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.

翻译：本文探讨了几何结构如何以指数级速度增长。研究过程从初始形状出发，通过施加一系列中心化增长操作来生成其他形状。我们重点研究初始形状仅为单一节点的情况。实现快速形状增长的技术挑战在于避免形状断裂、拉伸或自交导致的碰撞。我们定义了参数$k$，表示形状特定部分内的转折点数量。我们证明：若仅在生成新节点时允许形成边且边不可删除，则每条根叶路径上具有$O(k)$个转折点的树可在$O(k\log n)$时间步内生成，而具有$O(\log n)$个转折点的螺旋结构可在$O(\log n)$时间步内生成（$n$为最终形状尺寸）。针对此情形，我们还证明：树的根叶路径上最大转折点数是生成该树所需时间步的下界，且存在一类路径，该类中具有$\Omega(k)$个转折点的任意路径需$\Omega(k\log k)$时间步才能生成。若节点在相邻时即可连接，我们证明：当形状$S$的生成树在每条根叶路径上具有$O(k)$个转折点时，$S$的邻接闭包可在$O(k \log n)$时间步内生成。在我们研究的最强模型（允许删除边并将邻居移交给新生成节点）中，我们得到通用算法：对任意形状$S$，该算法均能给出从单一节点以指数级速度生成$S$的过程。