  Classical toric varieties are among the simplest objects in algebraic
geometry. They arise in an elementary fashion as varieties parametrized by
monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$.
They may also be constructed from a rational fan $\Sigma$ in $\mathbb{R}^n$.
The combinatorics of the set $\mathcal{A}$ or fan $\Sigma$ control the geometry
of the associated toric variety. These toric varieties have an action of an
algebraic torus with a dense orbit. Applications of algebraic geometry in
geometric modeling and algebraic statistics have long studied the nonnegative
real part of a toric variety as the main object, where the set $\mathcal{A}$
may be an arbitrary set in $\mathbb{R}^n$. These are called irrational affine
toric varieties. This theory has been limited by the lack of a construction of
an irrational toric variety from an arbitrary fan in $\mathbb{R}^n$.
  We construct a theory of irrational toric varieties associated to arbitrary
fans. These are $(\mathbb{R}_>)^n$-equivariant cell complexes dual to the fan.
Such an irrational toric variety is projective (may be embedded in a simplex)
if and only if its fan is the normal fan of a polytope, and in that case, the
toric variety is homeomorphic to that polytope. We use irrational toric
varieties to show that the space of Hausdorff limits of translates an
irrational toric variety associated to a finite subset $\mathcal{A}$ of
$\mathbb{R}^n$ is homeomorphic to the secondary polytope of $\mathcal{A}$.
