The pentaflake is a fractal with 5-fold symmetry. As illustrated above, five pentagons can be arranged around
an identical pentagon to form the first iteration of
the pentaflake. This cluster of six pentagons has the shape of a pentagon with five
triangular wedges removed. This construction was first noticed by Albrecht Dürer
(Dixon 1991).
For a pentagon of side length 1, the first ring of pentagons has centers at radius
Now, the length of the side of the first pentagon compound is given by
(7)
so the ratio of side lengths of the original pentagon to that of the compound is
(8)
We can now calculate the dimension of the pentaflake fractal. Let be the number of black pentagons and the length of side of a pentagon after the iteration,
An attractive variation obtained by recursive construction of pentagons is illustrated
above (Aigner et al. 1991; Zeitler 2002; Trott 2004, pp. 21-22).