TITLE: Cover of "Gödel, Escher, Bach"
QUESTION [95 upvotes]: Consider the cover image of the book "Gödel, Escher, Bach", depicted below. The interesting feature is that it shows the existence of a subset of $\mathbb{R}^3$ which projects onto $\mathbb{R}^2$ in three different ways to form the letters of the book's title. It is natural to ask for generalizations: for which subsets $A_1, A_2, A_3$ of $\mathbb{R}^2$ is there set $X \subset \mathbb{R}^3$ such that, with $\pi_1, \pi_2, \pi_3$ the projection maps $\mathbb{R}^3 \rightarrow \mathbb{R}^2$, $\pi_1(X) = A_1, \pi_2(X) = A_2$, and $\pi_3(X) = A_3$?
More generally, let $\{ \pi_i \}_{i \in I_{n,m}}$ be the canonical projection maps $\mathbb{R}^n \rightarrow \mathbb{R}^m$, where $m \leq n$. For which sets $\{ A_i \}_{i \in I_{n, m}} \subset \mathbb{R}^m$ is there a set $X$ such that $\pi_i (X) = A_i \forall i \in I_{n, m}$?
Other interesting considerations:
1) I do not require the set to be connected. Nevertheless this presents an interesting question as to when the set in question is connected.
2) Let $X$ be the largest possible set satisfying the question, supposing it exists. Is there a simple way to calculate its boundary, $\partial X$?
3) What is the volume of the largest possible set in question in terms of $A_i$? It's worth noting that, if $A_1, A_2, A_3$ are measurable subsets of $I^2$, then there is an interesting formula for the volume of $X$ plus the volume of $Y = I^3 - \pi_1^{-1}(A_1) \cap I^3 - \pi_2^{-1}(A_2) \cap I^3 - \pi_3^{-1}(A_3) \cap I^3$.

REPLY [2 votes]: Your generalization to triples $A_1, A_2, A_3$ for which there exists a single $X$ that projects to each of the $A_i$ is an example of the general phenomenon I have called $polysemy$. It's been explored in various applications in graph theory and order theory. 
A simple example question is, "For which pairs of partial orders $<_1$ and $<_2$ on a set $S$ does there exist a bijection $f:S\rightarrow{\mathbf R}^2$ such that for all $s,t \in S$,
$$s <_1 t \iff d(f(s),\mathbf{0}) < d(f(t),\mathbf{0})$$
and
$$s <_2 t \iff d(f(s),\mathbf{1}) < d(f(t),\mathbf{1}),$$
where $\mathbf{0}$ is the origin and $\mathbf{1}$ is the point (1,0)?"
For more info, see (of all places!) https://en.wikipedia.org/wiki/Representation_(mathematics)#Polysemy.