Return to Question

added 2 characters in body

view source

edited Nov 18 at 1:27

At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?

What is the strongest, most impressive statement one can make here? The Borsuk-Ulam Theorem applies, but I am uncertain of its full implications. Could one say that the two points are (1) separated by the same 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.

To what extent do these results extend to $\mathbb{R}^d$ for $d>3$? Thank you for your help!
_{(Wikipedia image)}

view source

asked Nov 18 at 1:13

Joseph O'Rourke
43.2k●1●49●214

There are two points on the Earth's surface that ... ?

At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?

What is the strongest, most impressive statement one can make here? The Borsuk-Ulam Theorem applies, but I am uncertain of its full implications. Could one say that the two points are (1) separated by the same 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.

To what extent do these results extend to $\mathbb{R}^d$ for $d>3$? Thank you for your help!
_{(Wikipedia image)}