  In this paper, we explore how geometric structures can be grown exponentially
fast. The studied processes start from an initial shape and apply a sequence of
centralized growth operations to grow other shapes. We focus on the case where
the initial shape is just a single node. A technical challenge in growing
shapes that fast is the need to avoid collisions caused when the shape breaks,
stretches, or self-intersects. We identify a parameter $k$, representing the
number of turning points within specific parts of a shape. We prove that, if
edges can only be formed when generating new nodes and cannot be deleted, trees
having $O(k)$ turning points on every root-to-leaf path can be grown in
$O(k\log n)$ time steps and spirals with $O(\log n)$ turning points can be
grown in $O(\log n)$ time steps, $n$ being the size of the final shape. For
this case, we also show that the maximum number of turning points in a
root-to-leaf path of a tree is a lower bound on the number of time steps to
grow the tree and that there exists a class of paths such that any path in the
class with $\Omega(k)$ turning points requires $\Omega(k\log k)$ time steps to
be grown. If nodes can additionally be connected as soon as they become
adjacent, we prove that if a shape $S$ has a spanning tree with $O(k)$ turning
points on every root-to-leaf path, then the adjacency closure of $S$ can be
grown in $O(k \log n)$ time steps. In the strongest model that we study, where
edges can be deleted and neighbors can be handed over to newly generated nodes,
we obtain a universal algorithm: for any shape $S$ it gives a process that
grows $S$ from a single node exponentially fast.
