Why there exist no void with co-ordination number 5
Trigonal bipyramidal complexes are abundant.  However, if you want a periodic crystal, you must tesselate volume with translation - the unit cell of the crystal lattice.  A five-fold symmetric figure cannot do this, for the angles do not add up without gaps.  2-, 3-, and 6-fold symmetries are allowed, offering 230 unique 3-D space groups overall. 

If you allow two kinds of unit cell, you can get approximate periodicity, as in quasicrystals and gas-water clathrates.  Both can have 5- or 10-fold symmetry overall and give sharp x-ray or neutron diffraction spots (the IUC definition of a crystal).