On this webpage
appears a radio antenna using a fractal shape for the various elements (reflector/driven/director).
By scaling from the photo, the basic snowflake shape has the following proportions:

The relation between any long and short dimension is 1.5 times (see diag')
the rest of the antenna is just a co-joining of the same basic snowflake shape.
Using a green spot to 'tag' the short-span and pink dot for long span;
we can do some simple arithmetic to determine the relative dimensions of the antenna
We are just measuring in "units"
Later; the operating frequency can be factored in, to give absolute dimensions, therby giving a plan for antenna fabrication.
Remember pink span = 1.5 times green (in length).

We can see the fractal shape occupies only 15.4% of the cross-sectional area of the conventional "quad" antenna
Using well regarded design formulae for a cubical quad (but using the snowflake fractal instead of the regular geometric square)

L reflector (feet)1030/f (MHz) G (feet)42.75/f (MHz)
L driven (feet)1005/f (MHz)W (feet) 13.74/f (MHz)
L director(s) (feet) 975/f (MHz) C (pF) 2930/f (MHz)
S (feet) 232/f (MHz)Dr Carl Jellnek N6YNG

I haven't yet tried this type of design; however it seems there would be a great saving in antenna volume
(viz wind loading, mechanical stability etc) over more traditional designs (cubical Quad).
We can hope that it would resonate and have similar (or better) gain and directivity than the older designs it might replace!

In 1988 ham radio enthusiast Nathan Cohen had to set up his short wave radio system in his flat in central Boston.
The lease stipulated no antennae on the outside of the building, so he had to be inventive.
He had read Mandelbrots fractal book and got the idea to try out a fractal antenna.
He cut out aluminium foil in the form of an inverse Koch curve and glued it to a piece of A4 paper.
It worked amazingly well.
Not until later did he realize how innovative he had been.
In the time that followed he tested the antenna properties for other fractals, with good results, and then he founded Fractal Antenna Systems in 1995.