March 2025
Choose some lattice points \(A\) and \(B\) and compute \(\tan(\angle AOB)\). The values of \(\tan(\angle AOB)\) are limited to some subset of the real numbers. Can you figure out what this subset is, and can you prove that \(\tan(60\degree)\) is not in that subset?
Compute the interior angle of a regular pentagon. Then compute \(\angle CBF\). Is there a theorem from geometry you can use to prove that two lines are parallel?
Suppose \(X\) and \(Y\) are points with coordinates \((x_1,x_2)\) and \((y_1,y_2)\). Let \(Z\) be the point with coordinates \((x_1 + y_1,x_2 + y_2)\). Plot the four points \(O, X, Y, Z\) on the Cartesian plane. Is there anything special about the quadrilateral with vertices \(O,X,Y,Z\)? Try it with some specific points \(X\) and \(Y\).
Start by computing the angle between \(L_1\) and \(L_2\).
Let \(A\) be the center of the \(n\)-gon \(B_1B_2\cdots B_n\) (that is, \(A\) is the point where all the lines \(L_i\) meet in the second image in the statement of the problem). Then \(\triangle AB_1B_2\) is an isosceles triangle with one of its side lengths equal to \(x\), and another one equal to \(y\).
Use Question \(1\) to rule out the existence of regular lattice triangles and hexagons. Use Question \(2\) to rule out the existence of regular lattice pentagons. Use Question \(3\) to rule out the existence of regular lattice \(n\)-gons where \(n \geq 7\).
As a general strategy, assume that there is a regular lattice \(n\)-gon, and try to construct a smaller regular lattice \(n\)-gon. If you can do this once, then you can do it again and again. Is it a problem to have smaller and smaller lattice polygons? Is this even possible?