41–50 of 84 posts
Along with short papers, there are short titles. I. I. Rabi tried to get a paper published in a German journal with a one-word title: "Molekularstrahlenablenkungsmethode" The journal turned it down. https://www.aip.org/history-programs/niels-bohr-library/oral...
Can someone explain the n^2 + 2 triangles paper?
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 (a…
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?
Though not quite as short, the Watson and Crick paper is another famously short paper: https://dosequis.colorado.edu/Courses/MethodsLogic/papers/Wa...
2×3×5×7×11×13 + 1 = 59×509
is a short counter example to the widespread misconception that adding one to the product of the first n consecutive prime numbers always yields a prime number.
The reason you get away with this in the infinitely-many-prime-numbers proof is that the new number may not be prime, but can be written as a product of primes that are distinct from the first n primes. Thus you still generate new prime numbers with this technique.
As a rule, when trying to convey information, I try to write and speak plainly. I try to avoid jargon. I have found many academic papers in the faux sciences to be extremely dense and full of terms that are only known to the priests of that arcane subject (still subsidized by taxes as if the result is a common good). If you have a point, say it. There is no need to write in legalese. When I see supposed research writ…
May I ask if these "faux sciences" contradict your political or ideological positions, and is it possible that is the real issue here?
Earlier quoted context omitted.
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 (a…
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?
As a rule, when trying to convey information, I try to write and speak plainly. I try to avoid jargon. I have found many academic papers in the faux sciences to be extremely dense and full of terms that are only known to the priests of that arcane subject (still subsidized by taxes as if the result is a common good). If you have a point, say it. There is no need to write in legalese. When I see supposed research writ…
Earlier quoted context omitted.
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?
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.
Can someone explain the n^2 + 2 triangles paper?
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…