# Sets of distances between points that do not form distinct shapes

If I gave you 3 points, they would have exactly 3 distances. With those 3 distances, one can create at most 1 shape, ignoring rotations and reflections. For example, the distances 3,4, and 5 always form the same right triangle shape.

I was making a program to determine if a set of 4 points created a certain shape. I thought checking the distances would be a good way to do it. However, contrary to what one might expect, the above correspondence is not always the case for 4 points. In this case, there will always be 6 distances. For example, consider the distances 1, 1, 2, 2, $\sqrt5$, and $\sqrt5$.

These 6 distances can form at least 2 2D shapes: a rectangle with side lengths 1 and 2, or a "T" shape.

However, another set of distances seem to only form 1 2D shape: 1, 1, 1, 1, $\sqrt2$, $\sqrt2$.

I naturally wondered if there were other sets of distances that could form multiple shapes. However, I was unable to find any other examples, which makes me believe that this is a strange phenomenon. And as the number of points increases, it seems less and less likely to occur (With n points, you'll have n(n+1)/2 different distances to satisfy).

There a lot of potential questions to ask about this (what is the maximum number of distinct shapes possible? Is there a way to generate such shapes? etc.)

However, what I am most interested in is if there is a certain number of points for which this event is guaranteed not to occur. If so, what is this number?

One thing that I did notice about my construction is that every point corresponds to the same 3 distances, 1, 2, and $\sqrt5$. I don't know if that's helpful or not, though.