The new sharp points will start a new ice crystal.
A new snowflake picture is formed by drawing a small star at the end of each exposed line of the previous picture:
Here (again) are the two things that recursion does:
Here are how this applies to the snowflake problem:
This snowflake drawing procedure stops when the sub-snowflakes reach the smallest size. In nature, if the flake remains in the atmosphere for long the ice crystals keep growing and the flake fills in. Or if flake reaches ground before this happens it has an open structure like our snowflake.