TITLE: I want to graph an equilateral triangle on graph paper
QUESTION [6 upvotes]: I want to graph an equilateral triangle. It would be ideal if I had a set of three points: $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ with $x_i, y_i \in \mathbb{Z}$ as the vertices. However, this is impossible. I am willing to "settle" for a triangle that is off by the width of a pencil lead. (That is, if we draw circles of radius $\delta$ around all of the lattice points, the true location of the vertices of the equilateral triangle are inside of these error circles.) 
For a given $\delta$ what is the smallest such triangle?
My instinct is to let the computer guess, is there a more elegant way?

REPLY [2 votes]: If you are interested in fast sketching of nearly equilateral triangles by hand, based on $5$mm graph paper and assuming a $0.5$mm pencil, then the triangle with corners on the grid coordinates $(0,0)$, $(1,4)$ and $(4,1)$ has an error of less than $0.7$mm (based on an upper bounds of two times the difference of the circumscribed circle based on the shortest and largest edge of this triangle). Even if you sketch without a ruler, the result is nearly indistinguishable from an equilateral triangle.