8 ms·

Can someone explain the n^2 + 2 triangles paper?

by quickthrower2 3y ago

brap 3y ago
Abstract: probably
khazhoux 3y ago
Yes
- quickthrower2 3y ago
  Good!
rawling 3y ago
The paper doesn't answer the problem in its title (n²+1) but demonstrates two different "advancements" towards it (n²+2) > We have posed a fine (in our opinion) open problem and reported two distinct “behold-style” proofs of our advance on this problem. There's also a linked PDF which, if I'm reading it correctly, trivially proves n²+1 is impossible.
- kevinwang 3y ago
  No, n^2+1 is an open problem (conjectured to be impossible though)
  - rawling 3y ago
    Ah yeah, the PDF shows it for n=2 and I assumed that scaled up but it doesn't.
scatters 3y ago
It exhibits two distinct constructions both of which demonstrate that n^2 + 2 unit equilateral triangles are sufficient to cover an equilateral triangle of side n + ε. The obvious area argument shows that at least n^2 + 1 are required. A small modification of the second figure can show that for any non-equilateral triangle, n^2 + 1 such triangles will cover a similar triangle of length ration 1 : n + ε; it remains (as of 2010, at least; see [1]) an open problem whether a construction of n^2 + 1 triangles exists in the equilateral case. 1. http://www.wfnmc.org/mc20101.pdf http://www.wfnmc.org/mc20101.pdf
- kristopolous 3y ago
  What's ε in this case? How is it bound? Also, they're permitting overlapping of triangles, right? If that's the case, why can't you just add an arbitrary + 1 wherever you please and call it a day?
  - mkl 3y ago
    ε > 0. Yes, overlapping triangles. Add an arbitrary + 1 to what? You need to arrange the small triangles so that they cover the big triangle of side length n+ε. n² unit equilateral triangles cover a big equilateral triangle of side length n without overlap, so at least n²+1 are needed for side length n+ε, and the paper shows (not especially clearly, IMHO) that at most n²+2 are needed.
    - deleted 3y ago
      [deleted]
    - crote 3y ago
      Figure 1 on its own is a pretty decent demonstration, once you zoom in quite a bit. The annoying part about the paper is figure 2: it shows a different method of doing so, without mentioning that it is unrelated to figure 1. It is also drawn in a less obvious style, which really hurts its readability.
      - mkl 3y ago
        Yes, Figure 1 is okay, but Figure 2 is not very clear; most of the triangles are missing and have to be imagined. I also think the examples for a single n are not the best argument for the general case - the reader can extrapolate other diagrams and a proof but I think a slightly longer paper could be clearer.
  - crote 3y ago
    ε is used to denote an arbitrary small value, which isn't zero. Overlap is indeed required here. Let's say you have a large equilateral triangle of side n. Covering it with triangles of side 1 is pretty easy: you build a pyramid out of them without any overlap. That requires n^2 smaller triangles. Now let's say you make the large triangle sliiightly larger, so it'll have sides of n+ε instead of n - for example we gone from 11.0 to 11.00001. How many smaller triangles do you need to cover it? Obviously n^2 isn't going to be enough - because that was exactly enough to cover a large triangle of side n. Our slighty-bigger triangle is slightly bigger, so it has a larger area. We're going to need at least one additional small triangle to cover the added area, leaving us with n^2+1 as an absolute lower bound. But just because it is a lower bound doesn't mean it is actually possible - you'd first have to demonstrate that it can actually be done. This paper demonstrates two different methods of constructing it with n^2+2 triangles, providing an upper bound which is definitely possible. This means we still don't know the exact number of triangles required, but we do know it is definitely bigger than n^2 and definitely smaller than or equal to n^2+2. This leaves the question: is n^2+1 possible?
tornato7 3y ago
I was curious if GPT-4 Vision could explain it: The paper presents a geometric problem centered on equilateral triangles. The key question is whether it's possible to use \( n^2 + 1 \) small equilateral triangles (each with side length of one unit) to cover a larger equilateral triangle that has a side length just slightly more than \( n \) (specifically, \( n + ε \), where \( ε \) is a small positive value). The two figures provided illustrate possible arrangements of the smaller triangles within the larger one: 1. *Figure 1*: This demonstrates that \( n^2 + 2 \) small triangles can cover an equilateral triangle whose side is \( 1 + ε \). It's evident that the small triangles fit neatly inside the larger triangle. 2. *Figure 2*: This shows a different configuration where the large triangle has a side length of \( 1 - ε \). It seems to suggest that with just one fewer triangle (i.e., \( n^2 \)), the tiling is not possible for a triangle of side length \( 1 + ε \), but it may be for \( 1 - ε \). The paper, although succinct, poses an intriguing tiling problem in geometry. The authors likely aim to stimulate thought and discussion on this particular geometric configuration and challenge readers to consider the conditions under which such tiling is feasible. Given the brevity, the paper might be a problem statement or a brief note, rather than a full research paper with exhaustive proofs.
- mkl 3y ago
  It couldn't.
  - IshKebab 3y ago
    Pretty damn close though! I haven't seen an explanation of what the second figure is trying to show so I'm not sure about that one. (And also their assertion that no further explanation is necessary is clearly bullshit.)
    - mkl 3y ago
      Its attempts at explaining both figures are totally wrong. Wrong side lengths, wrong assertion that the small triangles fit inside the large triangles, wrong relationship between the figures, complete misunderstanding of the second figure. The second figure is actually showing another arrangement of n²+2 small unit equilateral triangles covering an equilateral triangle of side length n+ε.
jacurtis 3y ago
I think the fact that you have to ask this proves that it is objectively a bad paper. The whole point of academic papers is to contribute to the larger global knowledgebase. You acknowledge the work that was done before, you submit your contribution and then you suggest how people can build or expand upon your work. This paper in question is just trying to be a mic-drop, like a middle-finger to academia. Generally research papers cover background context as their 1st and 2nd sections (at least IEEE format papers do). So normally a paper like this would start with an introduction section which explains what the paper is accomplishing and then the background section number two would explain context for where the author is coming from or what inspired research or background to justify its value. These sections would provide the context you are looking for and at the very least give references for you to go back and learn about it on your own. Even a few sentences would have been powerful here. This paper does fail to really provide value in my opinion and is objectively a bad paper. With some additional context from the introduction and background this could be much more valuable. Less critical, but also important is to acknowledge limitations and suggest future research. Now with all that being said, I'm not saying research papers are perfect. It is easy to find examples that go too far the other way, with far too much verbosity and pomp and circumstance. So I do at least acknowledge the statement being made with this paper that maybe all you need is two words. The reality is we should be somewhere in the middle. I read 3-10 academic papers per week, and the average page length is usually around 10 pages and really should be closer to 3-4. So i acknowledge the statement being made here, but this paper is clearly a protest, and not actually a productive example.
- Ar-Curunir 3y ago
  While providing introducing the problem and motivating itis common in CS paper, it's not a common practice in mathematics.

