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Figure Eights (Posted on 2005-02-15) Difficulty: 5 of 5
Suppose that I drew an infinite number of disjoint closed curves in the plane (such as circles, squares, etc.). Suppose that I then tell you that there is one curve for each positive real number.

You would not have too much trouble believing my assertions at this point. For example, I could have drawn all circles with center at the origin. They are all disjoint, and for each positive real number x, there is a corresponding circle - namely, the circle of radius x.

But suppose that I also tell you that all the curves I drew were figure eights. Can you believe my assertions now?

(A figure eight is a curve in the plane obtained from the basic "8" shape by any combination of translation, rotation, expansion, or shrinking.)

See The Solution Submitted by David Shin    
Rating: 4.2000 (5 votes)

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thoughts and cheats | Comment 1 of 34

This is tricky because the reals are uncountable. 

If they were numbered in a countable (orderable) way you could just put a little 8 at each whole number on the x-axis.  Another fun way would be to put each in one of the loops of the previous.

Some ways to cheat:

1) hide them at infinity: assign each positive real to a real from 0 to 360, then translate them each to an infinite distance from the origin at this angle.

2)Shrink each to a point (expansion of magnitude 0.) Then you can just line then up on the x-axis. 

If there is a solution (which there may, I think, be) the assertion I have a problem with is that you 'drew' it.

  Posted by Jer on 2005-02-15 16:43:39
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