  We paste together patches of $AdS_6$ to find solutions which describe two
4-branes intersecting on a 3-brane with non-zero tension. We construct
explicitly brane arrays with Minkowski, de Sitter and Anti-de Sitter geometries
intrinsic to the 3-brane, and describe how to generalize these solutions to the
case of $AdS_{4+n}$, $n>2$, where $n$ $n+2$-branes intersect on a 3-brane. The
Minkowski and de Sitter solutions localize gravity to the intersection, leading
to 4D Newtonian gravity at large distances. We show this explicitly in the case
of Minkowski origami by finding the zero-mode graviton, and computing the
couplings of the bulk gravitons to the matter on the intersection. In de Sitter
case, this follows from the finiteness of the bulk volume. The effective 4D
Planck scale depends on the square of the fundamental 6D Planck scale, the
$AdS_6$ radius and the angles between the 4-branes and the radial $AdS$
direction, and for the Minkowski origami it is $M_4{}^2 = {2/3} \Bigl(\tan
\alpha_1 + \tan \alpha_2 \Bigr) M_*{}^4 L^2$. If $M_* \sim {\rm few} \times
TeV$ this may account for the Planck-electroweak hierarchy even if $L \sim
10^{-4} {\rm m}$, with a possibility for sub-millimeter corrections to the
Newton's law. We comment on the early universe cosmology of such models.
