Red and Blue make Purple

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hethatcures

This legitimately upsets me.

hookteeth

… Y’see, now, y’see, I’m looking at this, thinking, squares fit together better than circles, so, say, if you wanted a box of donuts, a full box, you could probably fit more square donuts in than circle donuts if the circumference of the circle touched the each of the corners of the square donut.

So you might end up with more donuts.

But then I also think… Does the square or round donut have a greater donut volume? Is the number of donuts better than the entire donut mass as a whole?

Hrm.

HRM.

entropiaorganizada

A round donut with radius R₁ occupies the same space as a square donut with side 2R₁. If the center circle of a round donut has a radius R₂ and the hole of a square donut has a side 2R₂, then the area of a round donut is $πR 12 -$ $πr 22 . The area of a square donut would be then 4R 12 - 4R 22 . This doesn’t say much, but in general and throwing numbers, a full box of square donuts has more donut per donut than a full box of round donuts. The interesting thing is knowing exactly how much more donut per donut we have. Assuming first a small center hole ($ $R 2 =$ $R 1 /4) and replacing in the proper expressions, we have a 27,6% more donut in the square one (Round: 15$ $π$ $R 12 /16$ $≃ 2,94$ $R 12, square: 15$ $R 12 /4 = 3,75$ $R 12). Now, assuming a large center hole ($ $R 2 = 3 R 1 /4) we have a 27,7% more donut in the square one ($ $Round: 7 π R 12 /16 ≃ 1,37 R 12, square: 7 R 12 /4 = 1,75 R 12). This tells us that, approximately, we’ll have a 27% bigger donut if it’s square than if it’s round.$