Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Why is it impossible to cover a sphere that has radius $R$ with $3$ open semispheres of radius $R$? In my mind I have the pictorial image of the situation, but I can't find a formal proof.

share|improve this question
    
What do you mean by cover ? –  Amr Nov 10 '12 at 12:19
    
If $S$ is the sphere (so a suface), and $U_i$ are the semispheres (so surfaces), I mean that $S=\bigcup_{i=1}^3 U_i$ –  Dubious Nov 10 '12 at 12:24
    
ok. Now I understand. –  Amr Nov 10 '12 at 12:29
    
Open means an open set (topological)? –  Amr Nov 10 '12 at 12:31
    
yes I mean open set. –  Dubious Nov 10 '12 at 12:45

1 Answer 1

up vote 4 down vote accepted

The union of two such semi-spheres always leaves an antipodal pair uncovered (where their boundaries intersect). This pair cannot be covered by a third semi-sphere since that does not contain any antipodal pair.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.