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 Figure Eights (Posted on 2005-02-15)
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)

 Subject Author Date What am I missing? Harry 2010-07-24 20:52:05 full area fractal - but still countable number of curves ronen 2005-02-24 15:04:04 re: solution - the number of curves must be countable(?) David Shin 2005-02-23 07:19:29 solution - the number of curves must be countable(?) ronen 2005-02-22 21:40:31 re: A thought David Shin 2005-02-22 03:17:33 A thought owl 2005-02-21 04:50:08 re(11): I think it's... - Elaboration Tristan 2005-02-19 17:28:58 re(10): I think it's... - Elaboration David Shin 2005-02-19 15:28:11 re(9): I think it's... - Elaboration Tristan 2005-02-19 15:10:04 re(8): I think it's... - Elaboration David Shin 2005-02-19 04:36:45 re(7): I think it's... - Elaboration Tristan 2005-02-19 03:22:30 re(6): I think it's... - Elaboration David Shin 2005-02-18 22:40:33 re(5): I think it's... David Shin 2005-02-18 00:14:05 re(4): I think it's... Tristan 2005-02-17 23:42:23 re(3): I think it's... David Shin 2005-02-17 19:31:39 No Subject Vee-Liem Veefessional 2005-02-17 05:05:51 re(4): still thinking / cardinality Larry 2005-02-17 04:42:35 re(2): I think it's... Tristan 2005-02-17 03:15:00 re: I think it's... Avin 2005-02-16 13:30:14 re: This may work! Charlie 2005-02-16 13:27:11 This may work! np_rt 2005-02-16 08:38:55 re(3): still thinking David Shin 2005-02-16 06:49:44 re(2): still thinking Larry 2005-02-16 03:56:09 re: still thinking Charlie 2005-02-16 03:31:26 re: I think it's... David Shin 2005-02-16 01:08:36 I think it's... Tristan 2005-02-15 23:56:25 still thinking Larry 2005-02-15 21:50:40 re(3): additional thoughts Gamer 2005-02-15 20:51:33 re(2): additional thoughts Steve Herman 2005-02-15 20:08:30 Tips David Shin 2005-02-15 19:59:16 re: additional thoughts Charlie 2005-02-15 19:04:45 re: additional thoughts Steve Herman 2005-02-15 17:41:01 additional thoughts Larry 2005-02-15 16:52:06 thoughts and cheats Jer 2005-02-15 16:43:39

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