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2. Lebesgue's universal covering problem - Wikipedia

   en.wikipedia.org/wiki/Lebesgue's_universal...

   Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper bound of the distances between all pairs of points in the set. A shape covers a set if it contains a congruent subset.

3. Packing problems - Wikipedia

   en.wikipedia.org/wiki/Packing_problems

   An a × b rectangle can be packed with 1 × n strips if and only if n divides a or n divides b. [15] [16] de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.) [15]

4. Uniformization theorem - Wikipedia

   en.wikipedia.org/wiki/Uniformization_theorem

   Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as ...

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6. Universal covering group - Wikipedia

   en.wikipedia.org/wiki/Covering_group

   The rotation group SO(3) has as a universal cover the group SU(2), which is isomorphic to the group of versors in the quaternions. This is a double cover since the kernel has order 2. (cf the tangloids.) The unitary group U(n) is covered by the compact group T × SU(n) with the covering homomorphism given by p(z, A) = zA. The universal cover is ...

7. Covering space - Wikipedia

   en.wikipedia.org/wiki/Covering_space

   For every topological space , the identity map: is a covering. Likewise for any discrete space the projection : taking (,) is a covering. Coverings of this type are called trivial coverings; if has finitely many (say ) elements, the covering is called the trivial -sheeted covering of .