  A mathematically interesting hyperbolic solution to the Einstein field
equations is studied on an eight-dimensional pseudo-Riemannian manifold
$\mathbb{X}_{4,4}$ that is a spacetime of four space dimensions and four time
dimensions. [The signature and dimension of $\mathbb{X}_{4,4}$ are chosen
because its tangent spaces satisfy a triality principle \cite{Nash2010}
(vectors and spinors are equivalent).] This solution exhibits temporal
hyperbolic inflation of three of the four space dimensions and temporal
hyperbolic \textbf{deflation} of three of the four time dimensions. Comoving
coordinates for the \textbf{unscaled} dimensions are chosen to be $(x^4
\leftrightarrow \textrm{time}, x^8 \leftrightarrow \textrm{space})$, where the
$x^4$ coordinate corresponds to our universe's observed physical time dimension
and the $x^8$ coordinate corresponds to a predicted new physical spatial
dimension. This solution of the field equations manifests temporal hyperbolic
inflation $\cosh{({1}{3}\,H\,x^4)}$ of the scale factor associated with three
of the four space dimensions, and temporal deflation
$\textrm{sech}({{{1}{3}\,H\,x^4}})$ of the scale factor associated with three
of the four time dimensions. (Here $H$ is the Hubble parameter.) The scale
factors possess a more complicated dependence on the spatial $x^8$ coordinate,
which, however, turn out to be \textbf{periodic} in $x^8$ with period $\propto
1 / H$. \textbf{After "inflation" the observed physical macroscopic world has
three space dimensions and one time dimension.}
