The Sierpinski carpet is constructed by removing the enter...

The Sierpinski carpet is constructed by removing the enter one-ninth of a square of side $1,$ then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is $1 .$ This implies that the Sierpinski carpet has area $0 .$

Key Concepts

Convergence

Convergence refers to the condition under which the sum of an infinite sequence of numbers approaches a specific finite value. This concept is crucial in analyzing fractal constructions because, despite an infinite process of removal or addition, the total measure can converge to a finite number, which often has counterintuitive results.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio. In the context of fractal construction, such as with the Sierpinski carpet, geometric series are used to sum an infinite number of decreasing areas to determine the total area removed or remaining.

Fractals

Fractals are structures that display complex patterns formed by repeating a simple process over and over in an ongoing feedback loop. They are characterized by having self-similar patterns at various scales, meaning the whole structure can be seen in its parts, and they often have non-integer dimensions.

Self-Similarity

Self-similarity is a property wherein a shape or pattern is invariant under certain scaling operations, meaning smaller portions of the object are similar to the whole. This principle is fundamental in understanding fractals since it explains the repeated, scaled-down replication seen in their construction.

Transcript

00:01 Hello and welcome.

00:02 We were looking at chapter 2, section 1, problem 59.

00:08 So we're given a serpinski carpet, and we're basically trying to find the area of the serpinski carpet.

00:20 So they give you some nice pictures there, so i don't have to.

00:28 And they're removing the center one -ninth of a square, and then that kind of leaves you eight smaller squares, if you look at it carefully, and then they remove the center one -ninth of that square, and so on.

00:44 So the carpet is what's not being removed.

00:48 So one way you can solve this is using a geometric series.

00:55 So remember, the sum of a geometric series is is a, the first term, divided by 1 minus r.

01:02 So we have to, if you, you could do this one of two ways.

01:07 You could look at, you could look at the area that's being removed, or you could look at the area that's left.

01:15 I was looking at the area that was removed when i solved this.

01:19 So that's how i'm going to show you guys.

01:21 So the area that's removed, it starts at one -ninth.

01:26 It's the center of the big square.

01:29 But we're not done there.

01:30 After we remove the center of the big square, we then remove the center of the eight, so there's going to be eight of these, eight of the smaller squares.

01:39 And if you look, if you think about it, the area of those smaller squares will be one -ninth of one -ninth, or one over 81.

01:56 And then if you look at the even smaller squares, though the first two terms is all we really need for our pattern.

02:03 But there's going to be 64 of those smaller squares...

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