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JohnBStone
 Senior Boarder
Posts: 70 
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Is it possible to have polyhedron (3D) in which every edge has a length that is a whole number of units long, and every length is different? If so, can you give an example? If not, can you prove it?
Carl G.
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cosmoschaos
Senior Boarder
Posts: 62
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Sure, a tetrahedron with edge lengths 1001, 1002, 1003, 1004, 1005, and 1006. More generally this should be possible for any stacked polyhedron (polyhedron formed by starting from a single tetrahedron and repeatedly gluing tetrahedral pyramids onto its faces).
It might be harder for other polyhedra such as cuboids, though.
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mintgus
Senior Boarder
Posts: 76
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Yes, trivially.
Make a 3-4-5 triangle and a 3-6-7 triangle. Identify the edges of length 3, and operate it as a hinge. Work the hinge so that the other two vertices are 8 apart.
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Jaxler
 Senior Boarder
Posts: 67
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Why? It seems obvious that you can change the length of any edge in a cuboid without changing the length of the other edges, up to a limit.
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Chant Dhames
Senior Boarder
Posts: 77
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Because you would make the faces nonplanar if you did.
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Jim
Expert Boarder
Posts: 88
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How do you define cuboid?
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glundby
Expert Boarder
Posts: 81
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A convex polyhedron (i.e. convex hull of finitely many points, or equivalently bounded intersection of finitely many halfspaces) whose vertices, edges, and faces meet in the same pattern as a cube.
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