# How can I make lines of equal differences from a central point to all directions?

This sounds like an elementary school problem but I have a problem that I can’t solve. Also, the fact that I’m not a mathematician might contribute to this fact. This is the problem;

#### “How can I make lines of equal differences from a central point to all directions?”

Below I will describe what I tried through some (self-made) figures. For simplification, I did it two dimensions. But in reality there are more dimensions of course. I used a grid to count the distance in integers. The integers are rounded and are often very wrong based on the method I used.

Method one: Horizontally and vertically only
With the first method, I tried what I would have done in pre-school, With this method, I started with counting on the x-axes (0), and then the y-axes (0).  After that, doing the same thing starting at the (-1,0) axes, and the (0,1). But as you can see, that if you look diagonally at the graph, the distances are very much off. This figure will become a diamond like shape if you take all axes. Nevertheless, the more the value’s come closer to the straight lines, the closer the integers come to the exact value.

Method two: Horizontally, vertically, and diagonally
Here, I also added a diagonal line (at 45°) that supports the lines to be of equal distance. Then, I filled up the rest of the grid. This method comes closer to the real distance than Method 1 for all value’s. But it is still off. This structure becomes a square. As you can imagine, putting an extra line right between the lines that are drawn (at 22,5° and 47,5°), will improve the correctness of the distances, but you will never make it perfectly straight in each direction. This is an infinite problem.

Method three: spirally
In this case I put the “0”, the center at the middle of the graph for example purposes. Here, the value’s are circling and expanding around the center to make equal distances. The larger the structures becomes, the rounder it becomes, and the more “circly” it becomes thus making the distance equal to the center. It will never be perfect, but I believe that it comes closer to the solution than method 1 and 2.

Method four: just circles
You can also just use a compass and make circles on the grid and measure the distance on the grid. However, I wanted something that started at the center, that was always connected to the center. If it were real circles from expansion, than there is some “gap” created between each distance. I believe that only a spiral can be fully connected to the previous numbers. With circles you create a gap.

An infinite sides structure?
I can only imagine that an infinite sided spirally structure will come close to the real solution. Thus, a combination of method 2 and 3. The spiral structure of method 3 to get a circly like distance, while the spiral itself is expanding like putting the extra diagonals between the diagonals (like in method 2) until infinity. As the structure itself expands, each side should expand equally to keep the same distance from the center.