We report a micromagnetic numerical study on different quasi-crystal formations of magnetic vortices in a rich variety of dynamic transient states in soft magnetic nano-disks. Only the application of spin-polarized dc
currents to a single magnetic vortex leads to the formation of topological-soliton quasi-crystals composed of different configurations of skyrmions with positive and negative half-integer numbers (magnetic vortices and
antivortices). Such topological object formations in soft magnets, not only in the absence of Dzyaloshinskii- Moriya interaction but also without magnetocrystalline anisotropy, are discussed in terms of two different
topological charges, the winding number and the skyrmion number. This work offers an insight into the dynamic topological-spin-texture quasi-crystal formations in soft magnets.