Ok Ok, the point is this. We can ask this question for whatever dimension we like. Then we can ask the same question: How big an n-dimensional sphere can we fit in the gap in the middle? If we continued on to four dimensions and this is where I stop being able to draw things , it turns out that the sphere you can fit in the middle is the same size as the spheres you already have in the box!
It turns out that in ten dimensions coincidentally the number of dimensions required for a consistent string theory , stuff starts to get really strange. In ten dimensions, the sphere that you can fit in the gap in the middle is so large that the sides of it poke out of the sides of the cube you started with! Ok, now for the math. If we draw the following triangle, we can ask what are the lengths of the sides of the triangle? We know that the length of the long side is the radius of the big circle, plus the radius of the little circle.
In our case, is the length of our hypotenuse, and the smaller sides are both length 1, that is and. Putting this into the formula gives us:. But this means that , and we finally get , so the radius of the smaller circle is around 41 centimetres.
In three dimensions, we do the same thing. Indeed, you can probably already see how this is going to generalise. In four dimensions, the radius of the centre sphere is. So the centre sphere has a radius of one, just like the other spheres inside the cube.
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These applications will - due to browser restrictions - send data between your browser and our server. We don't save this data. Google use cookies for serving our ads and handling visitor statistics. AddThis use cookies for handling links to social media. Please read AddThis Privacy for more information. Maximum number of circle packing into a rectangle Ask Question. Asked 3 years, 11 months ago.
Active 2 years, 7 months ago. Viewed 26k times. Add a comment. Active Oldest Votes. We pick the first option, which gave us the greater value. Misha Lavrov Misha Lavrov k 10 10 gold badges 94 94 silver badges bronze badges. So this would be the only method to find the accurate value. However, it's great to know the 'limiting density' part, big help, :D even though i don't fully understand about it. Upcoming Events. Featured on Meta.
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