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Math Help - Lattice Problem!

  1. #1
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    Lightbulb Lattice Problem!

    Hey guys I also have this question that I have no idea on!! I am greatful for any help!

    Lattice pie

    Is it possible to have an equilateral triangle in the plane all of whose vertices lie on the integer lattice {(m,n) | m,n are integers }? If so, what is the shortest side length possible for such a triangle? What about other regular polygons?

    Thanks
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  2. #2
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    Dear jessismith,

    Let one of the vertex in the origin, the other is (n, m). Rotate on z-axis with 60, so we get:

    n' = cos(60)*n - sin(60)*m
    m' = sin(60)*n + cos(60)*m

    Since sin(60) is rational and cos(60) is irrational so n' and m' cann't be rational.
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  3. #3
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    Lexington, MA (USA)
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    My Calculus 3 professor posed this problem to us.
    Most of us came up with a proof of impossibility
    . . and were quite proud of ourselves.

    Then he asked, "Is there a regular tetrahedron with vertices
    . . having integer coordinates?"

    We worked on it over a weekend and gave up.
    On Monday, he gave us the answer (Yes)
    . . and an eye-opening example.


    If you know the solution, please don't give it way.

    .
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