Refer to the construction of the quadratic Koch island. The...

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule. Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.) Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$. Assume that the seed square of the quadratic Koch island has sides of length 16 . (a) Carefully draw the figures obtained in Steps 1 and 2 of the construction. (Hint: Use graph paper and make the seed square a 16 by 16 square. (b) Find the perimeter of the figure obtained in Step 1 of the construction. (c) Find the perimeter of the figure obtained in Step 2 of the construction. (d) Explain why the quadratic Koch island has infinite perimeter.

Refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.
Start: Start with a seed square [Fig. $12-37(a)]$. (Notice that here we are only dealing with the boundary of the square.)
Replacement rule: In each step replace any horizontal boundary segment with the "sawtooth" version shown in Fig. $12-37(b)$ and any vertical line segment with the "sawtooth" version shown in Fig. $12-37(c)$.
Assume that the seed square of the quadratic Koch island has sides of length 16 .
(a) Carefully draw the figures obtained in Steps 1 and 2 of the construction. (Hint: Use graph paper and make the seed square a 16 by 16 square.
(b) Find the perimeter of the figure obtained in Step 1 of the construction.
(c) Find the perimeter of the figure obtained in Step 2 of the construction.
(d) Explain why the quadratic Koch island has infinite perimeter.

Key Concepts

Fractals and Self-Similarity

This concept involves shapes that maintain a repeating pattern at every scale. The quadratic Koch island is a fractal, showing repeating, intricate patterns at increasingly smaller scales. Despite the complex appearance, a rule is applied uniformly at each iteration, illustrating the principle of self-similarity, where each part of the object resembles the whole.

Recursive Construction

In recursive construction, a simple initial shape is progressively transformed by repeatedly applying a specific rule. For the quadratic Koch island, the replacement rule is applied to every line segment at each stage, systematically increasing the complexity of the boundary. This recursive process is central in creating the fractal structure and leads to the elaborate 'sawtooth' patterns observed at every new step.

Geometric Replacement Rules

This key concept involves the method of replacing a line segment with a more complex pattern according to a fixed rule. In the quadratic Koch island, horizontal and vertical segments are replaced with distinct 'sawtooth' patterns, which not only alter the appearance but also impact the length of the boundary. Such replacement rules are critical in fractal geometry for generating detailed structures from simple starting forms.

Perimeter Calculation in Fractal Geometry

Calculating the perimeter in fractal constructions involves understanding how each iterative step affects the total boundary length. With each replacement, the number of segments increases while each segment is typically a fraction of the original length. Analyzing these changes often leads to a geometric series that explains the progression of the perimeter, which in many cases grows rapidly with each iteration.

Infinite Perimeter Concept

The phenomenon of having an infinite perimeter in finite fractal objects stems from the continual increase in boundary length with each iteration of the construction process. Although the area of the figure remains bounded, the repeated application of the replacement rule progressively adds more length, ultimately leading to a perimeter that diverges to infinity. This concept exemplifies one of the counterintuitive properties of fractal geometry.

Transcript

00:01 Okay, we're going to use our pythagorean theorem today.

00:05 And for those of you playing along, you've been doing the pythagorean theorem since probably sixth grade.

00:10 That's a great theorem.

00:13 And so they're wanting us to find these letters.

00:16 And these are all, are these a, b, c, d, e, f, g, you know, typical math book.

00:20 And these are all right triangles, so pythagorean theorem applies.

00:24 Let's start out with a here.

00:27 Here, we have 1 squared plus 1 squared equals a squared.

00:33 Because we've got that side and that side of the right triangle, and we want a squared.

00:39 So 1 squared is 1, plus 1 squared is 1.

00:43 That is, 1 plus 1 is 2 equals a squared.

00:49 So then a equals square root of 2.

00:54 All right.

00:55 Alright, and if you've done 45, 45, 90 right triangles, you know that the sides of those triangles are 1, you get 1, 1, root 2.

01:10 Same thing, this is why.

01:14 Okay, so now that was a.

01:18 Now let's look at, here i'm going to put this in red, we're going to look at this triangle here, this one here.

01:23 And we know that this leg is square root of 2 and the other one's 1 so we have 1 squared plus square root of 2 squared equals i'm guessing it's bb square let's check yeah we're going for b now we're going for b okay so let's just crank those out 1 squared is 1 plus 2 square root 2 squared is 2 equals b squared 1 1 plus 2 is 3.

01:57 That equals b squared.

01:59 So b equals square root of 3.

02:02 Notice it's not plus or minus because we're talking about a length of a side.

02:07 So you can't have a negative length.

02:11 That's weird.

02:12 So you only have the positive square root of 3.

02:15 So let's get that cranked in there.

02:17 So here is square root of 3.

02:21 Okay, so now let's work on c.

02:24 We'll do c in blue...