The knowledge of the nodes of the many-fermion wave function would enable exact calculation of the properties of fermion systems by Monte Carlo methods. It is proved that fermion nodal regions have a tiling property, there is only one distinct kind of nodal region. All others are related to it by permutational symmetry. For some free particle systems, it is shown that there are only two nodal regions. An explicit form for the nodes of the many-fermion density matrix would enable exact simulations to be carried out at finite temperature. In the high-temperature limit, its nodes are related to Voronoi polyhedra. Two-dimensional cross sections of nodes are depicted. General computable families of fermion wave functions and density matrices are discussed.