## Thursday, September 24, 2020

### [xdgetfvm] Equilateral triangles on a square grid

Connect lattice points (-1,0) (1,0) (0,2) to approximate an equilateral triangle. 1^2 + 2^2 ~= 2^2

Digression: one could slightly squash a square grid into a grid of rectangles: 1 by 0.83 = sqrt(3)/2.  This would make (-1,0) (1,0) (0,2) be an actual equilateral triangle.

Better (back on a square grid): connect (-4,0) (4,0) (0,7): 4^2 + 7^2 ~= 8^2

Can we do better if no edge is axis aligned?  5^2 + 5^2 ~= 7^2 + 2^2 comes from the hexagon below.  I don't think it is better than 4^2 + 7^2 ~= 8^2 above, though we haven't defined "better".

3^2 + 1^2 ~= 2^2 + 2^2.  A triangle with vertices (0,0) (6,18) (18,6) has edge midpoints (3,9) (9,3) (12,12) and centroid (8,8), all on lattice points.

Motivation was needing an equilateral triangle grid when all we have is square graph paper.

Draw an approximately regular hexagonal grid using only square lattice points.  3^2 + 3^2 ~= 1^2 + 4^2

Three vertices of cube all edge-adjacent to the same vertex form a perfect equilateral triangle.  Consider extending that cutting plane into a cubic tessellation.  What pattern do the cube vertices make on that cutting plane?  Probably equilateral triangular lattice.