welcome to find fewer!

It isn’t hard to see the five vertical bands in this figure — the first three look the same (although the second one appears to be reflected), and the last two also look the same, although reflections of each other.

The first band is all ternary expansions in this new set beginning with 0.10.  How do these relate to the whole set?  Well, 1/9 of the set consists of expansions beginning with 0.001… or 0.002…, and then adding digits different from those previous.  Adding 1/3 therefore gives all expansions beginning with 0.101… or 0.102…, and then adding different digits.  This implies that the self-similarity describing the first vertical band is

y=\dfrac 19x+\dfrac13.

The second band consists of those expansions in our strings beginning with 0.12.  But if x is an expansion in our set beginning with 0.10, then 1 – x must be an expansion in our set beginning with 0.12, since we may write 1 as .22222…, repeating.  Therefore, the second band is represented by the transformation

y=1-\left(\dfrac 19x+\dfrac13\right)=\dfrac23-\dfrac19x.

We can think of the third band just as we did the first — except that this band consists of number beginning with 0.20 (rather than 0.10).  So this band is represented by the transformation

y=\dfrac 19x+\dfrac23.

The last two bands consist of those expansions beginning with 0.21.  Here, we break into the two cases 0.210 and 0.212, and use our previous work.  For example, those beginning with 0.210 can be found by taking those beginning with 0.10, dividing by 3 to get expansions beginning with 0.010, and then adding 2/3 to get expansions beginning with 0.210:

y=\dfrac13\left(\dfrac 19x+\dfrac13\right)+\dfrac23=\dfrac1{27}x+\dfrac79.

We describe the self-similarity of the last band — those expansions beginning with 0.212 — analogously:

y=\dfrac13\left(\dfrac23-\dfrac 19x\right)+\dfrac23=\dfrac89-\dfrac1{27}x.

These are five transformations which describe the self-similarity of the new Cantor set.  I haven’t rigorously proved yet that you can’t do it in fewer, but I think this is the case.

Of course this is just the beginning — you can use the same rule in any base.  That is, begin with 0., and either add or subtract one from the previous digit.  In base 4, you get nice self-similarity, but it gets more involved in higher bases.  In bases higher than 3, you can also use the rule that the next digit in the expansion is different than the previous — and this gives yet another class of Cantor sets.  I’ll leave you to investigate, and perhaps I’ll write again about these Cantor sets when I find out more!

Imagifractalous! 3: Fractal Binary Trees

I’ve taken a break from Koch-like curves and p-adic sequences for an arboreal interlude….  Yes, there’s a story about why — I needed to work with Nick on a paper he was writing for Bridges — but that story isn’t quite finished yet.  When it is, I’ll tell it.  But for now, I thought I’d share some of the fascinating images we created along the way.

b17depth6-7v2

Let’s start with a few examples of simple binary trees.  If you want to see more, just do a quick online search — there are lots of fractal trees out there on the web!  The construction is pretty straightforward.  Start by drawing a vertical trunk of length 1.  Then, move left and right some specified angle, and draw a branch of some length r < 1.  Recursively repeat from where you left off, always adding two more smaller branches at the tip of each branch you’ve already drawn.

If you look at these two examples for a moment, you’ll get the idea.  Here, the angle used is 40 degrees, and the ratio is 5/8.  On the left, there are 5 iterations of the recursive drawing, and there are 6 iterations on the right.

Here’s another example with a lot more interaction among the branches.

2016-12-01-tree1.png

This type of fractal binary tree has been studied quite a bit.  There is a well-known paper by Mandelbrot and Frame which discusses these trees, but it’s not available without paying for it.  So here is a paper by Pons which addresses the same issues, but is available online.  It’s an interesting read, but be forewarned that there’s a lot of mathematics in it!

2017-01-20ris1.png

In trying to understand various properties of these fractal trees, it’s natural to write code which creates them.  But here’s the interesting thing about writing programs like that — once they’re written, you can input anything you like!  Who says that r has to be less than 1?  The tree above is a nice example of a fractal tree with r = 1.  All the branches are of the same length, and there is a lot of overlap.  This helps create an interesting texture.

But here’s the catch.  The more iterations you go, the bigger the tree gets.  In a mathematical sense, the iterations are said to be unbounded.  But when Mathematica outputs a graphic, it is automatically scaled to fit your viewing window.  So in practice, you don’t really care how large the tree gets, since it will automatically be scaled down so the entire tree is visible.

It is important to note that when r < 1, the trees are bounded, so they are easier to study mathematically.  The paper Nick and I are working on scales unbounded trees so they are more accessible, but as I said, I’ll talk more about this in a later post.

2017-01-21rgt1

Here are a few examples with r > 1.  Notice that as there are more and more iterations, the branches keep getting larger.  This creates a very different type of binary tree, and again, a tree which keeps getting bigger (and unbounded) as the number of iterations increases.  But as mentioned earlier, Mathematica will automatically scale an image, so these trees are easy to generate and look at.

Nick created the following image using copies of binary trees with r approximately equal to 1.04.  The ever-expanding branches allow for the creation of interesting textures you really can’t achieve when r < 1.

blackwhitetree

Another of my favorites is the following tree, created with r = 1.  The angle used, though, is 90.9 degrees.  Making the angle just slightly larger than a right angle creates an interesting visual effect.

2017-01-18binarytree

But the exploration didn’t stop with just varying r so it could take on values 1 or greater.  I started thinking about other ways to alter the parameters used to create fractal binary trees.

For example, why does r have to stay the same at each iteration?  Well, it doesn’t!  The following image was created using values of r which alternate between iterations.

2016-12-21-ralt.png

And the values of r can vary in other ways from iteration to iteration.  There is a lot more to investigate, such as generating a binary tree from any sequence of r values.  But studying these mathematically may be somewhat more difficult….

Now in a typical binary tree, the angle you branch to the left is the same as the angle you branch to the right.  Of course these two angles don’t have to be the same.  What happens if the branching angle to the left is different from the branching angle to the right?  Below is one possibility.

2016-12-25-2a1.png

And for another possibility?  What if you choose two different angles, but have the computer randomly decide which is used to branch left/right at each iteration?  What then?

2017-01-02randangles2.png

Here is one example, where the branching angles are 45 and 90 degrees, but which is left or right is chosen randomly (with equal probability) at each iteration.  Gives the fractal tree a funky feel….

You might have noticed that none of these images are in color.  One very practial reason is that for writing Bridges papers, you need to make sure your images look OK printed in black-and-white, since the book of conference papers is not printed in color.

But there’s another reason I didn’t include color images in this post.  Yes, I’ve got plenty…and I will share them with you later.  What I want to communicate is the amazing variety of textures available by using a simple algorithm to create binary trees.  Nick and I never imagined there would be such a fantastic range of images we could create.  But there are.  You’ve just seen them.

Once the Bridges paper is submitted, accepted (hopefully!), and revised, I’ll continue the story of our arboreal adventure.  There is a lot more to share, and it will certainly be worth the wait!