Comments on: The Limits of Flatness
http://ask.metafilter.com/220367/The-Limits-of-Flatness/
Comments on Ask MetaFilter post The Limits of FlatnessThu, 19 Jul 2012 13:27:57 -0800Thu, 19 Jul 2012 13:29:53 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: The Limits of Flatness
http://ask.metafilter.com/220367/The-Limits-of-Flatness
Are there an infinite number of 2D shapes? <br /><br /> I'm pretty sure--but at times not--and want to double-check. If you have the logic/science behind your answer, that'd also be cool.post:ask.metafilter.com,2012:site.220367Thu, 19 Jul 2012 13:27:57 -0800mwachsmathshapesgeometry2DlogicscienceresolvedBy: restless_nomad
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184319
If a shape has an infinite number of sides, it's a circle. If it has infinity - 1 sides, it's not. Therefore, yeah, there are an infinite number of shapes.comment:ask.metafilter.com,2012:site.220367-3184319Thu, 19 Jul 2012 13:29:53 -0800restless_nomadBy: tyllwin
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184323
Yes. Even just the set of "shapes of n sides with equal length sides" is infinite.comment:ask.metafilter.com,2012:site.220367-3184323Thu, 19 Jul 2012 13:32:17 -0800tyllwinBy: Pre-Taped Call In Show
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184324
Yes. Consider the shapes with n equal sides, where n is any integer. This already gives you as many unique shapes as there are integers, in other words, an infinite number.comment:ask.metafilter.com,2012:site.220367-3184324Thu, 19 Jul 2012 13:33:21 -0800Pre-Taped Call In ShowBy: DWRoelands
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184326
If a shape is simply the lines that connect the points on the corners...<br>
And there are an infinite number of possible x,y coordinates for those corners...<br>
Then there are an infinite number of shapes.comment:ask.metafilter.com,2012:site.220367-3184326Thu, 19 Jul 2012 13:33:34 -0800DWRoelandsBy: justkevin
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184327
Yes, there are an infinite number of 2d shapes. As proof by induction, take an isosceles triangle with a base of length 1. This triangle can have any real height between 0 and infinity, hence an infinite number of isosceles triangles.comment:ask.metafilter.com,2012:site.220367-3184327Thu, 19 Jul 2012 13:35:39 -0800justkevinBy: Sphinx
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184341
To answer the question: Yes. 2d, 3d, 4d, they all have infinite shapes. The # of dimensions are irrelevant.<br>
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Infinity has lost a lot of its luster over the years.comment:ask.metafilter.com,2012:site.220367-3184341Thu, 19 Jul 2012 13:46:32 -0800SphinxBy: arhammer
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184414
I do not mean to troll here, but as stated before, restless_nomad's answer makes no sense at all.<br>
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In general, the concept of a side, as in sides for polygons, does not apply to circles. <br>
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Also, it depends very much on the theory at hand what terms like "infinity" and "infinity - 1" denote. In all cases, they are not numbers. If at all meaningful, infinity - 1 would be regarded as infinity. Thus, to say that a property is satisfied for infinity (whatever that means), but not for infinity - 1 makes no sense.<br>
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Lastly, the conclusion does not follow from the previous statements at all.comment:ask.metafilter.com,2012:site.220367-3184414Thu, 19 Jul 2012 14:52:45 -0800arhammerBy: Hactar
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184420
There are not only an infinite number of 2D shapes, there are an infinite number of different triangles as well. There are an infinite number of numbers between 1 and 179 (1.01, 1.001, 1.0001, etc.) which are all angles that triangle can have at one of its points. There are an infinite number of regular 2D shapes, like justkevin said, and each one of these shapes can then be altered like the triangle. <br>
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Infinities of infinity.<br>
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Infinity is weird like that.comment:ask.metafilter.com,2012:site.220367-3184420Thu, 19 Jul 2012 15:03:16 -0800HactarBy: Chocolate Pickle
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184464
