What's the difference between 2 and 3? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T09:34:00Z http://mathoverflow.net/feeds/question/62130 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3 What's the difference between 2 and 3? Guntram 2011-04-18T15:09:11Z 2011-04-18T15:55:27Z <p>Here are two classical results which depend on whether a parameter is 2 or 3:</p> <ul> <li><p>It is possible to bisect an arbitrary angle with ruler and compass, but impossible to trisect it. </p></li> <li><p>While there are infinitely many Pythagorean triples, i.e. integer solutions to $x^2+y^2=z^2$, there are no non-trivial integer solutions to $x^3+y^3=z^3$. </p></li> </ul> <p>There are several other instances where the dividing line seems to be between 2 and 3:</p> <ul> <li><p>A 2-regular tree is countable, a 3-regular tree is uncountable.</p></li> <li><p>2SAT is solvable in polynomial time, 3SAT is NP-complete.</p></li> <li><p>A random walk on $\mathbf Z^2$ is recurrent, while a random walk on $\mathbf Z^3$ is transient. </p></li> </ul> <p>What other examples can you think of? </p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62132#62132 Answer by Hans Stricker for What's the difference between 2 and 3? Hans Stricker 2011-04-18T15:25:21Z 2011-04-18T15:25:21Z <p>In 2-dimensional Euclidean space every two lines intersect (maybe in the infinite), in 3-dimensional Euclidean space there are skew (?) lines.</p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62134#62134 Answer by subshift for What's the difference between 2 and 3? subshift 2011-04-18T15:29:50Z 2011-04-18T15:40:08Z <p>More examples are given as answers to <a href="http://cstheory.stackexchange.com/questions/5251/geometric-problems-that-are-np-complete-in-r3-but-tractable-in-r2" rel="nofollow">a similar question</a> about problems NP-hard in $\mathbb R^3$ but not in $\mathbb R^2$:</p> <ul> <li>Set-cover by half-spaces.</li> <li>Finding a shortest path between two points among polygonal obstacles.</li> <li>Determining whether a non-convex polygon/polyhedron can be triangulated without Steiner points.</li> <li>Realizability problem for $d$-dimensional polytopes is a candidate ($d \leq 3$ vs $d \geq 4$).</li> </ul> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62135#62135 Answer by Tom De Medts for What's the difference between 2 and 3? Tom De Medts 2011-04-18T15:30:35Z 2011-04-18T15:30:35Z <p>$SL_2(\mathbb{Z})$ is an amalgam whereas $SL_3(\mathbb{Z})$ is not.</p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62136#62136 Answer by Michael Renardy for What's the difference between 2 and 3? Michael Renardy 2011-04-18T15:33:33Z 2011-04-18T15:33:33Z <p>Autonomous systems of ODEs produce simple dynamics in two dimensions, but complex dynamics in three or more. This is directly related to the fact that curves in three or more dimensions can pass each other without crossing.</p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62138#62138 Answer by Roland Bacher for What's the difference between 2 and 3? Roland Bacher 2011-04-18T15:40:41Z 2011-04-18T15:47:43Z <p>$\mathbb R^3$ is much more rigid than $\mathbb R^2$ when considering conformality: Conformal transformations of $\mathbb R^2$ do not form a finite-dimensional Lie-group.</p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62139#62139 Answer by Roland Bacher for What's the difference between 2 and 3? Roland Bacher 2011-04-18T15:44:19Z 2011-04-18T15:44:19Z <p>The permutation group of two elements is abelian, the permutation group of three elements is not. There are thus non-galoisian number fields of degree $3$. </p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62141#62141 Answer by Roland Bacher for What's the difference between 2 and 3? Roland Bacher 2011-04-18T15:52:43Z 2011-04-18T15:52:43Z <p>There are infinitely many regular polytopes in $\mathbb R^2$ and only five in $\mathbb R^3$.</p> http://mathoverflow.net/questions/62130/whats-the-difference-between-2-and-3/62143#62143 Answer by Roland Bacher for What's the difference between 2 and 3? Roland Bacher 2011-04-18T15:55:27Z 2011-04-18T15:55:27Z <p>Elements of order $2$ in a group are the only non-trivial elements which their own inverse.</p>