THE SNOWFLAKE:

• At each stage (until we go back to the start) the snowflake's area increases. If we didn't stop, would it eventually get arbitrarily large? or is there an upper bound? and if there is a bound,what is the smallest such bound? And what about the perimeter (distance around the boundary) of the snowflake?

• If the process continues indefinitely, the snowflake has an infinite number of points (each with smaller points sticking out of its sides). How would you describe the relationship between any given point and its two "children")?

The Snowflake Curve is an example of a class of mathematical objects called "Fractals".
(That name is a bit of a play on words in that it suggests both the "fractured" looking shapes that are often involved and also that their fuzziness gives them an extra "fractional dimension".)

Such shapes often involve small parts which look like reduced images of the whole thing - a characteristic that mathematicians refer to as "self-similarity".

Many beautiful images have been constructed using these mathematical ideas. Some are abstract but others are strikingly similar to complicated structures in the world around us - and some of these figure in the creation of realistic textures and objects. The mathematics gives us both a tool for creating realistic images and perhaps some hints for understanding the processes by which some of the objects and beings around us come to take the forms they have.

There are a number of sites where you can learn more about fractals,
or about mathematics and art in general.

Another page devoted to the Snowflake Curve is at Bellevue Community College in Washington, and another is among the pages of "Mathman" Don Cohen who runs a private tutoring program in Illinois, (and if you search at the Math Forum you can find lots more!)