# If you place a map of your town on the ground, how much of it covers the area represented?

Asked by LostInParadise (30221) November 5th, 2011

Not too much to add. Assume the ground is flat. As a hint, you may want to consider the problem in one dimension.

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## 13 Answers

Hmm, wouldn’t it be just one point? The distance from one point to another on the map would represent a larger distance than the distance from one point to another in the town, so any area larger than a point wouldn’t correspond…right?

Mariah (25883)

I’m not at all understanding this question. When I have a map of the town, the whole town is included. If you’re asking how much of the town is actually physically covered when I put a map on the ground, well then it depends on the scale and size of the map.

zenvelo (38002)

What size is the map?

XOIIO (18328)

And, to waht scale is it?

XOIIO (18328)

@zenvelo I think he means, some portion of the map must be on top of the exact place in the town that is pictured in the map… so hard to articulate, but for example, if you do this in town hall, at some point on the map, the point where town hall is pictured will lie exactly on top of the same point of town hall IRL.

Mariah (25883)

@Mariah , Thanks for interpreting. You are correct. It would be just one point. It is easy to see in one dimension. Suppose I make a map of a part of the number line, say from 0 to 10 and scale by a factor of 1/10. I place the map on the number line starting at 2. 0 on the map corresponds to 2 on the line and 10 on the map corresponds to 3. At 0 the map coordinate is smaller and at 10 it is larger. At exactly one place they will correspond. It is easy to find that place. If x is the map coordinate then we have x = 2 + x/10. Solving we get x = 2 2/9.

For a 2 dimensional map, we can do the same thing for North/South coordinates to get a single coordinate. Then do the same thing with East/West coordinates. The North/South and East/West coordinates determine the point where the two coincide.

LostInParadise (30221)

@LostInParadise Cool question. Always a fan of math conundroms.

Mariah (25883)

Oh my..well…I live on the outskirts of my tiny tourist town, lets see…if the map was your average like 3×3 fold out it would cover the kiddie pool my goose swims in on my little mountain top. lol

Coloma (47180)

Long live linear transformations. I picture the actual paper map (I assume a rectangle) that you placed on the ground being represented (or imaged?) somewhere on the map as a miniscule rectangle. Your actual location is shown by this tiny rectangular feature. It’s recursive: an endless regress of nested tiny rectangles, converging exponentially to a limit point. Not unlike images in parallel facing mirrors. But then I’m not sure I understood the question.

gasman (11315)

@gasman , I think you got the question right. You show another way of looking at the problem that is a little more difficult to visualize. Start with the map on the ground. Then cut out the part of the map that represents the ground covered by the map and return it to its place on the ground. Keep repeating this process. The map keeps getting smaller and converges to a single point.

This problem is a special case of Brower’s Fixed Point Theorem

LostInParadise (30221)

@LostInParadise I think I’ve seen that before – very interesting. I never formally learned topology – I got heavy doses of applied math instead.

@hiphiphopflipflapflop Exactly.

gasman (11315)

“The consequence of the Brouwer fixed point theorem in three dimensions is that no matter how much you stir a cup of coffee, some point of the liquid will return to its original position. That is, assuming that none of the liquid was spilled.” Very counter intuitive but apparently true. Great question!

flutherother (33263)

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