\subsection{Hollow Cube Structure}
\label{hc-dsc}
	A natural approach to accommodating this 4:1 convergence in a world
limited to three-dimensions, is to build hollow cubes.  If we select one
side as the ``top'' of the cube, the four sides can accommodate four times
the surface area of the top, and naturally four times the number of routing
stacks of a given size.  As such, the ``sides'' contain the converging
stacks from one level, and the ``top'' contains the set of stacks at the
next level up the tree to which the ``side'' stacks are converging.  The
remaining side will remain open, free of stacks.  This last side could be
used to shorten wires slightly farther, but utilizing it in this manner
would decrease accessibility to the cube's interior.


\begin{figure}
\vspace{3 in}
\special{psfile=/home/wh/andre/thesis/PS/hc1-nobox.PS
         hoffset= -50 voffset= -75
         hscale = 0.5 vscale = 0.5}
\caption{First Level Hollow Cube Geometry (right)}
\label{hc-one}
\special{psfile=/home/wh/andre/thesis/PS/hc1-48-nobox.PS
         hoffset= 200 voffset= -63
         hscale = 0.5 vscale = 0.5}
\caption{Hollow Cube with Top and Side Stacks of Different Sizes (left)}
\label{hc-one-sm}
\end{figure}


\subsection{Features}
\label{hc-features}


This solution seems practical while giving reasonable performance for the
sizes of interest for the next decade.  It is not known to be optimal for
minimizing the inter-stage wiring distances.  Finding an optimal solution
that retains adequate accessibility to be of practical interest is still an
open issue.  While this hollow cube structure may not be the most compact
structure, it does exhibit a number of nice properties.