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 lengh possible for sucha triangle? what about other regular polygons?

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- Nov 26th 2008, 11:39 AMsonia1lattice pie - another puzzle stuck on
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 lengh possible for sucha triangle? what about other regular polygons?

- Jun 20th 2011, 02:20 PMAlso sprach ZarathustraRe: lattice pie - another puzzle stuck on
The answer is no. There is no such triangle exists.

It easy to prove that fact knowing that the area of regular triangle with side is that is irrational.

For , is a square which we can describe on system of lattice points.

For we can not describe the regular polygon on system of lattice points.

Here is a proof for this fact:

Suppose that there is regular polygon with n sides, that his edges are on lattice points. Now, from all regular polygons that can be describe on system of lattice points we choose the one with the minimal side length.

Say that the chosen one is: .

Now, we look at the reflections of for vis-a-vis (where and ).

So, we got a regular polygon with sides .

for is lattice point, why?

Due to our contraction is a rhombus.

Now, let and .

Let say that is intersection of the diagonals in rhombus.

We calculate that with two ways,

First with :

Ans with :

Hence,

We deduce from the above that and are integers.

Finally, he have regular polygon with sides witch his edges on lattice points.

But, (By our constraction) and this is a contradiction to the fact that chosen with minimal . - Jun 20th 2011, 06:49 PMDrexel28Re: lattice pie - another puzzle stuck on
In fact, it's impossible for an equilateral triangle to have all three vertices lying on . There are a few high-tech proofs (most notably a counting argument) but it can be done with simple geometry

__Spoiler__: - Jun 21st 2011, 06:25 AMSorobanRe: lattice pie - another puzzle stuck on

My Calculus 2 professor posed that problem on a Friday.

I'm proud to say that, over the weekend, our class

. . found a variety of ways to to prove it impossible.

On Monday he asked, "Is it possible to have a reqular

. . tetrahedron whose vertices have integer coordinates?"

None of us had a satisfactory proof (either way).

The next class, he blew us away with a simple sketch.

If you've never considered this problem, give it a try.

I'll post my professor's solution later.

- Jun 22nd 2011, 06:24 AMSorobanRe: lattice pie - another puzzle stuck on

Here is my professor's solution . . .

__Spoiler__: