The relation between escape time fractals and chaos game fractals

I had the insight that both escape time fractals and "chaos game" fractals involve some kind of iteration. Some fractals can even be visualized by both ways: the escape time algorithm and the chaos game. One example of this is the Sierpinski triangle, which (or its approximation) I've visualized by the chaos game in a previous blog post, as well as by the escape-time fractal algorithm of the fractal zoomer software called XaoS (although the latter actually showed the areas that are NOT part of the fractal, with different colors depending of the escape time). Yesterday I had the idea what if I tried to visualize the Mandelbrot set by the chaos game, especially the points that are part of that fractal by different colors (so points that are usually black in other fractal generators, as those software visualize those points that are not part of the fractal instead, by different colors).

I had the insight that we could continue the iteration of the escape-time algorithm of the Mandelbrot set in the opposite direction! So, instead of taking:

z=z^2+c

we could take:

z=sqrt(z-c)

Okay, but there are two square roots of a complex number! Well, it is just handy for us, because we can introduce a random element in the iteration that is required by the chaos game anyway. So we take one of the square roots randomly. Thus I tried to implement it by JavaScript yesterday, but always the same picture appeared, no matter what was the initial value of z and what was the constant c:

Soon I realized that I've forgotten to subtract c in the algorithm, so different pictures appeared afterwards, depending on the random c value (but note that they did not much depend on the random initial value of z):

Still, the images did not look nice enough, as you can see, because there were too little amount of points belonging to the resulting fractals. So I had the idea what if I take:

z=cbrt(z-c)

Where cbrt is the cube root of its argument (I just took the complex cube root formula from WikiPedia, just like the complex square root formula). Again, there are three cube roots of a complex number, and it is good for us. This was handy for two reasons: it produced more points, and it made it easier to color the points, as there are 3 main colors: red, green and blue, just like there are 3 possible ways of taking the cube root. Now this method produced nicer pictures than the previous one:

Notice that the second picture of this kind is zoomed. I've implemented basic zooming, but I emphasize that the chaos game is generally poor in zooming, because the random iteration always wanders in the full area of the fractal, so we cannot zoom deep. That's why my method will probably not be a very popular one, and it will not replace the escape time fractal algorithm.

By the way, these pictures look like the Julia sets of the fractal z=z^3+c.

The previous pictures are of 640x480 resolution, but I've made two pictures for those who like changing wallpapers. So:

 

 

That's about the relationship of the escape time fractal iteration and the chaos game fractal iteration.

More playing with the Chaos Game

This blog post is a follow-up for two previous blog posts: "I was programming fractals a bit" and "More fractal wallpapers related to the Sierpinski Triangle". First it came to my mind that I could transform the coordinates of the next iteration point by taking the distance between 0 and 1, then executing "y=Math.sqrt(x*(2-x))" or "y=1-Math.sqrt(1-x*x)" on the distance, and then getting the transformed point based on the new distance. However, this idea did not provide nice pictures, unless I made a mistake in the algorithm:

1. Fractal-like picture by mistake:

2. Fractal-like picture by mistake:

After I've figured the mistake out, I realized that I can make nicer fractals if I get a weighted arithmetic mean between x and y as described above (because my original transformation was too much). So here are some nicer pictures:

3. (Math.sqrt(num*(2-num)) + 4*num) / 5:

4. ((1 - Math.sqrt(1-num*num)) + num) / 2:

5.  The same as above, but added triangles in the centers:

So these are the ideas I had for triangles, not as nice as expected. Another idea has come to my mind that I could also do the same with regular heptagons and the colors of the rainbow instead of just red, green and blue in the corners. I did not feel like to spend the time computing every ratios exactly, so the following pictures are the result of experimentation with shrinking values:

1. Regular heptagon fractal wallpaper on black background:

 

2. Regular heptagon fractal wallpaper on white background:

I think that's all for today... but I also have other ideas for programming math pictures, not only fractals... I may implement them if I have free time for it, and feel like to do it.

I didn't win on the Gravity Research Foundation 2020 Awards for Essays on Gravitation

I've found this essay among my old emails (in PDF), and I thought why not publish it here?

