Can an equilateral triangle have vertices at integral lattice points?
Integral lattice points are such points as (101, 254) or (3453, 12), but not points such as (123.4, 1) or (√2, 5)
If you can't find a solution in the 2D Cartesian plane, can you find one in a 3 (or more) dimensional space?
As Richard said, (1,0,0), (0,1,0), and (0,0,1) are vertices of an equilateral triangle. If we add to this list (1,1,1), we get four vertices of a regular tetrahedron with integral lattice points as vertices!
Makes me wonder. What platonic solids can be placed in 3-space so that their vertices are integral lattice points? Obviously the cube works, and, from the above, the tetrahedron works. I suspect that even other platonic solids also work.
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Posted by McWorter
on 2005-03-03 22:26:40 |
Regular tetrahedron too!
