  We present a novel $N$-body simulation method that compactifies the infinite
spatial extent of the Universe into a finite sphere with isotropic boundary
conditions to follow the evolution of the large-scale structure. Our approach
eliminates the need for periodic boundary conditions, a mere numerical
convenience which is not supported by observation and which modifies the law of
force on large scales in an unrealistic fashion. We demonstrate that our method
outclasses standard simulations executed on workstation-scale hardware in
dynamic range, it is balanced in following a comparable number of high and low
$k$ modes and, its fundamental geometry and topology match observations. Our
approach is also capable of simulating an expanding, infinite universe in
static coordinates with Newtonian dynamics. The price of these achievements is
that most of the simulated volume has smoothly varying mass and spatial
resolution, an approximation that carries different systematics than periodic
simulations.
  Our initial implementation of the method is called StePS which stands for
Stereographically Projected Cosmological Simulations. It uses stereographic
projection for space compactification and naive $\mathcal{O}(N^2)$ force
calculation which is nevertheless faster to arrive at a correlation function of
the same quality than any standard (tree or P$^3$M) algorithm with similar
spatial and mass resolution. The $N^2$ force calculation is easy to adapt to
modern graphics cards, hence our code can function as a high-speed prediction
tool for modern large-scale surveys. To learn about the limits of the
respective methods, we compare StePS with GADGET-2
\citep{Gadget2_2005MNRAS.364.1105S} running matching initial conditions.