On the gravitational effects of third bodies on two specified bodies in space, and their consequences

Abstract:

In a system of many objects or bodies, the movements of two bodies close to each other are determined not only by their own masses, locations and velocities, but also by the masses, locations and velocities of all the other (third) bodies in the system. In some cases the gravitational fields of these third bodies act to increase attraction between the two specified bodies, but in some other cases they act as if to cause some degree of repulsion between the two bodies, like a tidal force. The author argues that this might be the real cause of many flat structures in the Universe, like the rings of Saturn, the planets of the Solar System, or the stars of the Milky Way.

Content:

Gravity is not as simple as calculations with the center of mass. Even in case of three bodies, the resulting problem of calculating their motions turned out to be very difficult, called the three-body-problem [1]. For the same reason, and also because I am not a Physicist (etc), I cannot come up with final calculations on the subject in the meantime. I just have one simplified example as follows: Let A and B be two bodies (actually point masses) of mass m, and their distance be 2s. Let O be the midpoint of the AB segment. Let c be a circle with midpoint O and radius 3s. For the sake of simplicity, our third bodies will be on the circle c. If the circle had a larger radius, our results would be similar, just different in scale. Let C1 and C2 be the intersections of the line AB and the circle c. Let e be a line through O perpendicular to the line AB, and let C3 and C4 be the intersections of the line e and the circle c. Let C1, C2, C3, and C4 also be point masses, or third bodies of mass m, and of course, let the initial velocities of A, B, C1, C2, C3 and C4 all be zero. Here is the figure:

Now we could apply the Newtonian equation of universal gravitation [3] to (approximately) get the forces which would move our bodies or point masses (F=G*m^2/r^2). C1 attracts A by F_A=G*m^2/(2s)^2, while C1 attracts B by only F_B= G*m^2/(4s)^2. As a result, C1 will cause A and B to increase their distance (of course, this will not be realized as A and B also attract each other, but still, their distance will be greater with C1 than without it). C2 will behave the same way as C1 did. However, C3 and C4 are in another position, and they will make A and B to be closer to each other than without C3 and C4. As for other possible bodies, on the circle c or farther, their effect on A and B will be somewhere in-between the repulsive (tidal [2]) effect of C1 and C2, and the attractive effect of C3 and C4. What does this mean? To me, it probably means that three-dimensional structures (with alike sizes in each dimension) tend to collapse, but flat structures tend to be preserved more. This could be the cause of why many structures in the Universe are (somewhat) flat: the Milky Way, the Solar System, and the rings of Saturn. Of course, this is harder to see in three dimensions with all the bodies having initial velocities (in different directions) and different masses, so I may let the evaluation of this to my Physicist readers. (Note that the WikiPedia article on „tidal force” [2] also mentions that the ring system of Saturn is caused by the tidal force, however, another WikiPedia article says that there is still no consensus as to how the rings of Saturn were formed [7].) Why is this important? Because otherwise the flatness of the celestial structures may be explained by other theories, like that which I have watched in a YouTube video by „minutephysics” entitled „Why is the Solar System Flat?” [4]. It says that every three-dimensional cloud of matter has a plane which defines its total angular momentum, which is preserved due to the law of conservation of angular momentum [5]… that's OK... but it also says that the matter not on this plane gets closer to it due to collisions between celestial bodies (maybe coming from different sides of this plane). Here I would argue that no collisions are necessary, but celestial bodies might arrange themselves near a plane for reasons I described above, i. e. the gravity of third bodies (but afterwards, they might collide as well, of course, especially when they enter into the stronger gravitational field of one another). Also, the „collision theory” might not adequately explain why the rings of Saturn are composed of so small objects, or why the asteroid belt [6] is there between the orbits of Mars and Jupiter. Well, I have come up with this idea of mine in 2014 or 2015, and included it in my math blog, and I might also have shared it on UseNet. Since then, I deleted that math blog of mine, and it was offline for about 5 years. Recently I have put some parts of it on the net again, with a different address, and some edition, so you could read my brief mentioning of this topic there [8]. I am not a Physicist, so I think it would be good if this topic were reviewed by some Physicists competent on this area.

