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I guess you can use the formula for the sum of a geometric progression for $k$ non equal to 0.

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Ok, finally solved it in a very simple geometrical way. IF we take a square slanted lattice in the hexagonal lattice, like in the image, which is $N$ particles along each side. Then the number of particles inside is $N^2$. The volume of that area is just $V=(Na)^2\cos(30)$, and so: $$\rho a^2=\frac{2}{\sqrt 3}$$. I'm sorry for asking, I was frustrating ...

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My educated guess is that a large block of ice, delivered to space somehow, would last quite a while. If we assume it is in Earth orbit, the side facing the sun would sublimate (go directly from solid to gas) and dissipate. The rate of sublimination would depend on the insolation (power per unit area), which is about 400 $W/m^2$ and the absorption ...

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In real life most fractures occur at defects. Even such everyday materials as cement can have their strength increased many times by reducing the defect density within them. You'll often see claims for the incredible strength of nanostructures, but the strength is just due to the fact that these structures are free of defects. It's a lot easier to make a ...

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Graphene is also very thin. According to this article the force required to break a sheet of perfect graphene by pulling it apart in such a way that all bonds break at the same time is 42 N per meter. If the width of your tape is 1cm you would need to apply 0.42 N. It is not surprising that you were able to. Even if the sheet would be perfect, you would pull ...

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