As a student in Lafayette’s first joint mathematics/art course, you are given the project of painting each point of an equilateral triangle (with side length 1) one of four colors.  After a ludicrous number of hours you somehow manage to paint each of these infinite points! Excited to hand in your masterpiece, you realize (having not carefully read the directions) that the triangle needed to be colored so that there are at least two points of the same color that are at least d units apart!

For what values of d are you guaranteed that your (current) coloring satisfies this additional requirement?