A sphere is painted black and white. Show that there are 3 equidistant points of the same color.

(In reply to Simplify by Jer)

By "arbitrarily small", you must mean "infinitely small", in which case, all the points are the same point.

On the sphere, the excess of the total angles over 180 degrees, of any equilateral triangle are proportional to the area of the sphere, so the excess of each angle over 60 degrees is also proportional to the area of the triangle.  For example, an equilateral triangle with all 90-degree angles is one octant of the full sphere.  Only as the size of the triange approaches zero does each angle approach 60 degrees. In the construction defined, the triangles have different sizes so the angles are different sizes, none of them 60 degrees, even though the differences be ever so slight.

Posted by Charlie on 2007-02-13 17:43:43