References:
[1] Three-body problem, WikiPedia
https://en.wikipedia.org/wiki/Three-body_problem
[2] Tidal force, WikiPedia
https://en.wikipedia.org/wiki/Tidal_force
[3] Newton's law of universal gravitation, WikiPedia
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
[4] „Why is the Solar System Flat?” by „minutephysics”, YouTube
https://www.youtube.com/watch?v=tmNXKqeUtJM
[5] Conservation of angular momentum, WikiPedia
https://en.wikipedia.org/wiki/Angular_momentum#Conservation_of_angular_momentum
[6] Asteroid belt, WikiPedia
https://en.wikipedia.org/wiki/Asteroid_belt
[7] Rings of Saturn, WikiPedia
https://en.wikipedia.org/wiki/Rings_of_Saturn
[8] Noble challenge: search for an error in Mathematics (my blog post mentioning this idea) from the blog „Recreation in Mathematics by Árpád Fekete” (dead link removed)

More fractal wallpapers related to the Sierpinski Triangle

You may have seen the result of my playing with the Chaos Game, to make some fractal wallpapers related to the Sierpinski Triangle.

Today I've felt like experimenting a bit more with this, using some newer ideas. I'm going to share the result here, 4 new fractal wallpaper pictures. (The JavaScript code is not looking very nice, but if I'll feel like to clean it, I may put it on GitHub in the future...)

1. Filling the center of the Sierpinski Triangle as well, but change the ratio of shrinking so that there should be black areas in the fractal as well.

2. Partially filling the center of the Sierpinski Triangle.

 3. Rotate the smaller triangles in the Sierpinski Triangle.

4. Use different shrinking ratios, so this looks like like cell division / fission.

Summary: I could not make more interesting images than in my previous blog post about this (linked in the beginning), but at least I played a bit and used some Mathematics (without having proof about it in my mind).

I did not find interesting polyhedra containing the regular heptagon

As I was thinking more about this, what I also mentioned in an earlier blog post of mine (about polyhedra), I realized that I cannot find interesting new polyhedra that are not yet discovered, and somewhat regular, and maybe contain the regular heptagon. For example, I was thinking about three different regular heptagons ABCDEFG, HIJKLMN, OPQRSTU in 3 dimensions where A=H, B=I, O=K, P=J, Q=C, R=D. Then I conjectured that this way of putting together regular heptagons (so that at each edge, two-two heptagons should meet this way, by this angle) could be continued (in a way that these heptagons can intersect), but would end after some time, and the result (all of the points and the heptagons) would make an interesting (probably concave) new polyhedron. Whether or not the process ends, I planned to check by computer (approximately).

However, today I approached the problem differently. I asked the question: on the plane, can regular heptagons be put together at their edges, so that two-two regular heptagons should meet at each edge, and in a finite part of the plane there should be a finite number of heptagons? Then I realized that there will be 14 edges around a vertex of a heptagon, and the angles between these edges will be π/7. The angles of the heptagon are (2*π)*5/14, or π*(7-2)/7, by the way (and 5 and 14 are relatively primes). Then I found that two adjacent edges (around a vertex) will form a triangle whose angles are (π/7, 3π/7, 3π/7) and therefore, the third edge of this triangle will be shorter than the edges of the heptagons. However, these shorter segments are also around a vertex of the heptagons, connecting other vertices of the heptagons to it, and there are again 14 of them, with angles π/7 between them. We can do this process again and again, getting smaller and smaller segments. Therefore, there should be an infinite number of vertices, if we try to put together heptagons on a part of the plane (even if they intersect).

Well, it has come to my mind that I could try to reason somewhat like this (by analogy), about heptagons in the 3D space (or on a spherical surface). Of course, this problem is harder, so I shall suppose that the edges of the heptagons are the smallest distances between two points of the supposed resulting polyhedron. Otherwise I may find smaller and smaller segments by a way analogous to the previous reasoning, and conclude that the process of forming the polyhedron will not end. Then I realized that it is not possible for 3 or 4 regular heptagons to meet at one vertex. It is possible for 5 or more heptagons to meet at one vertex, if these heptagons intersect, and also intersect with the diagonals of the pentagon (or more) that is projected around this vertex. However, 6 or more heptagons meeting at one vertex will make smaller segments (as distances between vertices) than the edges (sides) of the heptagons. The only possible way seemed to be 5 heptagons meeting at one vertex, but I checked this possiblity and it looked impossible. Of course, all of this reasoning of mine is not Mathematically precise or perfect, but it is enough for me to reject and cancel further work on this topic, as my time is very precious.

What's next for me in Mathematics? I think I'll train myself in Number Theory (both by reading books about it and by solving KÖMAL exercises of type B, or later even of type A). (Fortunately, I have a lot of books and ebooks about Mathematics now, I do not need more for a long time.) Then I'll probably try to prove or disprove the Beal conjecture or the Goldbach conjecture... but if Mathematics remains hard for me, maybe I'll try to get rich quick by other means, too (e.g. writing books or composing music by LMMS).

