So young, yet so long in the tooth

This maple leaf budded out in the last day or two. Although quite tiny, about 2cm long as shown, the length of the edge is a different matter. I would hazard a guess the length of the edge is already at least a meter long, maybe even several meters, depending on how we measure it.

This is an instance of the problem discovered by British cartographers who tried to measure the length of the coast of Britain. With more detail, measurements doubled, and the closer they looked, the longer it became. They had encountered a property of fractals.

The figure drawn on the white area in the photo above is an illustration of the first four steps in the construction of a Koch snowflake. Starting with an equilateral triangle, you continue to build smaller equilateral triangles on the middle third of each line segment. It is easy to see that every time you do this you increase the length by 4/3. After an infinite number of iterations, the length of the edge will be infinite, but curiously the area enclosed is quite finite, less than the area of a circle enclosing the figure.

So our little maple leaf is was getting (before I picked it) longer by the minute. Fern leaf edges are particularly long. Who knows, they may be longer than the distance between me and you.

And for something completely different...