Sierpinskis Gasket - Linear Algebra

[Upsidealien]

Homework Statement

Where
:- ∆0 = ∆ is the original triangle ABC.
:- DEF are the midpoints of AB, BC, AC respectively.
:- f1, f2, f3 map the triangular region ABC to the triangular region ADF, DBE and FEC respectively.
:- ∆n+1 = f1(∆n) ∪ f2(∆n) ∪ f3(∆n) for n≥0.

(these are just definitions of Sierpinskis gasket)

1. We need to first sketch ∆1 and ∆2 and prove that ∆n+1 ⊆ ∆n for all n ≥ 0.

We then define S = intersection of ∆n from n=1 to ∞.

2. We then need to prove that S is non-empty and that S = f1(S) ∪ f2(S) ∪ f3(S).

Homework Equations

Given above.

The Attempt at a Solution

I attempted 1. by using induction but did not get very far and for 2. I have proved that S ⊇ f1(S) ∪ f2(S) ∪ f3(S) but I am struggling to prove S ⊆ f1(S) ∪ f2(S) ∪ f3(S) Thanks

Tom

 

[MathematicalPhysicist]

So you basically at the n+1 step map the n-th triangle to three of its interior triangles, and leaving the middle triangle intact. Doesn't this by itself prove the inclusion, even proper inclusion.

For the intersection, take x\in S thus for every n \geq 1 x is in delta(n), thus x\in f_1(\Delta n)\cup f_2(\Delta(n)\cup f_3(\Delta(n)) for every n>0, thus you can keep iterating it ad infinitum, thus x\in f_1(S)\cup f_2(S)\cup f_3(S).