I didn't win on the Alpine Fellowship Writing Prize, 2025 (On Fear)

I didn't win on the Alpine Fellowship Writing Prize, 2025 (on the topic "Fear"), I got a notification about it today. Thus I'm going to share my essay here, because it is non-fiction (but I do not plan to share my losing poetry or fiction):

VIPs can fear

I fear of dogs, especially the large ones like a pit bull, especially when there are more of them. I would like to show you that I am right. Flies do not seem to fear, neither do locusts. That is because it is a good strategy for them not to fear in the struggle for existence. Many other animals fear, however, humans included. That is because evolution favours fear in many cases. Thus it is reasonable to fear sometimes, and it is reasonable not to fear at other times. Still, there are people who tend to fear more often, they are the cowards. Traditionally, cowardice is considered a bad trait, while its opposite, courage is considered a virtue since at least Aristotle. Indeed, Aristotle taught courage (and not audacity), and his disciple, Alexander the Great successfully applied it to spread the ancient Greek language and culture more in antiquity. However, this fact does not justify the claim that courage is better than cowardice in general, or in our case of fear of dogs. We need to examine the question further.

As far as we believe, Socrates, Aristotle's teacher's teacher, said that "Virtue is knowledge", and Aristotle himself taught that "Courage is a virtue", so if both Socrates and Aristotle were right, then we could infer that "Courage is knowledge". According to this theory, the people who know more about dogs and about fighting may be usually less afraid of dogs than those who only have their instincts of fear. For instance, news about someone strangling a cougar has given me some more courage against dogs. Other examples may be found in the novel "Call of the Wild" by Jack London, so this could also help, at least when we have clubs. (People with clubs could easily beat dogs in that novel.) Still, there are many times when I fear of dogs and I don't think that more knowledge could help, except for knowledge about Heaven. There is also knowledge that increases fear, like the Wikipedia List of fatal dog attacks including a lot of cases about pit bulls. Simply, there may be cases where the chances of dogs are better in a fight. Consequently, cowardice in these cases may be a virtue, if virtue is still knowledge.

Perhaps it will be enlightening to you if I share the details of some specific cases where I fear of dogs. The shortest route from/to the city center passes by a house with a German Shepherd dog there. The gate is usually closed, but sometimes it was open and the owner of the dog was there. Therefore, I usually take another route. (Once I went that way in spite of the dog and the gate open, because I was in a depressed mood.) In other cases, we went to my grandmother's house by bike. There is a house with many big dogs on the way, where the gate is usually closed, but sometimes it was open and the owner of the dogs was far away in the house. Therefore, I prefer not going that way, if I can... and I do not bless those who keep big dogs.

Am I a coward? This really seems to be so, because I am also timid when approaching women. But I can explain. One of my acquaintances has told me that he drank alcohol to pluck up courage to talk to a girl. Our fear of courtship might be justified when we unconsciously feel that the girl or woman is not the right companion for us. Otherwise, our Nature may allow us to overcome this fear anyway. Now if not everyone can be our partner, then we are VIPs (very important persons), at least to ourselves! Thus the lack of bravery might mean something positive, something of value! VIPs also deserve more protection from dogs and from other attackers. Fear can mean such a protection. VIPs should not depend on the mercy of dogs!

Once I walked in the city, and I saw a German Shepherd dog (or a similar one) on the other side of the street closed in a large yard. As far as I remember, it saw me, too. I raised my arm and shaped my fingers as if I were holding a sword. I imagined that it was the Sword of God (Sword of Attila) or a lightsaber (from Star Wars). Then the dog seemed to fear and whine.

I was programming fractals a bit

It is easy to draw a Sierpinski triangle by programming, there is an easy algorithm for that called the Chaos game. By enhancing this method, I generated some fractals today afternoon. My main goal was to prepare a picture for a physical present that I'm going to give to someone. However, since I've written the program, I've also generated some wallpapers for myself. I'm also going to share these pictures on this blog. First comes the original Sierpinski triangle, colored by my idea:

Next comes the "Golden Sierpinski triangle" that I also added earlier to the fractal drawing software XaoS (but XaoS uses a different algorithm and different coloring). It uses the reciprocal of the golden ratio for shrinking instead of 1/2:

Next comes the same fractal, if I do not run the Chaos game long enough:

 

Happy PI Day, 2025!