There are two points on the Earth's surface that ... ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:51:51Z http://mathoverflow.net/feeds/question/112735 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112735/there-are-two-points-on-the-earths-surface-that There are two points on the Earth's surface that ... ? Joseph O'Rourke 2012-11-18T01:13:11Z 2012-11-19T05:03:40Z <p>At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...? </p> <p>What is the strongest, most impressive statement one can make here? The <a href="http://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" rel="nofollow">Borsuk-Ulam Theorem</a> applies, but I am uncertain of its full implications. Could one say that the two points are (1) separated by a specific geodesic distance, (2) have the same temperature, and (3) have the same barometric pressure? For example...? I pose this question for its pedagocial import, but it clearly follows from known theorems.</p> <p>To what extent do these results extend to $\mathbb{R}^d$ for $d>3$? Thank you for your help! <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Hairy_ball_one_pole.jpg/220px-Hairy_ball_one_pole.jpg" /> <sub>(<a href="http://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Hairy_ball_one_pole.jpg/220px-Hairy_ball_one_pole.jpg" rel="nofollow">Wikipedia image</a>)</sub></p> http://mathoverflow.net/questions/112735/there-are-two-points-on-the-earths-surface-that/112738#112738 Answer by Eric Naslund for There are two points on the Earth's surface that ... ? Eric Naslund 2012-11-18T01:49:10Z 2012-11-18T01:49:10Z <p>We can say the following:</p> <p>"At any given time, there are two points on the earth exactly 20 000 km apart with the same exact same temperature <em>and</em> barometric pressure."</p> <p>I am making a few assumptions, but do note that the distance from the north pole to the south pole is 20 000 km. Indeed, in the <a href="http://en.wikipedia.org/wiki/Borsuk%25E2%2580%2593Ulam_theorem" rel="nofollow">Wikipedia article</a> which you linked to in your question, we find the quote:</p> <p>"The case n = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures. This assumes that temperature and barometric pressure vary continuously."</p> http://mathoverflow.net/questions/112735/there-are-two-points-on-the-earths-surface-that/113795#113795 Answer by Igor Rivin for There are two points on the Earth's surface that ... ? Igor Rivin 2012-11-19T04:55:18Z 2012-11-19T05:03:40Z <p>One of the standard generalizations is Knaster's conjecture: for every function $f: \mathbb{S}^{n-1}\rightarrow \mathbb{R}^m, m\lt n,$ and $k=n-m+1$ points $p_1, \dots, p_k \in \mathbb{S}^{n-1}$ does there always exista rotation $\rho \in SO(n),$ such that $f(\rho(p_1) = \dots = f(\rho(p_k)).$ That this is true for $k=2$ is a theorem of H. Hopf (which generalizes Borsuk-Ulam). It turns out that Knaster's conjecture is true for some $m, n$ and false for others. See this <a href="http://kam.mff.cuni.cz/~matousek/cla/hinrich-richter-counterknaster.pdf" rel="nofollow">nice paper by Hinrich and Richter</a> for more results and references.</p>