Fractals as a GP Sequence - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

Transcript

Fractals as a GP Sequence A fractal is a never-ending pattern. In regular geometry, if you zoom in close enough to the edge of a circle, the curve eventually flattens out and just looks like a straight line. But if you zoom in on a fractal, the pattern never smooths out. Instead, you will see the exact same complex shapes repeating over and over again, shrinking down infinitely. WHAT IS A FRACTAL? A never-ending pattern where the parts look like the whole.Let’s look at an example of a fractal – Sierpiński Triangle Sierpiński Triangle We have two patterns here Pattern 1: The Number of Triangles Increases Rapidly Pattern 2: The Area of the Triangles Decreases Let’s look at them in detail Pattern 1: The Number of Triangles Increases Rapidly At each stage, every solid black triangle is broken into 4 smaller triangles, and the middle one is completely removed. This means 1 triangle becomes 3 smaller triangles. Stage 0: 1 triangle Stage 1: 1 × 3=𝟑 triangles Stage 2: 3 × 3=𝟗 triangles Stage 3: 9 × 3=𝟐𝟕 triangles So, our number of triangles form a GP 1, 3, 9, 27, …. With First term = a = 1 Common ratio = r = 3 But, the first term is Stage 0 So, we can write Number of triangles in Stage n = (n + 1)th term of GP = an + 1 = 𝑎 × 𝑟^((𝑛 + 1)−1) = 𝑎 × 𝑟^𝑛 Putting values = 1 × 3^𝑛 = 𝟑^𝒏 Pattern 2: The Area of the Triangles Decreases Now, let's look at the area (the total amount of black space). Let's assume the very first triangle at Stage 0 has a total area of 1 square unit When we move to Stage 1, we divide that area into 4 equal pieces, and we throw 1 piece away This means we are only keeping 3 out of the 4 pieces Mathematically, we are multiplying the area by 𝟑/𝟒 Thus, we can write Stage 0: Area = 1 Stage 1: 1×3/4=3/4 Stage 2: 3/4×3/4=(3/4)^2=9/16 Stage 3: 9/16×3/4=(3/4)^3=27/64 The Math (A shrinking GP) This is also a Geometric Progression, but our common ratio is a fraction: 𝒓=𝟑/𝟒. Because we are multiplying by a number smaller than 1 , the total area keeps shrinking. Now, At Stage 4, the area is (3/4)^4=𝟖𝟏/𝟐𝟓𝟔 At Stage 5, the area is (3/4)^5=𝟐𝟒𝟑/𝟏𝟎𝟐𝟒 The 𝒏^𝒕𝒉 Stage Rule Just like before, the exponent matches the stage number. The rule for the area at stage 𝑛 is 𝒔_𝒏=(𝟑/𝟒)^𝒏. Thus, we can write Stage 0: Area = 1 Stage 1: 1×3/4=3/4 Stage 2: 3/4×3/4=(3/4)^2=9/16 Stage 3: 9/16×3/4=(3/4)^3=27/64 The Math (A shrinking GP) This is also a Geometric Progression, but our common ratio is a fraction: 𝒓=𝟑/𝟒. Because we are multiplying by a number smaller than 1 , the total area keeps shrinking. Now, At Stage 4, the area is (3/4)^4=𝟖𝟏/𝟐𝟓𝟔 At Stage 5, the area is (3/4)^5=𝟐𝟒𝟑/𝟏𝟎𝟐𝟒 The 𝒏^𝒕𝒉 Stage Rule Just like before, the exponent matches the stage number. The rule for the area at stage 𝑛 is 𝒔_𝒏=(𝟑/𝟒)^𝒏.