Sierpinskis Gasket - Linear Algebra

In summary, the conversation discusses the definitions of Sierpinski's gasket and the process of mapping triangles to their interior triangles. It also mentions the need to prove the inclusion and intersection of these triangles.
  • #1
Upsidealien
8
0

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
 
Last edited:
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  • #2
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 [tex]x\in S[/tex] thus for every [tex]n \geq 1[/tex] x is in delta(n), thus [tex]x\in f_1(\Delta n)\cup f_2(\Delta(n)\cup f_3(\Delta(n))[/tex] for every n>0, thus you can keep iterating it ad infinitum, thus [tex]x\in f_1(S)\cup f_2(S)\cup f_3(S) [/tex].
 

Related to Sierpinskis Gasket - Linear Algebra

1. What is the Sierpinski Gasket?

The Sierpinski Gasket is a fractal shape constructed by recursively dividing an equilateral triangle into smaller triangles and removing the central triangle in each iteration.

2. What is the significance of Sierpinski Gasket in Linear Algebra?

In Linear Algebra, the Sierpinski Gasket is often used as an example of a self-similar shape, which is a shape that can be divided into smaller copies of itself. This concept is important in understanding transformations and matrices in Linear Algebra.

3. How is the Sierpinski Gasket related to matrices?

The Sierpinski Gasket can be represented by a matrix known as the Sierpinski matrix, which is a 3x3 matrix with 1s on the main diagonal and 0s everywhere else. This matrix can be used to generate the Sierpinski Gasket through matrix multiplication.

4. Can the Sierpinski Gasket be extended to higher dimensions?

Yes, the concept of the Sierpinski Gasket can be extended to higher dimensions, such as the Sierpinski Pyramid in 3D or the Sierpinski Cube in 4D. These shapes can also be represented by matrices and have applications in Linear Algebra.

5. What are some real-world applications of the Sierpinski Gasket in Linear Algebra?

The Sierpinski Gasket can be used to model natural phenomena, such as the branching patterns of trees or the formation of coastlines. It also has applications in computer graphics and image compression, where the self-similar properties of the gasket can be utilized to reduce storage space and processing time.

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