In
this nice picture you see the *golden
(Sierpiński) gasket*.

Here
is how it is defined. Let *p*_{0},
*p*_{1},
*p*_{2}
be
the vertices of the equilateral triangle Δ.

Put
*λ
= *(√5
- 1)/2 ≈ 0.618 (the
golden mean!) and define
the three contracting similitudes *f*_{0},
*f*_{1},
*f*_{2}
as
follows:

*f*_{i}
(* x*)
=

Now define the sets by induction:

Δ_{0
}:=
Δ, Δ_{n
}:=
U_{0
≤ }_{i
}_{≤
}_{2 }
*f*_{i
}(Δ_{n}_{-1}),
*n
*=
1, 2,...

The
sets Δ_{n}_{
}are
nested, and the golden gasket is defined as their *intersection*.
In other words, it is the invariant set for the iterated
function system constituted
by the *f*_{i}*.*

Despite
the fact that the Open
Set Condition is
not
satisfied
by the golden gasket, it (perhaps surprisingly) possesses the
important property of *total
self-similarity*,
which means that a suitable arbitrarily small bit of it not just
resembles the original gasket, but is in fact similar
to
it (*i.e.*,
after an appropriate rescaling it coincides with the original
gasket). This allows one to deduce a closed formula for its *Hausdorff
dimension*,
which turns out to be approximately equal to 1.93063
(and
of course strictly
less than
the similarity dimension!).

Let
*u*_{n
}denote
he number of holes on the *n*'th
level. As we see in this picture (hopefully), *u*_{0}
= 1,
*u*_{1}
= 3, *u*_{2}
= 9, *u*_{3}
= 24, *u*_{4}
= 63... Can
you now guess the recurrence relation?

One
may ask whether the fact that *λ
*equals
the golden mean is important. Well, it is! In fact, there are other
values that yield the total self-similarity of the invariant
set, but only a discrete set, and they all algebraic integers which
resemble the golden mean (s.c. multinacci
numbers).

Want to know more? You may download my paper (joint with the late Dave Broomhead and James Montaldi).