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Hello everyone, hope you are doing well. Since we have a huge spectrum of people here including high schoolers to professional mathematician, I want to ask you what’s a math theorem(s) that you like the most. Be it any field but the statement, brief explanation of the statement and why you like it would be appreciated.

I personally have a lot of favourite theorems but I’d like to quote this one.
Theorem: Every oriented 3-manifold is parallelizable.
Which means the tangent bundle TM of M^3 is always trivial.

Do share yours, looking forward to reading yours.

There seems to be some breakthrough (non-AI) by well known chinese mathematicians.

Who's a mathematician from the past, and what's a modern mathematical fact that they could have understood but that would have absolutely blown their mind if you'd told them about it?

edit: I originally meant the post to be specific. For example, Conway, a specialist in tilings, would have been delighted to know that an aperiodic monotile exists, and a very simple one at that. Or it could be Euclid and arbitrarily long arithmetic progressions of primes (Green-Tao theorem).

Can anyone explain the basic concepts, differences, and similarities? I know it is kind of like using one to describe the other

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

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If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

When you define a mathematical object, it is usually a good idea to consider what kind of sub-objects (instances of the same type contained inside it) it might have---so is considering what methods might exist for generating examples of such a mathematical object from some minimal information. So it is in linear algebra: we know these as subspaces and spans.

This is a good opportunity to start learning about how the concepts of linear algebra actually connect with applications of various sorts. The most obvious example (and one covered by every linear algebra course in existence, I suspect) is that in Euclidean space, subspaces correspond to points, lines, and planes through the origin. This is already reason enough for this to be potentially important, but since I want to lay the groundwork for some applications that are less often featured in linear algebra courses, we'll also talk about how, say, the space of continuous functions is a subspace.

Read the full post (for free) on Substack: Subspaces and Spans

Only focussing on the polyominoes, there are 12 classes that exist as shown with each of the 12 polyominoes present. But what is a class? And how can a polyomino belong to one?

A class is named like so 1/2/4 where each number denotes the kind of squares a polyomino may only have. 1 means there are squares with only 1 orthoganol squares (neighbors), 2 means there are squares with only 2 orthoganol squares and so on to 3 and 4 as squares may only have a maximum of 4 orthoganol squares neighboring it.

Another note about a class is that there must be an equal number of each square type. so for 1/2/4, there must be N squares with 1 neighbor, N squares with 2 neighbors, and N squares with 4 neighbors.

Now that you know what a class is, it can be asked, how many classes are there? 31. 15 of which ask for squares with 0 neighbors, which is impossible for polyominoes as every square is connected. Therefore class 0 is the only existing class which asks for squares with no neighbors. The rest of the impossible classes are slightly less trivial: 3, 4, 1/4, 3/4. That leaves 12 which hey look at that they've all been found as shown in the first image!

Same logic goes for polyiamonds except there's 15 possible classes, 7 of which are trivial and 1 that's not that being class 3. That leaves 7 classes left which has polyiamonds shown to prove the existence for each class.

What else can be said?

1. How many polyominoes/polyiamonds are in a class?

2. What's the smallest polyomino/polyiamond in each class?

3. What symmetries can be achieved in each class?

Other problems that are more challenging

Polyominoes but harder, they way to count neighbors is now the 8 surrounding squares around one square kinda like a king's move or how minesweeper counts mines. Should be 60 remaining classes yet to be solved or disproven.

https://www.reddit.com/r/mathematics/comments/1x9akue/list_of_the_unproven_classes_i_currently_have/

Polyhex version where each hexagon counts the surrounding 6 hexagons around itself. The remaining unproven classes are 1/2/3/5/6, 1/2/4/5/6, 1/3/5/6, 1/4/5/6, 1/2/3/5, 1/2/4/5, 1/2/4/6, 1/3/5.

Polyiamonds but harder, counts all 12 triangles around a central triangle! There should be around 8191 classes which I have not touched as I am honestly afraid to touch that. But what's cool is that classes with 0 or 1 cannot exist leaving around like 2049 nontrivial classes.

What can be said about those problems?

1. Again how many, what's the smallest, what symmetries?

2. Can a class only exist if holes are allowed in a polyform?

3. What impossible sets can exist if the equality requirement is ignored?

4. Where/how can these solutions be documented online?

Things I find interesting but are unsolved

1. Can a class have a finite number of polyforms thats not just one?

2. May a class exist where every polyform of that class doesn't have holes? Or may some classes also have a maximum number of holes?

