Question regarding polygonsCan you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?

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zoranovxp · Accepted Answer

Step 1The set of possible side lengths is exactly the set S of all positive numbers of the form             a      2        +          b      2       for integers a and b. In increasing order, its elements are   {  1  ,      2    ,  2  ,      5    ,      8    ,  3  ,      10    ,  …  }  .This set S is well-ordered, which means that every nonempty subset of S contains a minimal element. (It is well-ordered because it is order-isomorphic to       N  , the positive integers, which is also well-ordered.)Step 2Since any constructible lattice n-gon must have a side length from S, the set of constructible side lengths C is a subset of S. Since S is well-ordered, C is either empty or has a minimum element, say m. So either there is no constructible lattice n-gon at all, or else there is at least one with the minimum side length m.

Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?

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