Can someone explain the n^2 + 2 triangles paper?

brap 3y ago

Abstract: probably

khazhoux 3y ago

Yes

quickthrower2 3y ago
Good!

rawling 3y ago

The paper doesn't answer the problem in its title (n²+1) but demonstrates two different "advancements" towards it (n²+2) > We have posed a fine (in our opinion) open problem and reported two distinct “behold-style” proofs of our advance on this problem. There's also a linked PDF which, if I'm reading it correctly, trivially proves n²+1 is impossible.

kevinwang 3y ago
No, n^2+1 is an open problem (conjectured to be impossible though)
- rawling 3y ago
  Ah yeah, the PDF shows it for n=2 and I assumed that scaled up but it doesn't.

scatters 3y ago

It exhibits two distinct constructions both of which demonstrate that n^2 + 2 unit equilateral triangles are sufficient to cover an equilateral triangle of side n + ε. The obvious area argument shows that at least n^2 + 1 are required. A small modification of the second figure can show that for any non-equilateral triangle, n^2 + 1 such triangles will cover a similar triangle of length ration 1 : n + ε; it remains (as of 2010, at least; see [1]) an open problem whether a construction of n^2 + 1 triangles exists in the equilateral case. 1. http://www.wfnmc.org/mc20101.pdf http://www.wfnmc.org/mc20101.pdf

