Good good stuff, [Dfox]. I wonder if an enormous fractal could be stamped in a huge rubber sheet, then soles cut from this. No two soles would be alike, though all would be similar. If there was someway to generate the fractal directly on the rubber (perhaps diffusing oil on the rubber, or a growing crystal) that would be a magnitude cooler.

Wouldn't the "infinite length" in this case be analagous to an infinite geometric sequence, in that it does converge to a number, but never actually reaches it?

I can see this working down several levels of fractility, but after about ten levels, wouldn't the texture of the rubber soles exceed the fractal pattern? +

It would of course. Fractals are only infinite in length conceptually and mathematically. As soon as you introduce them to the real world all that infinite stuff gets ignored by atoms and the like.

A friend once claimed that the M25 is a fractal, for although it encloses a finite area (Greater London), it takes a near infinite amount of time to traverse, and must therefore be infinitely long.

I stumbled upon this yesterday. I am 4 years late, but I have
some feedback:

At the time this idea appeared, Shoes For Crews:

http://ShoesForCrews.Com/

Already had fractal treads. Shoes For Crews gloats that these
shoetreads reduces claims for slip-injuries in Workers’
Compensation by over half.

The good news is that fractal treads work. The bad news is that
someone else already patented it. The good news is that the
patent only covers a specific fractal pattern and that one can
create and patent a new fractal pattern for tire/shoe-treads.

Those shoes don't have fractal soles, nor do they remotely resemble
fractals in any way, which leads me to believe that new user [Walabio]
is just advertising for that company and should
have his/her comments removed – post-haste!

doubtful that a company shill would use the phrase "the company gloats that..." in a spam post. The word "fractal" doesn't appear on their website anywhere.

¡I am not a spammer! I just stumbled upon this site and noted
that fractal soles already exists. Indeed I wear them. I work in
a
restaurant. The claims about slipresistance is on the website:

http://tinyurl.com/m9tj6w

If one looks at the pattern it is an obvious fractal. It is
selfsimilar on multiple scales.

It is true that the site never uses. The word fractal. I see 2
reasons it might not state that the design is fractal:

It is a tradesecret.

Perhaps a genetic algorithm created the pattern and the
company does not realize that it is a fractal.

I suspect the former because the company calls it SFC Mighty
Grip. SFC is an acronym for Space-Filling Curve. Space-filling
curves are a subset of fractals and the pattern on the sole is a
space-filling fractal curve:

http://tinyurl.com/nywguf

This all seems to boil down to confusing about what a fractal is:

http://tinyurl.com/nofo5q

Many people are under the impression that fractals are always
chaotic. Many fractals are totally regular like the Sierpiński-
Triangle

http://tinyurl.com/m9gqvd

The only requirement for fractals is selfsimilarity.

The point is that for years, I wore shoes with fractal treads as
have millions.

Ever since I stumbled upon this page and decided to chimein, /i
thought about fractal treads and came up with mine own:

Start off with a right-angled isosceles triangles. The the groove
between the hypotenuse is 1 millimeter wide, 1 centimeter
deep and 1 decimeter long. The groove between the legs is the
squareroote of the hypotenuse in width, length. and depth.
Start cutting then in half with grooves the squareroot of the
legs. Iterate until the one makes hypotenuses less than 1
millimeter long. This should be near the limits of what one can
do with a mold. One should get great traction with that space-
filling fractal curve.