# Cover of “Gödel, Escher, Bach”

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 $Y = I^3 - \pi_1^{-1}(A_1) \cap I^3 - \pi_2^{-1}(A_2) \cap I^3 - \pi_3^{-1}(A_3) \cap I^3$. The question is if the projections of this maximal set are as desired. This is the case for the first projection if and only if for each $(y,z)\in A_1$ there exists $x\in\Bbb R$ such that $(x,y)\in A_3$ and $(x,z)\in A_2$. Similarly for the other two projections.