Geometrically speaking, a positive number is a perfect cube if and only if one can arrange solid unit cubes into a larger, solid cube.
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The pieces of the Soma cube consist of all possible combinations of three or four unit cubes, joined at their faces, such that at least one inside corner is formed.
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These overlapping circles can also be seen as a projection of an " n "-unit cube of spheres in 3-dimensional space, viewed on the diagonal axis.
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To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with the lower face, the right with the left face, and the front with the back face.
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For higher dimensions a Monte Carlo approach can be used where random Cartesian coordinates are chosen in a unit cube, points falling outside the unit ball are discarded, and the remaining points are normalized to lie on the unit sphere.
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Surface area should be rotationally invariant, but this calculation is not, with a unit cube having a value ranging from its actual surface area up to twice that figure, depending on how it is rotated with respect to the axes of the cuboids.
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While the latter is less obvious than the former for the general case, it is easily seen for the one-sixth of the unit cube formed by a face ( area 1 ) and the midpoint of the cube ( height 1 / 2 ).
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If points outside the unit cube were allowed, however, a number of them, including s = (-0.5, 0, 0 ), would do the trick . talk ) 21 : 10, 6 August 2007 ( UTC)
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Before or around this time, Nieuwland had found the largest cube that can pass through a hole in a unit cube, a problem that had been posed 100 years earlier by Prince Rupert of the Rhine and given an inferior solution by English mathematician John Wallis.
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So what I'm looking to create, ultimately is a normalized dataset of points with 3D Cartesian coordinates, such that each point represents a super-cluster, and the whole set is scaled to fit inside the unit cube ( or unit ball or something similar ).