Роман Михайлов сейчас в своем тгк открыл анонимные вопросы и вот что я решил задать(см. ниже). Он пока не отвечает, но может тут у кого то есть мысли по этому поводу? Хотя если честно я думаю что те кто могут ответить на этот вопрос на реддите не сидят. Буду благодарен любым конструктивным ответам.

В один момент возникло такое ощущение что математика бессмысленна. то что в конечном итоге это просто абстрактная пустота. а потом я вспомнил про то какая вокруг этого всего была эзотерика раньше и насколько она наполняла математику, например у Пифагорейцев. они могли создавать гармоничную музыку с помощью математики. и такое ощущение что математика отделилась от духа и потеряла смысл.

и вообще логическая система в которую изначально завернута математика довольно ограничивающая. переживаемый опыт часто не вписывается в эту структуру. и соответственно есть более творческие способы воспринимать мир, например musicology.

что вы думаете по этому поводу ?

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

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All types and levels of mathematics are welcomed!

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My friend’s 7yo is already a math-enjoyer—he loved the book “Math for Curious Kids” and keeps wanting to find out more. I’ve found a few things that have been hits (a graph theory zine for kids, a group theory coloring book), but I’m always on the look out for more. If anyone has some other fun math books for kids, especially that have some game/problem-solving/recreational aspect, I’d love to recommend more things for him!

Hello, I've started to study italian and I'd like to read math books or lecture notes by Italian authors (and written in Italian).

Do you have suggestions of books that you find particularly interesting?

I'm mostly interested in linear algebra, analysis, differential equations, dynamical systems and probability.

Thanks

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

• informal announcements of AI-assisted discoveries, such as those not yet published in a peer-reviewed journal, or not uploaded as a paper to arXiv;

• informal announcements of discoveries related to AI architecture (if relevant to mathematics);

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AI-assisted mathematical papers published in peer-reviewed journals or as arXiv preprints may be submitted as their own posts.

Please keep in mind rules 1 and 6 of our subreddit.

Hello, I'd like to showcase some new results in tilings.

This summer, I was contacted by a student named Alessandro Longo who tried to enumerate k-Euclidean tilings for his thesis. He eventually found online the algorithm I made for this problem several years ago. He used his thesis to rigorously prove what I had only believed, i.e. that the algorithm is truly capable of enumerate all tilings that satisfy set conditions. With that in hand, he built this website:

https://www.thetilingatlas.com/

I am really humbled that my amateur algorithm could get this far! We have tilings of all kinds, but mostly hyperbolic ones (they are just easier to find), and now we even have a stand-alone search program. Not very user-friendly yet, I'm afraid.

I wanted to share some results. They are from recent searches, so not uploaded yet.

This is a very complex hyperbolic tiling, but you need more slices to see it. So if we going straight up, we get

And moving aside to better capture the apeirogons,

It contains 8 kinds of regular polygons with 4 different edge lengths.

For Euclidean tilings, I have tried to look at 14-Archimedean ones. The first one, Amazing Amalgam, was found using my algorithm in 2022, I think:

This one is 24-uniform, but the search program is now fast enough to go to k=40 in reasonable amount of time, so here's a 40-uniform result:

We hope that enyone who's interested in tilings will give the site a look :)

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

There is Pascal triangle where the 2 numbers above are used to get the number below, then there is Delannoy triangle where the 3 numbers above are used to get the number below

Based on that I thought of a stronger triangle where as many numbers above as possible are used to get the number below and I got this triangle

The rule is 1st number in a row is always 1, 2nd number comes from the 3 numbers above, 3rd number from the 10 numbers above, 4th number from 21 numbers above and so on till we reach the mid point of any row after which the numbers get repeated in reverse order to get this triangle

I couldn't find this triangle anywhere but this still seems to be interesting

Lately I've been enjoying exploring the connections between algebra in the natural numbers and combinatorics. It's a really rich vein of mathematics to play with and even though it's not an area of math I've studied formally I'm constantly surprised that I haven't heard of some of the patterns I've spotted.

