Ridiculously easy Maths question

[Vivitrix]

Sweet Jesus...
What.... What have i done...!

[Skraa]

Are you glad now you got the easy question sorted out? What if it had been some difficult one?

[maky]

Square tiles = normal mode

Instead of those, fill that same room with circles of diameter 300 mm ! How many?

edit: Clarification. Fit as many circles as possible, without any 2 overlapping. Of course you can`t cover the whole area, there will be gaps.

[Posix]

With circles you have the same basic layout as the squares, but the difference is that you can fit another tile in between 4 other tiles. You wouldn't be able to put the around the outside edge though, so it would be

n = number of tiles in the square.
x = width of square

xn+(n(x-2))

What I thought was the case initially, doesn't seem to be true as you can't actually fit another circle in the middle. So assuming the same 'dimensions' - that is the same width and height as the squares, doesn't it mean you have the same amount of tiles?

[Skraa]

What I thought was the case initially, doesn't seem to be true as you can't actually fit another circle in the middle. So assuming the same 'dimensions' - that is the same width and height as the squares, doesn't it mean you have the same amount of tiles?

You can for sure fit in more circles if you shift every other row half circle down and pack the gaps and ridges alternatively, but that is way harder math problem than I'm willing to go for little forum amusement.

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[maky]

Skraa wins some internets. Yeah, the thing is ridiculously hard, I don`t even remember. Probably your drawing does actually present the proper solution, but I don`t know a proof - kinda beats everything in the math geek thread anyways

[Posix]

Ah yeah, I didn't think to take that into consideration at all.

[Ensar]

Given an infinite plane to place the circles, then Skraa's diagram will be optimal with each circle having six around it in a hexagon. But in a bound space, it probably won't be, but it'll give you a good lower bound to work with. I'll have a play with this later...

[Skraa]

Given an infinite plane to place the circles, then Skraa's diagram will be optimal with each circle having six around it in a hexagon. But in a bound space, it probably won't be, but it'll give you a good lower bound to work with. I'll have a play with this later...

Skraa wins some internets. Yeah, the thing is ridiculously hard, I don`t even remember. Probably your drawing does actually present the proper solution, but I don`t know a proof - kinda beats everything in the math geek thread anyways

I vaguely remember that it was proven very recently, only few years ago, that it is the best way to pack circles inside square, and the mathematical proof of that was ridiculously complicated and hard thing to do. Although it has been used to pack bottles and cans into crates for very long time.

[Ensar]

It can't have been proven, as a trivial contradictions exist. Take a square of length 3cm, and nine circles with diameter 1cm. You can fit all nine in to to the box by lining them in a 3x3 grid, but using the hexagonal method you'll only fit eight. This'll be true for and square number.

As for packing bottles, then you have practical reasons to go for hexagons due to structural strength and stability as they're removed. No use having complex optimal patterns when you may end up with free moving bottles or ones that'll collapse when key bottles are removed.

[Katimi]

You all get Kati maths love for this thread. It has made me smile <3

[Idreil]

from tiles to circles to bottles...

How many circles can you fit inside a bottle being imploded into a black hole!?