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!

Many ways to practice Mathematics (some interesting links)

I thought that I knew all the best English-language and German-language podcasts about Mathematics, but some days ago I've found two that are not only alternatives, but may also be better than the others! The first one is called "My Favorite Theorem", and it is accessible on the following link:

https://kpknudson.com/

The other is called "Breaking Math", and it is found here (Maybe I knew the title of this one, but I could not access it on another location for dead links):

https://www.breakingmath.io/

Maybe there will be more. Thus I'm going to listen to these podcasts (EDIT: except Breaking Math, as it is only accessible through service providers which prohibit use for commercial purposes), even if I did not plan to listen to new podcasts in 2025. Earlier I have also started to read free google books, and I first read about the history of Mathematics. As far as I remember, I read "A General History of Mathematics" by John Bossut in 2024, and I'm currently reading "A Short Account of the History of Mathematics" by W. W. Rouse Ball, and I like the latter more than the former, so I can recommend the latter (maybe it is best to read both). I've also read "A History of the Mathematical Theory of Probability" by I. Todhunter in 2025, but although it has broadened my horizons, it was not entirely intelligible to me, due to its complicated integrals and other things hard to understand. It did not make me feel like doing this kind of Mathematics, although I still had a takeaway: the St. Petersburg Paradox. Apart from these resources, I've also found a very good resource for Mathematics, which can be found here:

https://planetmath.org/

It has a similar licence as WikiPedia, so maybe these two are the best places to read about advanced Math. I especially liked the proof of Lagrange's Four Square Theorem here:

https://planetmath.org/proofoflagrangesfoursquaretheorem

Recently I was also interested in Symmetry Groups and Point groups in three dimensions, but it is still hard to understand for me, whether these theories prevent me from discovering such polyhedra that are highly symmetric, but yet unknown. After these articles, it seems to be harder to imagine that I will be able to discover anything new in this field. That's why I'm also turning part of my attention to Number Theory, first by solving easier exercises, e.g. from KÖMAL. By the way, I have also received an old Number Theory textbook for Christmas (translated to Hungarian from Russian).

By the way, after listening an episode of the "My Favorite Theorem" podcast about the Gauss-Bonnet Theorem, I have found (again) that my discovery that I mentioned earlier on this blog is already known as Descartes's Theorem on the "total defect" of a polyhedron. So this was a summary of my recent Math life.

About Writing, I have also some things to share. Nowadays there are less international essay contests for adults (with acceptable terms) than there were earlier, but maybe it's good to know that there may also be opportunities to publish works in journals, and the best articles in those journals may win prizes. I've found the "Royal Economic Society Prize" and the prizes of "The American Finance Association". Apart from these, I could also subscribe to the newsletter of the "Independent Social Research Foundation", maybe there will also be some good opportunity there.

The Brahmagupta–Fibonacci identity and "Binary quadratic forms"

I really like Fermat's theorem on sums of two squares... and its proof probably requires the Brahmagupta–Fibonacci identity (or by another name, Diophantus identity), so I like that as well. When looking at this identity and its generalizations, it has come to my mind that I may be able to generalize it as well, or at least discover something similar. The Brahmagupta-Fibonacci identity says that the product of two integers that are each expressible as the sums of two squares, so this product is also expressible as the sums of two squares. This can be checked by practicing basic algebra:

(a²+b²)(c²+d²) = (ac-bd)²+(ad+bc)² = (ac+bd)²+(ad-bc)²

Brahmagupta actually knew a more general identity:

(a²+nb²)(c²+nd²) = (ac-nbd)²+n(ad+bc)² = (ac+nbd)²+n(ad-bc)²

So I made the conjecture that the product of two integers that are each expressible as (x²+xy+y²), for different integer values of x and y, so this product is also expressible as (x²+xy+y²) for some integer values of x and y. My conjecture was justified by checking it for small values of x and y. But it was harder to prove the conjecture, as I had no formula to check. However, by looking at many examples, I managed to guess the formula, so I could check it then:

(a²+ab+b²)(c²+cd+d²) = (bd-ac)²+(bd-ac)(ad+bc+ac)+(ad+bc+ac)² = e²+ef+f²

where:

e = bd-ac and f =ad+bc+ac

Then I was thinking about how useful this result might be... and I have also checked WikiPedia more, maybe I find something about it... and I've found! Actually, Gauss and Lagrange were working on something even more general, i.e. (integral) binary quadratic forms, so they probably also knew what I "discovered" here. Still, it was good to take care of it, as this feels better than solving crossword puzzles, for example.

By the way, I also checked numbers of the form x²-xy+y², and I have found that:

• They are always positive or zero, as when xy is positive, then either x²>xy or y²>xy
• They are exactly the same numbers as those of the form u²+uw+w², because we can convert them this way: x²-xy+y² = (-x)²+(-x)y+y² = u²+uw+w² where u=(-x) and w=y. Still, it is interesting that they also seem to be the same numbers for only non-negative values of x, y, u and w.

What do I do in Mathematics in these times?

Although today I had other things to do (visit my relatives), on most Saturdays I'm planning to do some Math and try to do it offline without computers, if possible. Thus I would observe the "Sabbath" this way (ideally, I should not turn on my laptop on Saturdays, and Mathematics can be an ideal activity I could do on those days). What these Math activities can be?

• I've downloaded a lot of lecture notes and books about Mathematics from the Internet (mostly in English), but only those which have a good license like CC-BY, CC-BY-SA, GNU FDL, public domain, Project Gutenberg license or downloaded from "free Google books" (which I assume should be almost public domain). These files occupy about half the size of a DVD on my disk (and more, since I have more copies of them). Thus I'm occasionally also reading some of these books, or their printed versions, which I received as presents. (So they asked me what should they give me as a present and I wrote that they could print some of these books for me like university students print their lecture notes.) Earlier I also printed some of these myself, but those are just separate pages in folders.
  

