Notating the set of all points were each point is same distance from all others

Question

The set of all points with same distance to one point can be written as $\{ (x,y)\in\mathbb R : r^2=\sqrt{(x-x_0)^2+(y-y_0)^2}\} $ I am trying to write the set of all points were each point is same distance from all other points but I have not seen something similar that donates a relation between all the objects in the set to all other objects in the set. it seems that it is true that in $N$ dimensions there are $n+1$ points that satisfy can satisfy being from same distance to all other points.

Answer 1

You are right. In 2D, you have an equilateral triangle. But that's not unique. If I give the center of the triangle, and even the distance between the points, you can still rotate the triangle by any angle around the center. If you go to 3D, you get a tetrahedron, where you have even more degrees of freedom. So if you want to specify the set, just give the elements. That seems to be the simplest approach.

Answer 2

In the notation you’re using to describe the circle, there are three free variables: $r$, $x_0$ and $y_0$. Only once you’ve specified what their values are do you get a concrete set. It’s the same here: there are many sets where all points have the same distance from one another (just like there are many circles); to use set builder notation to denote one of them, you need to provide the information which one you want.

So let us think about what this information might be: You choose the first point however you want (just like you can choose an arbitrary center $(x_0, y_0)$ for your circle). The second point is also arbitrary (except that it has to be different from your first). Now your distance is fixed, but there are still choices for the third point: If your surrounding space is 2D, you can make a equilateral triangle to the left or to the right. In 3D, you can choose your third point from an entire circle. (In higher dimensions you have even more choices.) In 2D you’re done after the third point, as you noted. In 3D, there are still two choices for the fourth (and last) point. Again, in higher dimensions you can go on. With each point you’ve chosen, the options for the remaining points shrink, but you always have some choice left. (Note that this is different from the circle where you are done after choosing the center and the radius.)

What I’m getting at: There is no easy way of describing these sets using set-builder notation because you choices to make for every point in your set.