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 R3 which projects onto R2 in three different ways to form the letters of the book’s title. It is natural to ask for generalizations: for which subsets A1,A2,A3 of R2 is there set XR3 such that, with π1,π2,π3 the projection maps R3R2, π1(X)=A1,π2(X)=A2, and π3(X)=A3?

More generally, let {πi}iIn,m be the canonical projection maps RnRm, where mn. For which sets {Ai}iIn,mRm is there a set X such that πi(X)=AiiIn,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, X?

3) What is the volume of the largest possible set in question in terms of Ai? It’s worth noting that, if A1,A2,A3 are measurable subsets of I2, then there is an interesting formula for the volume of X plus the volume of Y=I3π11(A1)I3π12(A2)I3π13(A3)I3.

enter image description here

Answer

The obvious (and maximal) candidate for the 3d object is
{(x,y,z)R3(y,z)A1,(x,z)A2,(x,y)A3}
obtained by intersecting the maximal sets that give one of the three projections each.
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)A1 there exists xR such that (x,y)A3 and (x,z)A2. Similarly for the other two projections.

Hofstadter’s examples work because already in the vertical bar of the E, there is so much material in the B (its lower line with final arc) that the G is guaranteed to work; and similarly, in the lower bar of the E, there is so much material in the G (its almost straight lower line) that the B is guaranteed to work; and finally the vertical bar of the B and the left end of the G are material enough to guarantee the E to work. So in a way, the trick is that the B and the G are less round than you might normally write them.

Attribution
Source : Link , Question Author : Kind Bubble , Answer Author : Hagen von Eitzen

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