# Overlapping maps

I have two maps that are exactly the same, except of different sizes (the maps are proportional). I place the smaller of the two maps onto the larger, both face up so that the smaller map is completely within the borders of the larger map.

Is there always an overlapping point corresponding to the same geographic place on both maps? Can you show why or why not?

• I'm not sure if this counts as a puzzle. It's like a math question. Feb 23 '18 at 15:55

Yes. Here's a simple proof.

The smaller map defines a region within the larger map. You can draw that region on the smaller map too. That region you've just drawn is a smaller version of the map. In this smaller version of map you can draw that same region again, which is an even smaller version of the map. You can repeat this ad infinitum, creating smaller and smaller drawings, until the drawing is infinitely small, i.e. the size of a single point. That point is in the same location on every version of the map, so in particular it is the same point on the two largest maps.

It is essentially the Droste Effect, of an image containing itself.

## Yes.

This is a special case of Brouwer's fixed point theorem:

Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point.

The function sending one of your maps to the other is a simple isometry - an enlargement, combined with whatever rotations or translations are described by your positioning of the smaller map on the larger one - so it's definitely continuous. And the maps are presumably rectangular, or at least homeomorphic to closed discs, and therefore convex and compact. Thus, by BFPT there is a fixed point: a point which is in the same place on both maps.

Incidentally, this result can be used to show that if you have a map of a place in the place itself, then there must be a point which has the same location on the map and in real life - in other words, it must be possible to put a You Are Here sign on the map.

• Huh, interesting. TIL. Feb 23 '18 at 16:09
• Brouwer's fixed point theorem seems like a overkill here; it suffices to note that there's a spiral similarity mapping one map to the other and the center of that is a fixed point. Feb 26 '18 at 16:25
• @Ankoganit True, but I wanted to use BFPT because proving this simple result about maps was actually precisely how I first heard about BFPT all those years ago :-) Feb 26 '18 at 16:38

Yes,

Theoretically its /they are the centroids (G) of them, physically/mathematically.

• That is not true. The small version of the map could be placed in a corner of the larger map, away from the centroid of the larger map. There is no guarantee that the centroid of the large map and the small map coincide. Feb 23 '18 at 16:16