A golden gasket is a self-similar fractal generated by three similarities centered at the vertices of an equilateral
triangle, with a specially chosen common contraction ratio (Broomhead et al.
2004). For an integer 
, let 
be the unique positive root of
 |
(1)
|
If 
, 
, and 
are the vertices, define
 |
(2)
|
for 
, 1, and 2. The golden gasket 
is the unique
nonempty compact set satisfying
 |
(3)
|
The left and right panels show seventh-step approximations to the Sierpiński sieve with contraction ratio 
and to 
, respectively. The middle panel shows the three first-step
images for 
, with their pairwise overlaps in orange and the central
hole outlined.
Different compositions can define the same map, with 
for 
,
where the powers denote repeated composition. The
Hausdorff dimension and capacity
dimension of 
are
 |
(4)
|
where 
is the smallest positive root
of 
(Broomhead et al.
2004). For 
,
is the reciprocal
of the golden ratio, and the Hausdorff
dimension is 
.
As 
, 
and the dimensions tend to 
, the dimension of the Sierpiński
sieve.
