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How does one formalize the notion of "fitting together" objects in space? For example, in 3D Euclidean space, fitting 6 square pyramids with a common apex together into a cube, or fitting 4 equilateral triangles together into a regular tetrahedron. (The first having internal faces and the second not; I'm not sure of the notation to distinguish between these.)

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Hilbert's Third Problem and Dehn's solution of it may be relevant: en.wikipedia.org/wiki/Hilbert's_third_problem –  deoxygerbe Apr 21 '12 at 3:33

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For these examples you can show the coordinates of the points of interest: the corners and center of the cube or the corners of the tetrahedron, then show that those points make the shape claimed. For the cube you can then identify for any point which pyramid it is in so you know there isn't anything left over. For a regular tetrahedron you then need that the angles are the same (because of the requirement of regular) but symmetry takes care of that.

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Okay so define the cube as $-0.5 \leq x,y,z \leq 0.5$ and square pyramids $z \leq x,y \leq -z$, $-z \leq x,y \leq z$, $-x \leq y,z \leq x$, $x \leq y,z \leq -x$, $-y \leq x,z \leq y$, $y \leq x,z \leq -y$. Now I'm trying to see an easy way to show that any point $-0.5 \leq a,b,c \leq 0.5$ is in one of those. –  AbstractionOfMe Apr 27 '12 at 19:22
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@AbstractionOfMe: The largest coordinate in absolute value tells you which pyramid the point is in. For example, (.1,-.2,-.3) is in the pyramid that uses the -z face of the cube. –  Ross Millikan Apr 27 '12 at 19:55
    
Thanks, that's very helpful. –  AbstractionOfMe Apr 28 '12 at 18:28

Can someone confirm whether the following method is correct?

Regarding the 6 square pyramids:

To show that their volumes are all pairwise disjoint except for faces: For any 2 (of the 6) pyramids, call them A and B, there is a plane onto which A and B can be projected such that the projections of A and B do not intersect at any point except on the faces. If A and B did intersect at some other point, their projections would intersect at the projection of that point, so this proves that the volumes of A and B are disjoint except for the faces.

To show that their volumes entirely fill up the cube: Note that each internal face of each pyramid exactly intersects one other internal face of another pyramid. So if you move a point around the inside of the 6 external faces (the cube), then when it leaves one pyramid it's sure to enter another.

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This should probably be a new question... –  J. M. Apr 27 '12 at 16:50

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