*Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). Please try to adjust the answer to my level.*


A map is a smaller version of the land: rotated and scaled down. The prerequisite is that the map (say of USA) lands entirely inside the land (and not partly in Canada or in the ocean). Another prerequisite might be that the map has no holes (***is that really necessary?***) and perhaps that it must be a convex region (***however I doubt that this is needed***.) Please help me eliminate these doubts and prove that $p \in M, p \in L$. Correct my notation if needed by editing this post.


Here is the notation that I came up with (let's see if LaTex likes me):


* $L$ = land (region in $R^2$)
* $M$ = map (region in $R^2$)
* $T$ = transformation in $R^2$ (scalar times a rotation matrix)
* $\vec{s}$ = shift vector (since the overall transformation is generally non-linear)
* $\vec{p}$ = "pivot" - the point that does not change.


Now, I can solve for p uniquely by picking a non-trivial triangle on the land denoted by vectors $\vec{l\_1}, \vec{l\_2}, \vec{l\_3}$, locating the corresponding vectors $\vec{m\_1}, \vec{m\_2}, \vec{m\_3}$ on the map and writing out:


$\vec{m\_1} = T \vec{l\_1} + \vec{s}$, $\vec{m\_2} = T \vec{l\_2} + \vec{s}$, $\vec{m\_3} = T \vec{l\_3} + \vec{s}$


I solve for T:
 $T = [ m\_1 - m\_2 \; \; \; m\_1 - m\_3 ] [ l\_1 - l\_2 \;\;\; l\_1 - l\_3 ]^{-1}$ 
Because we picked a non-trivial triangle, it's area will be non-zero and the matrix on the right will be invertible. So, we can always solve for T.
We can also solve for s: $\vec{s} = \vec{m\_1} - T \vec{l\_1}$, and finally for the pivot p: $\vec{p} = (I - T)^{-1} \vec{s}$. Since T = c \* rotation matrix, where $c \leq 1$, the only time when (I - T) does not have an inverse is when I = T (details omitted).


***So, it seems that I can solve for a unique p.***


* Now, how do I prove that pivot $\vec{p}$ will be inside both shapes/regions L and M?
* Finally, what assumptions must I make about convexity of M, absence of holes, etc ?


P.S. Which undergraduate classes might have this as a homework problem?



Help me with this proof: Drop a printed map of the land on the land and there must be some common point.
This is usually quoted as an "application" of Brouwer's fixed point theorem: 
<http://en.wikipedia.org/wiki/Brouwer%27s_fixed_point_theorem>


which says that a continuous map from a closed ball (in any number of dimensions) to itself must have a fixed point.


You definitely need it to not have holes, because otherwise you could just rotate around the center of the hole and no fixed points will exist. However, convexity is not relevant. The main point is that the United States (minus Alaska and Hawaii and whatever holes it might have) is homeomorphic to the closed disk, so we can apply the homeomorphism, find the fixed point for the closed disk, and then reverse the homeomorphism (apply its inverse) to get the fixed point of the original function (which is your laying of the land on itself).


I'm not really sure how to finish your proof, but I would think this would appear in an algebraic topology course.


