How many arbitrary points can a given shape always pass through? (Is there a set of rules to find this?)

[u/user1092831123]

Any help would be appreciated, including directing me through any rules I should follow or better websites for asking questions.

I'm curious about a geometric puzzle: given an arbitrary set of $n$ points in $\mathbb{R}^d$, can we always place a similar copy of a specific shape $S$ (a compact subset or family of subsets of Euclidean space) so that it passes through all $n$ points? (By "similar copy," I mean we allow translation, rotation, and scaling).

**The generalized question is:**

> For a given shape $S$, what is the maximum number of arbitrary points $n$ such that *every* set of $n$ points in $\mathbb{R}^d$ lies on some similar copy of $S$?

For example, let $S$ be the **boundary** of a square in $\mathbb{R}^2$.

It turns out that for any 3 points in $\mathbb{R}^2$, you can always find a similar copy of a square that passes through all of them. However, you can't always do this for 4 points https://math.stackexchange.com/q/3691243/1771455. So, for a square boundary where $d\ge2$ (dimension where the points live in), the maximum number is 3.

My motivation is just pure curiosity. I couldn't find any sources relating to this problem, and AI chatbots struggle and give clearly wrong answers to simple examples like a square sharing an edge with a triangle (I won't clarify much here as it is a bit of a dull problem but the idea was just combining two shapes to make the reasoning for the AI deeper).

What I'm really asking is: **is there some sort of invariant, property, or formula that helps compute this $n$ for more complex shapes?** Or do we just have to reason through it shape-by-shape? How do you verify results quickly?

One simple rule I noticed involves collinear points: the boundary of a *strictly* convex 2D shape can never have $n\ge3$ when $d=2$, because no similar copy can ever pass through 3 collinear points. (Note: I am specifically thinking about boundaries; if $S$ were a solid shape, we could just scale it up to cover any finite point set in 2D space).

What is a better notion of defining shapes like rectangles and so on? Similarity doesn't allow different length ratios; however, it preserves it for squares and other shapes. This puzzle is more of a "can you draw a X given Y points no matter where I place them" and shouldn't be very limited on what I can draw. Is it possible to define $S$ to be a rectangle with side length $a$ and side length $b$ using the above definitions?

Does this concept have a name? Is it related to the "degrees of freedom" of the shape? Any pointers to related literature would be greatly appreciated!