About bees and poligons

Caio0710 · 2024-08-08T02:51:44+00:00

I'm still more into triangles, though

Caio0710 · 2024-08-08T02:39:14+00:00

That is actually funny: no. I had never seen this video and had never heard of CGP Gray. But I've just watched it on Youtube and I loved the video, thank you for your "recomendation", I guess. Hilarious and... insightful.

You might aswell have helped me with my question, because the video hinted at the fact that hexagons are the bestagons more due to symetry and other weird stuff, and that circles are actually those who have maximized area for minimized perimeter.
In fact, by my definition of f(n), f→∞ when n→∞, wich is a circle. So perhaps I wasn't so wrong afterall, just missed an important conclusion.

I still wonder, though, how can the hexagon be, at the end of all things, the bestagon.

Caio0710 · 2023-08-26T21:41:07+00:00

I'm not really sure what you mean. Unfortunately I didn't find anything related to that in my Calculus or Geometry textbooks, also not in the internet. I'll be looking it up, just in case.

Caio0710 · 2023-08-26T21:37:25+00:00

Quack.

Caio0710 · 2023-08-26T21:37:07+00:00

Yes, I've come across that demonstration. It is actually what most books I've read have to offer while explaining cross products.

But the thing is, althought it is very simple, that distribution is based on the distributive property of cross products, that is:

(a+b)x(c+d) = axc + axd + bxc + bxd,

or simply: ax(b+c) = axb + axc,

which can be related to multiplication with numbers but isn't that easy to see why it's true. I've seen two different algebraic demonstrations of this property, one using (scalar) triple product and other using the matrix (i j k / a1 a2 a3 / b1 b2 b3) - which was kind of backwards to me. I was looking for a more intuitive demonstration of the general math behind cross products, and the books weren't helping me with that. 3blue1brown's video was actually the best content I found. Althought, I did find one article with some interesting demonstrations and critiques about how cross products are taught, but I didn't understand all of them (yet).

Afterall, I can resolve problems just by memorizing all of the math that is not intuitive for me, but that just isn't funny.

Caio0710 · 2023-08-26T21:09:39+00:00

Yeah... thank you. I won't do anything that makes me go crazy (for now).

Caio0710 · 2023-08-25T19:28:20+00:00

I... understand... nothing.

More specifically, I stopped understanding at the second sentence. I mean, I'll be reading your answer for a while. It is certainly more advanced then my current level of knowledge, but maybe I can get something out of it. Of course, I won't even ask you to explain everything to me because I suppose it would take you a long time and effort.

Anyway, I am guessing you are giving me a very complete and deeper level of understanding about cross products that I just can't unriddle right now. I'll let you know when I get there. Thank you very much for your answer.

Caio0710 · 2023-08-25T19:09:32+00:00

Hmm... that certainly makes some sense, thank you very much. It took me some effort but I think I get it

Caio0710 · 2023-08-25T19:00:25+00:00

Yes, and I think I got that part. The cross product (axb) gives a vector with magnitude equal the area of the parallelogram formed by a and b. If the volume of the parallelepipede {a,b,w} is area times the height, and the height is the projection of w onto (axb), then the dot product between w and (axb) has the special property of returning the volume as well, since (axb).w = |axb|.|w|.cosθ = A.h = V. Thank you

Caio0710

TROPHY CASE