In this nice picture you see the golden (Sierpiński) gasket.
Here is how it is defined. Let p0, p1, p2 be the vertices of the right triangle Δ.
Put λ = (√5 - 1)/2 ≈ 0.618 (the golden mean!) and define the three contracting similitudes f0, f1, f2 as follows:
fi (x) = λx + (1 - λ) pi, i = 0, 1, 2.
Now define the sets by induction:
Δ0 := Δ, Δn := U0 ≤ i ≤ 2 fi (Δn-1), n = 1, 2,...
The sets Δn 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 fi.
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 un denote he number of holes on the n'th level. As we see in this picture (hopefully), u0 = 1, u1 = 3, u2 = 9, u3 = 24, u4 = 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 recent paper (joint with Dave Broomhead and James Montaldi).