09/02/2017, 12:10 AM
(This post was last modified: 09/02/2017, 06:59 AM by stephrenny.)

If any of you are curious, this is the story of how I came across these fractals, if any of you are interested (it is quite long):

I had a math problem that was an age old nemesis of mine. When I first learned about power towers (which I now know as tetration) in the sixth grade, I was shocked at how ridiculously quickly the numbers grew. I figured there was no way that 2^(2^(2^(2^2))) was as large a number as my teacher claimed it to be, nor that adding another two to that expression would make its decimal representation too large to write down given the physical constraints of the universe. After a little computation, I found myself humbled and intrigued.

After that, I would occasionally play with power towers in my calculator, mainly because I liked to see it trip over itself with overflow errors. Eventually, I picked a small enough number as a base, and was quickly disappointed as no matter how many times I applied the tetration it would never seem to get anywhere. After a lot of experimentation, I found that 1.44 converged while 1.45 grew off to infinity, and the same for 1.444 and 1.445, respectively. However, not very surprisingly, this did not hold for 1.4444 and 1.4445. It was at this point that I wondered what the largest number such that repeated tetration would converge. I had an elementary understanding of limits at the time, but it did not extend much beyond things such as "the limit of x^2 as x -> 3 was 9", so I figured they were quite useless and common sense worked as well or better most of the time. I also did not know calculus, so I never solved that problem until I revisited it this year.

For the longest time I figured I would never find the answer to this problem (I did not learn calculus until 10th grade). Once in a while, I would sometimes think about this problem as I pondered the other mysteries of life, and even after I learned calculus I didn't think to apply it. However, one day, I finally, finally had a breakthrough. I tried visualizing the problem graphically, with the equation y = n ^ x representing the tetration, and the line y = x to change the output of the function into the new input. For example if you had n = 2, first you would start at x = 1 on the exponential function, giving you the point (1, 2). Then, you "move" horizontally until you hit y = x, so that now your x value was your previous y value, in this case two. Next you move vertically up so that you hit 2 ^ x again, and since this time x was the output of the previous iteration, you have reached the next step in the power tower. This "staircase"-like process repeated over and over represents the creation of higher and higher tetrations.

The solution was immediately obvious, what I was looking for was an n such that n ^ x was tangent to y = x. Solving the system, the answer comes out to be n = e ^ 1/e ~= 1.4447. I was quite ecstatic. I didn't know the term "tetration" at the time, but of course, the Wikipedia article on tetration mentions that Euler had proved this conjecture. Though honestly I am glad I didn't find that Wikipedia page until later, because the sense of accomplishment and discovery at the point of finding the answer was amazing.

How did this tie into fractals? Well during that week, aside from solving this problem that haunted me for several years, I developed an interest in complex fractals (I clicked on a Mandelbrot zoom on Youtube for some reason, and became intrigued). Astonishingly, two days after I solved my problem, I stumbled into this Mathologer video: https://www.youtube.com/watch?v=9gk_8mQuerg , and the visualization at 8:55 was almost exactly how I had solved my problem. I was very, very much intrigued. That ultimately led me to try to create my own fractals based on tetration (I had been considering writing a fractal generating program, but at that moment I dropped everything and began working).

As I was working on my program, the first step was defining an imaginary class for objects that acted like imaginary numbers. Everything was straightforward enough, however, I soon ran into a problem - complex exponents, which formed the basis of tetration fractals.

I figured the way to do this was through Demoivre’s, by turning the base complex number into exponential form and keeping the exponent in rectangular form and simplifying from there. It turned out to be simple algebra, but, to me at least, it was an "ooooh" moment, as though I had previously used Demoivre's for solving a variety math problems, I never really dealt with complex powers, but seeing it applied this way made me appreciate even more how flexible it was. After making sure it works by verifying it with several online sources and testing with many different complex numbers, I continued with the rest of the program, which was more or less straightforward.

Once I finally got rid of all the random bugs and had my program up and running, it was pretty late into the night, but I was extremely excited when the tetration fractal showed up on my screen (these were the normal tetration fractals, not the new kind I stumbled across). For the next few days I played around with the fractal in my spare time, and implemented something akin to a zoom method.

Then I started to experiment. My main goal when I went into this was wanting to create tetration fractals, and though I was extremely happy that it worked out, after the initial euphoria I started to wish they looked a just little bit... better. So I messed with the things that gave the Mandelbrot and Julia sets their iconic look, such as escape and period. Eventually, I tried changing the representation from convergence/divergence to the angle of the point as tetration is applied, not knowing what to expect. This is where things got interesting.

These images are very stunning, and you could imagine my shock when they first shown through, especially contrasted with the rather bland style of normal tetration fractals in black and white. Of course, when I first used this method, the mapping of angle to color was linear, and the fractal was in a sense a lot thicker and darker, but the overall shape was there and it was different enough that I dove into it. I played around with converting between angle and color with a straightforward function. The current mapping I use for these images is based on 1/x.

Here, the x represents one minus the ratio of theta (the maximum angle previously mentioned) to pi/2. On the y scale, 1 represents an rgb of (255, 255, 255), and 0 is (0,0,0), with a linear relationship in between.

Though you use simple algebraic transformation to modify the "thickness" of the branches in the fractal, the construction of the fractal itself is based just on tetration and argument of the point, and no complex algorithms such as fractal overlaying or anything like that was used, which is why I'm surprised it came out as detailed as it did.

