09/02/2017, 12:09 AM
(This post was last modified: 09/02/2017, 12:33 AM by stephrenny.)

Hello everyone,

My name is Stephen Ren, I am currently a high school student, and I have always loved mathematics. Several months ago, I learned of the connection between complex numbers and fractals, and wrote a Java program to try it out for myself. I had recently been tackling an algebra problem regarding tetration, and decided to play around with tetration fractals using my program. After producing the standard tetration fractal, I decided to experiment. I changed the representation of the fractal from convergence/divergence of the point to how "steep" the point gets as tetration was applied (explained in greater detail later), and to my surprise, produced very interesting and beautiful fractals, several of which I have attached to this post. Many have been resized to fit the restraints of the post. For full sized images I just created an Imgur album yesterday at https://imgur.com/a/D5g78, and I will be adding more and more images to it as time goes on.

I have not been able to find these fractals or style of fractals anywhere else, which is a shame. The closest I have come to finding one is called Pickover Stalks, which is based on the distance the point gets to the imaginary axis. Though similarities are apparent, it is not quite the same, and I had not found examples of it with tetration fractals. I feel this is kind of like a "hidden gem" for tetration fractals, as I applied the same method to the Mandelbrot set and the result was very, very underwhelming. In any case, I seems to me that tetration fractals are often overlooked when compared to the Mandelbrot, Julia sets, and the like, and I wish this may at least do something to change that.

As for the generation of these fractals:

Normal tetration fractals are generated by coloring each point based on convergence/divergence. However, for these fractals, each pixel is colored based on the maximum reference angle as tetration was applied to the point it represents in the complex plane. For example (and I apologize for any errors in notation I may have made), if F^n(z) = z^(z^(z^(z ... n times))) represents tetration, the color of the pixel representing a point z would be based on the maximum value in the set of the reference angles for arg(F^k(z)) for k = 1 ... n, where n was the number of iterations of tetration (as with all fractals, more is needed to see detail at greater zoom, for these images n ranged from 100 to 400). At first I started k at zero but as you would expect all it does is overlay a gradient over the fractal, otherwise it looked exactly the same.

Conceptually, it is simply colored based on how "steep" the point gets along its orbit.

For reference, the following image is of this fractal centered on the origin, with the real axis ranging from -6 to 6. This was generated with 100 iterations of tetration

For comparison, this next image is of "normal" convergence/divergence tetration fractals with the same center and scale.

The following images are several more portions of the fractal, all of which are zoomed in snapshots of the first image. I hope you all enjoy.

This image is centered on -4.125, magnified 40x from the original fractal. The overall shape of this figure is just barely visible in the complete depiction, but, as shown in this image, it is heavily laced with detail. This figure is particularly interesting because it is set apart from the rest of the “main” fractal body, and further so far I had not found it repeated elsewhere in the fractal. This image was created with 100 iterations of tetration.

This image is centered on 0.9073 + 2.354i, and is magnified 50x from the original fractal. This figure shows up repeatedly within the fractal, not only in isolation, but often within in other patterns and figures. I think there was a post on here about Kneser’s Chi-Star function? I don't quite understand what was but the pattern bears some resemblance to this, though it may be coincidence. This image was created with 100 iterations of tetration.

This image is centered on 0.170938545, and is magnified 6000x from the original fractal. Patterns similar to this can be found repeated in many areas along the real axis, though it is often deeply burred in a larger figure. This image is generated with 400 iterations of tetration.

Please do note, however, that I have only a basic understanding in tetration, though I have a decent amount of math knowledge for a high schooler (I was 1 point away from qualifying for usamo). I am afraid I may have to leave much of the theoretical heavy lifting to all of you, however if possible, I am always looking to be able to learn more things. If you have any questions or insights feel free to email me or comment on this post.

Thank you all very much!

My name is Stephen Ren, I am currently a high school student, and I have always loved mathematics. Several months ago, I learned of the connection between complex numbers and fractals, and wrote a Java program to try it out for myself. I had recently been tackling an algebra problem regarding tetration, and decided to play around with tetration fractals using my program. After producing the standard tetration fractal, I decided to experiment. I changed the representation of the fractal from convergence/divergence of the point to how "steep" the point gets as tetration was applied (explained in greater detail later), and to my surprise, produced very interesting and beautiful fractals, several of which I have attached to this post. Many have been resized to fit the restraints of the post. For full sized images I just created an Imgur album yesterday at https://imgur.com/a/D5g78, and I will be adding more and more images to it as time goes on.

I have not been able to find these fractals or style of fractals anywhere else, which is a shame. The closest I have come to finding one is called Pickover Stalks, which is based on the distance the point gets to the imaginary axis. Though similarities are apparent, it is not quite the same, and I had not found examples of it with tetration fractals. I feel this is kind of like a "hidden gem" for tetration fractals, as I applied the same method to the Mandelbrot set and the result was very, very underwhelming. In any case, I seems to me that tetration fractals are often overlooked when compared to the Mandelbrot, Julia sets, and the like, and I wish this may at least do something to change that.

As for the generation of these fractals:

Normal tetration fractals are generated by coloring each point based on convergence/divergence. However, for these fractals, each pixel is colored based on the maximum reference angle as tetration was applied to the point it represents in the complex plane. For example (and I apologize for any errors in notation I may have made), if F^n(z) = z^(z^(z^(z ... n times))) represents tetration, the color of the pixel representing a point z would be based on the maximum value in the set of the reference angles for arg(F^k(z)) for k = 1 ... n, where n was the number of iterations of tetration (as with all fractals, more is needed to see detail at greater zoom, for these images n ranged from 100 to 400). At first I started k at zero but as you would expect all it does is overlay a gradient over the fractal, otherwise it looked exactly the same.

Conceptually, it is simply colored based on how "steep" the point gets along its orbit.

For reference, the following image is of this fractal centered on the origin, with the real axis ranging from -6 to 6. This was generated with 100 iterations of tetration

For comparison, this next image is of "normal" convergence/divergence tetration fractals with the same center and scale.

The following images are several more portions of the fractal, all of which are zoomed in snapshots of the first image. I hope you all enjoy.

This image is centered on -4.125, magnified 40x from the original fractal. The overall shape of this figure is just barely visible in the complete depiction, but, as shown in this image, it is heavily laced with detail. This figure is particularly interesting because it is set apart from the rest of the “main” fractal body, and further so far I had not found it repeated elsewhere in the fractal. This image was created with 100 iterations of tetration.

This image is centered on 0.9073 + 2.354i, and is magnified 50x from the original fractal. This figure shows up repeatedly within the fractal, not only in isolation, but often within in other patterns and figures. I think there was a post on here about Kneser’s Chi-Star function? I don't quite understand what was but the pattern bears some resemblance to this, though it may be coincidence. This image was created with 100 iterations of tetration.

This image is centered on 0.170938545, and is magnified 6000x from the original fractal. Patterns similar to this can be found repeated in many areas along the real axis, though it is often deeply burred in a larger figure. This image is generated with 400 iterations of tetration.

Please do note, however, that I have only a basic understanding in tetration, though I have a decent amount of math knowledge for a high schooler (I was 1 point away from qualifying for usamo). I am afraid I may have to leave much of the theoretical heavy lifting to all of you, however if possible, I am always looking to be able to learn more things. If you have any questions or insights feel free to email me or comment on this post.

Thank you all very much!