Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 204

New comment
 
You're crazy, mate!
 
TheXpert:
You're crazy, man!
Yeah.
on your whole, pardon me, head.
 
TheXpert:
You're a psycho, you bastard!

Yeah, I know. ;)

But you have to prove it correctly....... :) :)

joo:
yeah.
on the whole, pardon me, head.
Damn it..... You can't make me laugh like that in the middle of the night... I've got the peepers asleep in the next room. ))))
 
MetaDriver:

By dividing the main trapezoid into smaller trapezoids, you can divide the bottom base into any number of parts... But how to prove the original problem I still don't understand. Analytical solution via vertex coordinates and straight line equations is realistic, but there is a lot of writing...

 
MigVRN:

By dividing the main trapezium into smaller trapeziums, you can divide the lower base into any number of parts... But how to prove the original problem I still don't understand. Analytical solution via vertex coordinates and straight line equations is feasible, but it's a lot of writing...

I can voice some of my general considerations.

  • Sometimes it is easier to understand (and even prove) a general principle than a particular case. (с)
  • (a) the construction and (b) the proof of its correctness are different tasks. there is no requirement to use the same instrumental constraints in both cases. (с)
  • The joy of finding a general principle should not be tempered by a temporary inability to immediately prove it to be rigorously and scientifically correct. (с)

;)

 

avtomat: кстати говоря, верхнее основание трапеции также разделено на три равные части.

Until this "solution" is proven, it is not a solution.

I understand all the steps, apart from the last one. But on the last one, I can't understand why it's the way it is. And I cannot refute it.

MetaDriver: // If you're really smart, develop an algorithm for dividing the base of trapezium into arbitrary number of equal parts using "a ruler without divisions".

I, too, can easily divide a trapezium into arbitrary parts. But I can't understand algorithm with MigVRN and avtomat's drawings... and it's shorter than mine for trisection.

MetaDriver:
  • (a) construction and (b) proving it correct are different tasks. There is no requirement to use the same toolkit constraints in both cases. (с)

That is correct in principle. But considerations of aesthetics, which are not alien to mathematicians at all, require proof by methods of the same part of mathematics with which the construction is done. And here it is projective geometry.

But at the moment I'm interested in at least some proof of the correctness of the algorithm proposed by MigVRN.

P.S. By the way, one fact from the history of mathematics: not a single proof of a basic algebra theorem is algebraic. They are all topological. And mathematicians stress this all the time. I don't know if the proof can't be algebraic.

 
Mathemat:

But I am currently interested in at least some proof of the correctness of the algorithm proposed by MigVRN.

I'll see what I can do.... :)

P.S. By the way, one fact from the history of mathematics: Not a single proof of the main theorem of algebra is algebraic. They are all topological.

It's legitimate, Gödel rules.

And mathematicians stress this all the time. I don't know if the proof can't be algebraic.

I wouldn't be surprised if such a proof is impossible.... Which in turn is also impossible to prove... Law of hole abstractions, sir...

However you can try. At least you'll broaden your mind, at most you'll find a proof and get a prestigious award... :)

--

It is useful to have several ways of looking at the same entity. It opens up thinking. For example, a trapezoid can be defined in different ways:

  • (school classics): A quadrilateral whose two sides are parallel
  • a quadrilateral cut off by two parallel lines from an angle
  • a quadrilateral cut from a pair of parallel lines by lines drawn from a single point
  • etc.

Each definition fixes some reference "coordinate system" of thinking. But when you compare them or just change them several times, a "more volumetric" abstraction emerges, which can be navigated using potentially more powerful mechanisms of systemic consideration (which our brains actually have naturally available).

 
Mathemat:

But at the moment I'm interested in at least some proof of the correctness of the algorithm proposed by MigVRN.

I'm still working on the proof, but I made a nice generator for dividing the bases (both, of course) of a trapezoid into consecutive fractions:


Very nice scheme.

In fact, it geometrically represents one of my very favorite functions - rational sigmoid: y = x / (1 + |x|)

The picture shows division up to 1/11 inclusive (red dot) // all divisions are correct and accurate - tested electronically.

 

Of course, that's not the only generator possible. Here's another one on top, check it out:

:

And there should be at least three in total (I have an example of dividing by seven in three ways).

However... it's time to get on with the proof.

 
MetaDriver: Of course, it's not the only possible generator. Here's another one on top, check it out:

Yes, it's beautiful. But I don't yet understand why this is the exact algorithm.

I'm thinking of a proof.

1...197198199200201202203204205206207208209210211...229
New comment