Making none of the sides meet is also be possible if you prefer that. There are actually multiple ways. It can be explained by dividing the plane into hexagonal, each consisting of six large triangles. Each hexagonal itself has again six neighbors. Let's focus on one, H, and its six neighbors, named N1 to N6 going clockwise.



There are two main options:

1) Make the outside of H a single color, e.g. red. This is for example the case in my drawing above. Its neighbors should also have a single outside color, but different ones. E.g. N1, N3 and N5 can have a white outside, and N2, N4 and N6 a grey outside. You can continue filling the entire plane this way.

2) Alternatively, let each hexagonal have two sides of every color. E.g. going clockwise, red, white, grey, red, white, grey. The neighbors can be colored the same, but should be rotated such that no sides meet, nor for H and any of its neighbors, but also not for neighboring neighbors. You can also continue filling the entire plane this way.



Hopefully this is clear. If not, let me know and I can make an image illustrating this, but that will be a bit more work...