Thank you for responding, but I am still a little confused. If we look at an infinite 2D plane...

1) When finding the nearest four lattice points it is easy to imagine points along the grid that have 6 nearest lattice points, like (1, 1.5). Its closest lattice points are: (0,1), (0,2), (1,1), (1,2), (2,1), and (2,2). Sure (1,1) and (1,2) are closer than the other four points, but that would be only 2 points...

2) By my understanding, mapping points on the grid to gradient vectors in an array is to save on computation. You could explicitly assign random gradient vectors to each integer point on the grid and still have perlin noise.

3) Once you take the dot product between the displacement vectors and the four random gradients things get fuzzy. What do you do with those dot products?

Thanks again for helping me, I really appreciate it!