Use this Demonstration to design your own snowflakes. Locators are indicated by the points, and you can create more (or delete some) with Alt+click (or Cmd+click). The points connect to form a line that is reflected and rotated to make a snowflake-like design. You can select new random starting points with the "new points" button.
If you have a snowflake that you would like to save, the "copy" button copies a design to the clipboard (using CopyToClipboard) that you can then paste into a new notebook.
This Demonstration also features a custom control modeled after ColorSlider.
When animated, the points each travel along a local Lissajous curve. The corresponding locators remain fixed during the animation. To realign the points and locators, reset the animation to the beginning of the "animate" slider.
This Demonstration is partially inspired by the snowflake activity within Jacquie Lawson advent calendars (www.jacquielawson.com).