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Equilateral Triangle (Posted on 2004-06-20) Difficulty: 4 of 5
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?

No Solution Yet Submitted by SilverKnight    
Rating: 2.6000 (5 votes)

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Some Thoughts Regular tetrahedron too! Comment 21 of 21 |

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.

  Posted by McWorter on 2005-03-03 22:26:40
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