I had a math problem that was an age old nemesis of mine. When I first learned about power towers (which I now know as tetration) in the sixth grade, I was shocked at how ridiculously quickly the numbers grew. I figured there was no way that 2^(2^(2^(2^2))) was as large a number as my teacher claimed it to be, nor that adding another two to that expression would make its decimal representation too large to write down given the physical constraints of the universe. After a little computation, I found myself humbled and intrigued.

After that, I would occasionally play with power towers in my calculator, mainly because I liked to see it trip over itself with overflow errors. Eventually, I picked a small enough number as a base, and was quickly disappointed as no matter how many times I applied the tetration it would never seem to get anywhere. After a lot of experimentation, I found that 1.44 converged while 1.45 grew off to infinity, and the same for 1.444 and 1.445, respectively. However, not very surprisingly, this did not hold for 1.4444 and 1.4445. It was at this point that I wondered what the largest number such that repeated tetration would converge. I had an elementary understanding of limits at the time, but it did not extend much beyond things such as "the limit of x^2 as x -> 3 was 9", so I figured they were quite useless and common sense worked as well or better most of the time. I also did not know calculus, so I never solved that problem until I revisited it this year.

For the longest time I figured I would never find the answer to this problem (I did not learn calculus until 10th grade). Once in a while, I would sometimes think about this problem as I pondered the other mysteries of life, and even after I learned calculus I didn't think to apply it. However, one day, I finally, finally had a breakthrough. I tried visualizing the problem graphically, with the equation y = n ^ x representing the tetration, and the line y = x to change the output of the function into the new input. For example if you had n = 2, first you would start at x = 1 on the exponential function, giving you the point (1, 2). Then, you "move" horizontally until you hit y = x, so that now your x value was your previous y value, in this case two. Next you move vertically up so that you hit 2 ^ x again, and since this time x was the output of the previous iteration, you have reached the next step in the power tower. This "staircase"-like process repeated over and over represents the creation of higher and higher tetrations.

The solution was immediately obvious, what I was looking for was an n such that n ^ x was tangent to y = x. Solving the system, the answer comes out to be n = e ^ 1/e ~= 1.4447. I was quite ecstatic. I didn't know the term "tetration" at the time, but of course, the Wikipedia article on tetration mentions that Euler had proved this conjecture. Though honestly I am glad I didn't find that Wikipedia page until later, because the sense of accomplishment and discovery at the point of finding the answer was amazing.

How did this tie into fractals? Well during that week, aside from solving this problem that haunted me for several years, I developed an interest in complex fractals (I clicked on a Mandelbrot zoom on Youtube for some reason, and became intrigued). Astonishingly, two days after I solved my problem, I stumbled into this Mathologer video: https://www.youtube.com/watch?v=9gk_8mQuerg , and the visualization at 8:55 was almost exactly how I had solved my problem. I was very, very much intrigued. That ultimately led me to try to create my own fractals based on tetration (I had been considering writing a fractal generating program, but at that moment I dropped everything and began working).

As I was working on my program, the first step was defining an imaginary class for objects that acted like imaginary numbers. Everything was straightforward enough, however, I soon ran into a problem - complex exponents, which formed the basis of tetration fractals.

I figured the way to do this was through Demoivre’s, by turning the base complex number into exponential form and keeping the exponent in rectangular form and simplifying from there. It turned out to be simple algebra, but, to me at least, it was an "ooooh" moment, as though I had previously used Demoivre's for solving a variety math problems, I never really dealt with complex powers, but seeing it applied this way made me appreciate even more how flexible it was. After making sure it works by verifying it with several online sources and testing with many different complex numbers, I continued with the rest of the program, which was more or less straightforward.

Once I finally got rid of all the random bugs and had my program up and running, it was pretty late into the night, but I was extremely excited when the tetration fractal showed up on my screen (these were the normal tetration fractals, not the new kind I stumbled across). For the next few days I played around with the fractal in my spare time, and implemented something akin to a zoom method.

Then I started to experiment. My main goal when I went into this was wanting to create tetration fractals, and though I was extremely happy that it worked out, after the initial euphoria I started to wish they looked a just little bit... better. So I messed with the things that gave the Mandelbrot and Julia sets their iconic look, such as escape and period. Eventually, I tried changing the representation from convergence/divergence to the angle of the point as tetration is applied, not knowing what to expect. This is where things got interesting.

These images are very stunning, and you could imagine my shock when they first shown through, especially contrasted with the rather bland style of normal tetration fractals in black and white. Of course, when I first used this method, the mapping of angle to color was linear, and the fractal was in a sense a lot thicker and darker, but the overall shape was there and it was different enough that I dove into it. I played around with converting between angle and color with a straightforward function. The current mapping I use for these images is based on 1/x.

Here, the x represents one minus the ratio of theta (the maximum angle previously mentioned) to pi/2. On the y scale, 1 represents an rgb of (255, 255, 255), and 0 is (0,0,0), with a linear relationship in between.

Though you use simple algebraic transformation to modify the "thickness" of the branches in the fractal, the construction of the fractal itself is based just on tetration and argument of the point, and no complex algorithms such as fractal overlaying or anything like that was used, which is why I'm surprised it came out as detailed as it did.