• I've also picked some good-looking Math exercies from the website of KÖMAL (I linked the English-language version for you, but I picked them in Hungarian, of course) (EDITED 2024-12-07). Thus I can think on them on Saturdays. Many people regularly solve crossword puzzles, Sudoku and the like, but in my opinion these KÖMAL exercises may be much more interesting and challenging than those. As a compensation, I'm planning to order 1% of my taxes to the foundation that runs KÖMAL (MatFund) in the next year (in Hungary, we can order 1% of our taxes to a NGO we choose, and another 1% to a Church we choose). In this year, however, I already ordered this 1% to an environmentalist organization (so 1% of my taxes for the year 2023 is given to Environmentalists, but 1% of my taxes for the year 2024 will probably be given to Mathematicians in 2025).
  

• Sometimes I am also thinking on really hard Math problems, like the Goldbach conjecture (by the way, I prefer those which can be formulated by relatively simple statements, so I'll probably not be thinking on the Riemann Hypothesis in the near future, because it is not "elementary"). Lately I had not many ideas for them, so they were neglected... also because I realized that the KÖMAL exercises are still hard for me, so it is not likely that I can solve these much harder problems in the meantime. (But I acknowledge that even if I cannot solve the Goldbach conjecture, I can become wiser if I try it and learn new things along the way.)
  

• ... and the following is what I am writing this blog post for: I've begun to think about new types of polyhedra, are they possible or not? This is what I could try to research. Thus I have some ideas as seeds (top secret), and I could try to do Math to decide if these ideas lead to new types of polyhedra or they fail. During this research I've found the Virtual Polyhedra website of George Hart, which is very interesting and well done! It lists a lot of polyhedra types I previously knew nothing about (although I knew the Archimedean solids and their duals, for example), and the interactive visualizations are outstanding. However, my ideas are not there, so they will either fail or they are still very well concealed by God. Mathematicians teach that the symmetry types of possible polyhedra may not be very different from what we already know about, but I would still like to check this myself. However, I may not be skilled enough in Mathematics to do it, so it is possible that I will use computers to help me... but then I should probably not do this on Saturdays. On other days, I still focus on writing a new book about the critique of Christianity, so it is possible that this polyhedra goal of mine will progress very slowly, and it will still be an open question for me in the next year... but at least I wrote about this today so that my readers know that I may be about to discover something, which may be more worthwhile for me than thinking on the Goldbach conjecture.
  

By the way, I could also do some other random things related to Mathematics not listed here. For example, I recently realized that I couldn't have beaten the best 8th graders in Mathematics on the Zrínyi 2021 competition for elementary schools (so I am probably weaker in this type of Mathematics than I was in elementary school). I've also recently mentioned some of my activities or plans related to Mathematics on my Hungarian blog (including some logic puzzles on computer or physically). Since then, I've also found a new game called "Chroma" in Ubuntu Linux, that is also looking nicely (if I had time for it)! On my Hungarian blog, playing chess is also mentioned, which I try to do regularly on LiChess (usually about one game per day). It would be good to level up my chess skills to be able to win a competition in my town, but I will probably not go to local chess competitions until I play good enough on LiChess or against a computer on Normal level.

By the way, knowledge in Mathematics is something that is good to have. Even if this World is evil, Mathematics is good, so we can build up knowledge about Mathematics in our brains, and this way we can get close to the universal values that are true in all possible worlds, not just ours. Thus I've also begun to read a book about Mathematics that contains many things I already know, but it was still good to read it, because it was organized in a special way. (E.g. it has chapters about the different proof techniques, which is a good summary of them.) Thus Mathematics is beautiful.

The Mandelbrot VS the Triceratops fractal in XaoS: what makes the difference?

Many years ago I played with the source code of the XaoS fractal zooming program... I searched for new (fractal) formulae, and viewed how they look like as fractal images. I had the idea that instead of f(z)=z²+c (the iteration formula for the Mandelbrot set), I could use the generalization of triangular numbers (for complex numbers), so I used triangular numbers instead of squares. The formula for triangular numbers is n*(n+1)/2, so the iteration formula for my new fractal has become f(z)=(z²+z)/2+c. A very interesting fractal formed, which I called Triceratops. This was interesting because it was very different from the Mandelbrot set...

The Mandelbrot set

The Triceratops fractal

Of course, we need to zoom in to see the interesting parts of the fractal, like:

These are kinds of images we do not usually see in the Mandelbrot set. Later I realized that this is because the escape condition was too low for the Triceratops fractal... the escape condition for the Mandelbrot set is 4, which means that the iteration stops when |z|²>4, or abs(z)^2>4. In the case of the Mandelbrot set, this is equivalent to saying that the point is not part of the fractal, because it diverges to (some) infinity... and therefore, it is colored by some color different from black (depending on the number of iterations needed to get this result). However, in the case of the Triceratops fractal formula, this condition of |z|²>4 was not enough, so we excluded some points which might otherwise be parts of the fractal (and be black). Increasing that to e.g. 32, we might get a fractal very similar to the Mandelbrot set:

The f(z)=(z²+z)/2+c fractal with escape condition |z|²>32

Zooming in this fractal makes images similar to those of the Mandelbrot set... and conversely, if we reduce the escape condition of the Mandelbrot fractal (to e.g. 1 or 0.8), we may get images similar to those of the Triceratops fractal... Thus the secret of these interesting images lies in the escape condition, not really in the formula.

Zoom in the Mandelbrot fractal but with escape condition |z|²>0.7