Snowflake island fractal The fractal called the snowflake...

Key Concepts

Geometric Series

The analysis of fractals often involves summing contributions from each iteration, which frequently leads to the appearance of geometric series. These series help in calculating measures such as perimeter and area, especially when the additions at each stage form a predictable pattern. Understanding the convergence or divergence of these series is crucial for determining whether properties like total area remain finite or total perimeter diverges to infinity.

Convergence and Divergence of Sequences

When studying iterative constructions, one must analyze the limits of sequences that represent physical attributes, such as perimeter or area. In fractal constructions, it is common to encounter situations where one measure (for example, the added perimeter at each stage) grows without bound (diverges), while another measure (such as the total area) converges to a finite value. This contrast forms an important part of understanding the metric properties of fractals.

Fractal Geometry

Fractal geometry studies structures that exhibit self-similarity on multiple scales, meaning their construction repeats a pattern over and over at different levels of magnification. This branch of mathematics often explores objects that have counterintuitive properties, such as having an infinitely long perimeter while enclosing a finite area, which challenges conventional ideas of dimension and measurement.

Iterative Processes and Self-Similarity

The construction of many fractals, including variants of the Koch snowflake, is based on iterative processes in which a basic pattern is repeatedly replaced or modified at every stage. This repeated process creates self-similar structures, where each magnified section resembles the whole, and is essential to understanding how complex, irregular forms can arise from simple rules.

Transcript

00:01 Hi, so today in this question we're looking at something that could be called snowflake island.

00:08 So what it is is a equilateral triangle that has, of course, three sides of the triangle, and each side is then split into a third, and the middle third is then created into another equilateral triangle.

00:29 And then one third of that size of each length of that size is again created into another equilateral triangle.

00:43 As i showed over here, i did a rudimentary drawing of it, just really showing one side, and i wrote down what each length of each side of these types of triangles would be.

00:59 So for the large equilateral triangle, we have that the side length is equal to one.

01:04 For the second size, we have that the side length is equal to a third.

01:09 And for the next one, the side length is equal to 1 over 3 to the n plus 1.

01:15 So first, we're going to be dealing with the parameter of this island.

01:24 So let's say that we have i to the, not i to the, sorry, i of n plus one.

01:38 Specifically, we can look at the large equilateral triangle in this case.

01:43 And that is, of course, obtained by dividing, sorry, not n plus one, the large one, the i one.

01:54 Look at i1 for this case, the blue one.

01:58 So the i n plus 1 can of course be obtained by dividing the larger equilateral triangle.

02:07 In this case we could say i0 into three each side into three equal parts and then removing that middle part.

02:16 Then we added two parts to that.

02:21 So from that we now have, we went from, i will say, in this case, let's say we went from i0 to i1, and we originally had three sides for our triangles, that was three equal parts, and we went from that to having one, two, three three three.

02:52 Or on each side that is equal parts.

02:57 So each side that we had for i0 we had three equal parts and now we have each side has four equal parts and then if we were to go from our i1 to our i2, we could see that from our i1 we again we had our our equilateral um triangle and we went from in this case having um our four equal parts to now having one two three four five six seven eight nine ten eleven 12, 13, 14, 15, 16 equal parts.

04:02 So now what we had was we had our, another four times our four.

04:17 We had an increase and if we were to continue, again, we'd keep having a multiplier of four.

04:25 So what's happening is we have three times four, and then we have three times four squared.

04:35 And if we'd have gone from our i -n to our i -n plus one, we would end up having three times four to the n, because it's always a multiple of four that we're getting an increase in the number of new sides that are created.

04:58 So that would tell us then because each of these we have, for this one we would have a length of n and we would have a length of n of l n plus 1.

05:19 And we know that for our length of l n plus 1, that is equal to 3 times 4 to the n.

05:34 Of our ln length.

05:42 So because the, this is a geometric series right here with a ratio greater to one, our nth term, as this our limit is going towards infinity.

06:01 So if we were to put this and just look at the limit of this part, we can see that this is always going to be going towards infinity.

06:15 This is always increasing.

06:17 So it increases without bounds.

06:19 So the perimeter is always increasing to infinity.

06:25 Now looking at this again and trying to figure out area with this island.

06:35 What we know is as we've gone over here is that our, so if we had our l, an l, sorry, i'm going to just put a b here, so we know we're on the second part.

06:48 Our i -n would have a length for a side of three times four to the n, let's say, with a side length.

07:09 We know that our side lengths are increased for each one.

07:16 So it's whatever the number, if we look here, so this is one third, and that's for i to the one and we know we have one over three to the n plus one for i to the two so that would be in that case that could be sorry that's not for the two actually this is for um every consecutive one so we actually should know that for i2 our n in this case would be one so it would be one ninth would be the size so we know we're constantly increasing by that amount so what we can say is then for our in, it would be 1 over 3 to the n is going to be equal to our side length.

08:13 And this is equal to the number of sides.

08:16 This is the number of sides for in.

08:24 And now from here, we're going to try and find our area.

08:29 So if we know that for this is our side length for i .n, we're going to.

08:37 Going to know that for i n plus one our side length is going to be one over three to the n plus one because it's just going to keep course increasing um and the area of the sides again we can write it in the same manner that we wrote the other one as three four and this case it would actually probably be best to say um it just has an n plus one type of thing, but we can just say n in that case.

09:17 So now let's look at our equilateral triangle.

09:20 So if we have an equilateral triangle, how we can compute our area, of course, of any triangle, we know it's one half base times height.

09:32 This would be our height.

09:33 And in an equilateral triangle, there is a formula.

09:38 So we would have s being our side length here, over 2 is equal to this portion, and then our height is equal to s times the square root of 3 over 2.

09:59 And so we would know that an equilateral triangle, and let's say the side is equal to x in this case, then the area would end up being equal to s, no, sorry, x times 3 squared over 2 times x over 2.

10:34 Because this is, of course, already halved.

10:37 This is already halve the base, so we don't have to worry about that.

10:40 So in this case, we're going to just have that x squared times the square root of 3 over 4.

10:50 Let me just say with side equal to x...

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