I wanted to share an algorithm I found that lists out prime factors of the natural numbers sequentially without using any algebra. I'm not sure how interesting it is as it's all derived from basic observations about primes but it's nifty nonetheless.

-----

Setup:

Start with a list L with an empty multiset L= [{}] and k = 0.

Iteration:

Take a copy of the L and reverse it for Lr, pair up the multisets in Lr with the multisets of L in order (the nth multi set of each list paired up).

Take the multiset union M of all the multisets of Lr that are sub multisets of the multisets in L which they are paired with.

If M is empty, increment k by 1 and add the multiset {k} the back of L.

If M is non empty but part of L then it has a unique element, increase it's multiplicity by 1 and add it to the back of L.

If M is non empty and not part of L add it to the back of L.

-----

The first few iterations give: [{}, {1}, {2}, {1, 1}, {3}, {2, 1}, {4}] which is the list of prime factors of 1 - 7, where i in the ith prime. (e.i. 2,1 -> 3*2 = 6).

I'd be curious to read up more about this if anyone might be able to point me in the right direction. I haven't proven it's validity yet but computationally it holds and I'm sure the proof is halfways straightforwards, I might try and work that out in the next few days.

I do have it written up as a python script too if anyone else wants to check that out.

This post is to announce that there will be a lecture course on General Topology (a.k.a. Point-Set Topology) conducted in Mandelcord (the blue math discord server). Everyone is invited to join: discord.gg/maths.

If any of you were present in the Real Analysis lecture course during the summer, you know what I'm talking about, and you might be pleased to hear that this course will also be run by Pear Category Theorem (the same person).

Why YOU should do topology

Point-set topology is one of the most important subjects that you learn, and mastering topological thinking is as important a milestone as learning to write a proof. This is a good subject to study alongside measure theory, or simply by itself after one has learned analysis or algebra.

Course Structure

The course will primarily follow the flow of the first half of Munkres. The plan is to go through all 8 chapters, spending maybe one to two weeks on each chapter, aiming for an approximately 16 week course. Also the instructor intends to supplement the book with his own intuition on the subject, as well as connections to other subjects that he thinks are of interest and possibly also other topics not covered in the book. Many examples of various pathological properties will be provided. The study of metric spaces and functions on them will be paramount; but where Rudin introduces metric spaces as merely a generalization of Euclidean spaces, we will explore just how pathological these spaces can be, or else what properties they must have.

The instructor also plans to put far more focus on the topics of topological groups and nets than Munkres does. Topological groups, because often in "the real world" we do not care about a topological space by itself, but rather some other more complicated structure given a topology, in order to see what the topology can tell you about the thing itself; topological groups provide a playground to learn that mode of thinking. Nets are also very important; where sequences are of importance in R, they can only tell you so much in general topological spaces, so nets therefore become the natural way to recover theorems about sequences.

Material and References

1. Topology - Munkres

2. Counterexamples in Topology - Steen & Seebach

Timeline

The course starts with the first lecture on 17th November at 9 PM CET (20:00 UTC), with subsequent lectures on Tuesdays and Thursdays. Lectures will be recorded and posted promptly.

Prerequisites/Who can join?

Everyone. It is completely free for anyone who wants to join. Just click the link and join the server: you can find the channel called topology-course in the section called General maths. As for prerequisites, a first course in Real Analysis and Group Theory is sufficent.

Looking forward to see all of you there. Thank you.

We know that a natural transformation between functors is a collection of morphisms which satisfies a certain commutative diagram.

I have my own ways to to understand this:

1. Naturality vaguely means that no artificial/inconsistent system of choices have been made, which means that the collection of morphisms are not biased towards any particular thing, which means that the collection of morphisms must behave " nicely " with everything else.

Now since the objects we are concerned with are a collection of morphisms, the term " everything else " just means " every morphism that can interact with the collection of morphisms " and this nice interaction ammounts to the naturality square being commutative.

2. In certain cases like G-Sets or morphisms of sheaves it's relatively easy to understand the definition of natural transformations because here they are just structure preserving maps, and for the general case we just generalize these specific observations.

3. In category theory/algebra one often defines a morphism to be something that commutes with everything else that already exists, for example morphisms of chain complexes, so the same goes for natural transformations.

What were Eilenberg and Mac Lane thinking while defining naturality? Do you guys have any nice motivation?