And thus Hactar has proved that it's aleph one, not just aleph zero.comment:ask.metafilter.com,2012:site.220367-3184464Thu, 19 Jul 2012 15:49:41 -0800Chocolate PickleBy: muddgirl
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184472
<i>In general, the concept of a side, as in sides for polygons, does not apply to circles.</i><br>
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It's just generalizing the concept of "slope of a line". How many slopes does a curve have?comment:ask.metafilter.com,2012:site.220367-3184472Thu, 19 Jul 2012 16:00:39 -0800muddgirlBy: naturetron
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184478
Surprisingly, infinities come in assorted sizes. Aleph-0 is the cardinality of integers; a bigger number called <em>c</em> is the cardinality of the real numbers. (Aleph-1 might be the same thing as <em>c</em>, but it's <a href="http://mathworld.wolfram.com/Aleph-1.html">undecideable</a>.)<br>
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There's a <a href="http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument">cool proof</a> illustrating how you can have <a href="http://en.wikipedia.org/wiki/Cardinality_of_the_continuum">a number bigger than aleph-0</a>.<br>
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Since you can represent (at least some) geometric curves parametrically with real-valued parameters, the number of geometric curves is at least <em>c</em>.comment:ask.metafilter.com,2012:site.220367-3184478Thu, 19 Jul 2012 16:07:03 -0800naturetronBy: weapons-grade pandemonium
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184603
Imagine a rectangle whose length is twice its width.<br>
Now imagine a rectangle whose length is three times its width. <br>
Now imagine a rectangle whose length is four times its width...<br>
Even between the first two there are an infinite number of shapes: <br>
Imagine a rectangle whose length is 2.10457 times its width.<br>
Imagine taking large sheets of paper and drawing closed doodles.<br>
Do you think you'd run out of shapes?comment:ask.metafilter.com,2012:site.220367-3184603Thu, 19 Jul 2012 19:02:29 -0800weapons-grade pandemoniumBy: weapons-grade pandemonium
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184631
Snowflakes.<br>
Fractal islands.comment:ask.metafilter.com,2012:site.220367-3184631Thu, 19 Jul 2012 19:30:24 -0800weapons-grade pandemoniumBy: Green With You
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184685
Speaking of <a href="http://en.wikipedia.org/wiki/Koch_snowflake">snowflakes</a>comment:ask.metafilter.com,2012:site.220367-3184685Thu, 19 Jul 2012 20:53:28 -0800Green With YouBy: windykites
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184699
I also wanted to mention fractals. Look 'em up, you'll like 'em.comment:ask.metafilter.com,2012:site.220367-3184699Thu, 19 Jul 2012 21:12:57 -0800windykitesBy: Aizkolari
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3185033
As naturetron pointed out, there are different kinds of infinities. For example, there's the <a href="http://en.wikipedia.org/wiki/Integers">integers</a> { ... -3, -2, -1, 0, 1, 2, 3...} which are <a href="http://en.wikipedia.org/wiki/Countably_infinite">countably infinite</a> and then there's the <a href="http://en.wikipedia.org/wiki/Real_numbers">reals</a>, which are <a href="http://en.wikipedia.org/wiki/Uncountably_infinite">uncountably infinite</a>.<br>
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Since you can draw justkevin's isosceles triangle with any real number greater than 0 as the height, the number of 2D shapes is uncountably infinite, which is bigger in some sense than being merely countably infinite.<br>
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I think you can use restless_nomad's idea to get a proof that the set of 2D shapes is at least countably infinite but it turns out it's significantly larger than that.comment:ask.metafilter.com,2012:site.220367-3185033Fri, 20 Jul 2012 08:31:37 -0800AizkolariBy: IAmBroom
http://ask.metafilter.com/220367/The-Limits-of-Flatness#3185172
<blockquote><a href="http://ask.metafilter.com/220367/The-Limits-of-Flatness#3184319">restless_nomad</a>: If a shape has an infinite number of sides, it's a circle.</blockquote><br>
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<a href="http://en.wikipedia.org/wiki/Julia_set">Julia</a> sniffs at being called a "circle", and turns her backs on you.comment:ask.metafilter.com,2012:site.220367-3185172Fri, 20 Jul 2012 10:31:04 -0800IAmBroom