The first element in understanding fractal antennae is the concept of
tortuosity in which the path length between two points is increased by
requiring that the small line elements are no longer colinear. A simple
tortuous model is displayed in Fig. 5, where the parameter 
represents the variation from the simple dipole model (line
radiator), i.e. the dipole is recovered as 
.
Figure 5: A simple tortuous variation of a line radiator. Note that the
antenna will radiate every time there is a change in direction.
Except for the propagation effect, we can observe that this antenna (Fig. 5) can be considered as the contribution from a long line element (a dipole) plus the contribution from a Cantor set of radiators as described in the previous section (see Eq. (3)). Therefore, the tortuosity naturally increases the radiation field intensity, at least in some direction, as compared with the single dipole element.
The field can be written for the structure of 5, with the help of Eq. (4), as the superposition of the 2N line elements, and is given by the normalized field
where 
is the length of the small segments composing the
tortuous path, and 
and 
. It is clear that in the
limit 
we recover the single dipole radiation
pattern. The effect of the tortuosity can now be posed as the behavior of
the normalized 
for 
. In general the
analysis can be simplified in the limit for small 
, i.e. 
. Of course P(0) is the dipole contribution, and 
is the change in the radiated power
density due to tortuosity. The dipole has a maximum in the radiated power
density 
, while the
tortuous contribution goes as 
. The function f depends on the given parameters, but its maximum is of the
order 
with clear regions in 
where it is
positive.
For our purposes, the most important contribution comes from the fact that 
is essentially independent of N and it
scales as 
which corresponds to the increase in the path length of the antenna due to
the tortuosity. Such technique can be applied to other geometries, giving
essentially the same scaling 
result.
This fact will be extremely relevant in our analysis since lightning has
naturally a tortuous path.
