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Manganese is an element with outstanding structural and magnetic properties. While most metallic elements adopt a simple crystal structure and order magnetically---if at all---in a simple ferromagnetic or antiferromagnetic configuration, the stable phase of manganese at ambient conditions, paramagnetic $\ensuremath{\alpha}\ensuremath{-}\mathrm{Mn},$ adopts a complex crystal structure with 58 atoms in the cubic cell. At a N\'eel temperature of ${T}_{N}=95\mathrm{K},$ a transition to a complex noncollinear antiferromagnetic phase takes place. The magnetic phase transition is coupled to a tetragonal distortion of the crystalline structure. In this paper we present an ab initio spin-density functional study of the structural and magnetic properties of $\ensuremath{\alpha}\ensuremath{-}\mathrm{Mn}.$ It is shown that the strange properties of Mn arise from conflicting tendencies to simultaneously maximize according to Hund's rule the magnetic spin moment and the bond strength, as expected for a half-filled d band. Short interatomic distances produced by strong bonding tend to quench magnetism. The crystal structure of $\ensuremath{\alpha}\ensuremath{-}\mathrm{Mn}$ is essentially a consequence of these conflicting tendencies---it may be considered as a topologically close-packed intermetallic compound formed by strongly magnetic (MnI, MnII) and weakly magnetic (MnIII) or even nearly nonmagnetic (MnIV) atoms. The noncollinear magnetic structure is due to the fact that the MnIV atoms arranged on triangular faces of the coordination polyhedra are not entirely nonmagnetic---their frustrated antiferromagnetic coupling leads to the formation of a local spin structure reminiscent of the N\'eel structure of a frustrated triangular antiferromagnet. Consequently, also the other magnetic moments are rotated out of their collinear orientation. The calculated crystalline and magnetic structures are in good agreement with experiment. However, it is suggested that the magnetism leads to a splitting of the crystallographically inequivalent sites into a larger number of magnetic subgroups than deduced from the magnetic neutron diffraction data, but in accordance with NMR experiments. In a companion paper, the properties of the other polymorphs of Mn and their relative stability will be discussed.