Tiling uniform grids using only one tile

See the pattern in the background? Guess how many different tiles are used to make it... If you guessed only one, you're totally right! Now before I show you how I made it, let's put up some rules.

One thing that comes to mind is to put one of the platonic solids in the middle of the tile. Now, it's very easy to realize that translating this platonic solid will only affect the offset of the tiling. So let's keep it in the center.

Now let's put the other platonic solids in the tile. One way of doing this that imediately comes to mind is to align them diagonally, like the points on the 5th side of a dice.

Here we can see that all the points aligned diagonally are equidistant, but the result does not contain any uniform grids. Let's try another configuration.

Now the points are equidistant on the horizontal and vertical axes, but stil no uniform grids emerge from the tiling. Maybe there is a rotation with a special angle that makes the points align into a grid. Let's find out.

Yes! There actually is an angle and it is or ≈26.57°. If you draw a line from the center of a tile and go up once and right twice (like in the figure below), you can place the points equidistantly and make a uniform grid. This explains the in the previous formula.

Here is a thing that I noticed but at first, I couldn't find any explaination/proof of how or why it works. Prepare to get your mind completely blown : if the distance between two tiles is square root of a whole number n, it is possible to make a uniform grid using a tile with n different objects in it. Since the distance between two cells can be defined as , this means that n must be the sum of two squares.

Here is why : In any whole number of squares, there are the same number of squares than points. (Note that a point in a corner counts for 0.25 points and 0.5 for points on edges). Now if you draw a square with a units long sides, the area of the square is 5. This means that you can place 5 equidistant points with a 1 unit of distance separating the points on each axis. And this is exactly what we want.

This is why you cannot make a uniform grid using one unique tile with 3 objects in it. Good thing there are exactly 5 platonic solids.

Here are all the possible uniform grid tilings with n going from 1 to 10.
n tile tiling
1
2
3 Impossible
4
5
6 Impossible
7 Impossible
8
9
10