kristopolous 3y ago
What's ε in this case? How is it bound? Also, they're permitting overlapping of triangles, right? If that's the case, why can't you just add an arbitrary + 1 wherever you please and call it a day?
- mkl 3y ago
  ε > 0. Yes, overlapping triangles. Add an arbitrary + 1 to what? You need to arrange the small triangles so that they cover the big triangle of side length n+ε. n² unit equilateral triangles cover a big equilateral triangle of side length n without overlap, so at least n²+1 are needed for side length n+ε, and the paper shows (not especially clearly, IMHO) that at most n²+2 are needed.
  - deleted 3y ago
    [deleted]
  - crote 3y ago
    Figure 1 on its own is a pretty decent demonstration, once you zoom in quite a bit. The annoying part about the paper is figure 2: it shows a different method of doing so, without mentioning that it is unrelated to figure 1. It is also drawn in a less obvious style, which really hurts its readability.
    - mkl 3y ago
      Yes, Figure 1 is okay, but Figure 2 is not very clear; most of the triangles are missing and have to be imagined. I also think the examples for a single n are not the best argument for the general case - the reader can extrapolate other diagrams and a proof but I think a slightly longer paper could be clearer.
- crote 3y ago
  ε is used to denote an arbitrary small value, which isn't zero. Overlap is indeed required here. Let's say you have a large equilateral triangle of side n. Covering it with triangles of side 1 is pretty easy: you build a pyramid out of them without any overlap. That requires n^2 smaller triangles. Now let's say you make the large triangle sliiightly larger, so it'll have sides of n+ε instead of n - for example we gone from 11.0 to 11.00001. How many smaller triangles do you need to cover it? Obviously n^2 isn't going to be enough - because that was exactly enough to cover a large triangle of side n. Our slighty-bigger triangle is slightly bigger, so it has a larger area. We're going to need at least one additional small triangle to cover the added area, leaving us with n^2+1 as an absolute lower bound. But just because it is a lower bound doesn't mean it is actually possible - you'd first have to demonstrate that it can actually be done. This paper demonstrates two different methods of constructing it with n^2+2 triangles, providing an upper bound which is definitely possible. This means we still don't know the exact number of triangles required, but we do know it is definitely bigger than n^2 and definitely smaller than or equal to n^2+2. This leaves the question: is n^2+1 possible?

tornato7 3y ago

I was curious if GPT-4 Vision could explain it: The paper presents a geometric problem centered on equilateral triangles. The key question is whether it's possible to use \( n^2 + 1 \) small equilateral triangles (each with side length of one unit) to cover a larger equilateral triangle that has a side length just slightly more than \( n \) (specifically, \( n + ε \), where \( ε \) is a small positive value). The two figures provided illustrate possible arrangements of the smaller triangles within the larger one: 1. *Figure 1*: This demonstrates that \( n^2 + 2 \) small triangles can cover an equilateral triangle whose side is \( 1 + ε \). It's evident that the small triangles fit neatly inside the larger triangle. 2. *Figure 2*: This shows a different configuration where the large triangle has a side length of \( 1 - ε \). It seems to suggest that with just one fewer triangle (i.e., \( n^2 \)), the tiling is not possible for a triangle of side length \( 1 + ε \), but it may be for \( 1 - ε \). The paper, although succinct, poses an intriguing tiling problem in geometry. The authors likely aim to stimulate thought and discussion on this particular geometric configuration and challenge readers to consider the conditions under which such tiling is feasible. Given the brevity, the paper might be a problem statement or a brief note, rather than a full research paper with exhaustive proofs.

mkl 3y ago
It couldn't.
- IshKebab 3y ago
  Pretty damn close though! I haven't seen an explanation of what the second figure is trying to show so I'm not sure about that one. (And also their assertion that no further explanation is necessary is clearly bullshit.)
  - mkl 3y ago
    Its attempts at explaining both figures are totally wrong. Wrong side lengths, wrong assertion that the small triangles fit inside the large triangles, wrong relationship between the figures, complete misunderstanding of the second figure. The second figure is actually showing another arrangement of n²+2 small unit equilateral triangles covering an equilateral triangle of side length n+ε.

jacurtis 3y ago

I think the fact that you have to ask this proves that it is objectively a bad paper. The whole point of academic papers is to contribute to the larger global knowledgebase. You acknowledge the work that was done before, you submit your contribution and then you suggest how people can build or expand upon your work. This paper in question is just trying to be a mic-drop, like a middle-finger to academia. Generally research papers cover background context as their 1st and 2nd sections (at least IEEE format papers do). So normally a paper like this would start with an introduction section which explains what the paper is accomplishing and then the background section number two would explain context for where the author is coming from or what inspired research or background to justify its value. These sections would provide the context you are looking for and at the very least give references for you to go back and learn about it on your own. Even a few sentences would have been powerful here. This paper does fail to really provide value in my opinion and is objectively a bad paper. With some additional context from the introduction and background this could be much more valuable. Less critical, but also important is to acknowledge limitations and suggest future research. Now with all that being said, I'm not saying research papers are perfect. It is easy to find examples that go too far the other way, with far too much verbosity and pomp and circumstance. So I do at least acknowledge the statement being made with this paper that maybe all you need is two words. The reality is we should be somewhere in the middle. I read 3-10 academic papers per week, and the average page length is usually around 10 pages and really should be closer to 3-4. So i acknowledge the statement being made here, but this paper is clearly a protest, and not actually a productive example.

Ar-Curunir 3y ago
While providing introducing the problem and motivating itis common in CS paper, it's not a common practice in mathematics.