Problem 104 - Algebra and Trigonometry

Question

Koch's Snowflake The area inside the fractal known as the Koch snowflake can be described as the sum of the areas of infinitely many equilateral triangles, as pictured below. For all but the center (largest) triangle, a triangle in the Koch snowflake is \(\frac{1}{9}\) the area of the next largest triangle in the fractal. Suppose the area of the largest triangle has area of 2 square meters. (a) Show that the area of the Koch snowflake is given by the series $$A=2+2 \cdot 3\left(\frac{1}{9}\right)+2 \cdot 12\left(\frac{1}{9}\right)^{2}+2 \cdot 48\left(\frac{1}{9}\right)^{3}+2 \cdot 192\left(\frac{1}{9}\right)^{4}+\cdots$$ (b) Find the exact area of the Koch snowflake by finding the sum of the series.

Step-by-Step Solution

Answer

6.8 square meters.

Step 1: Understanding the Series

The area of the Koch snowflake is given by the series \(A=2+2 \cdot 3\left(\frac{1}{9}\right)+2 \cdot 12\left(\frac{1}{9}\right)^{2}+2 \cdot 48\left(\frac{1}{9}\right)^{3}+2 \cdot 192\left(\frac{1}{9}\right)^{4}+\cdots\). Each term after the initial 2 is a progressively smaller set of triangles. Notice how the term \(2 \cdot 3\left( \frac{1}{9} \right)^n\) can be characterized by a geometric series.

Step 2: Rewriting the Series

To find and sum the series, rewrite it in a more standard form by factoring out the common term 2: \( A = 2 \left[ 1 + 3 \left( \frac{1}{9} \right) + 12 \left( \frac{1}{9} \right)^2 + 48 \left( \frac{1}{9} \right)^3 + 192 \left( \frac{1}{9} \right)^4 + \cdots \right] \).

Step 3: Identifying the Geometric Series

Notice the pattern in the series: the coefficients of each term (3, 12, 48, 192, ...) can be represented as \(3 \cdot 4^{n-1}\) for the nth term. Hence, the series can be written as \( A = 2 \left[ 1 + 3 \sum_{n=1}^{\infty} 4^{n-1} \left( \frac{1}{9} \right)^n \right] \).

Step 4: Simplifying the Sum

Recognize that the sum of \( 4 ^ {n-1} \left( \frac{1}{9} \right)^n \) for \( n = 1 \) to \( \infty \) is a geometric series with the first term \( \frac{4}{9} \) and the common ratio \( \frac{4}{9} \). The sum of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \) can be applied.

Step 5: Calculating the Series Sum

Calculate the series sum with \( a = \frac{4}{9} \) and \( r = \frac{4}{9} \): \[ \sum_{n=1}^{\infty} 4^{n-1} \left( \frac{1}{9} \right)^n = \sum_{n=0}^{\infty} \left( \frac{4}{9} \right)^{n+1} = \frac{\frac{4}{9}}{1 - \frac{4}{9}} = \frac{\frac{4}{9}}{\frac{5}{9}} = \frac{4}{5} \].

Step 6: Final Calculation

Plug the result back into the series: \[ A = 2 \left( 1 + 3 \cdot \frac{4}{5} \right) = 2 \left( 1 + \frac{12}{5} \right) = 2 \left( \frac{5+12}{5} \right) = 2 \cdot \frac{17}{5} = \frac{34}{5} = 6.8 \].

Key Concepts

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 called the ratio. In the Koch snowflake, we see this in the pattern of the triangles. Each term shows the addition of smaller and smaller triangles, reducing in size by a factor of \ \( \frac{1}{9} \ \). This makes it a classic example of a geometric series. The formula to find the sum of an infinite geometric series for \ \( |r| < 1 \ \) is given by:

\ \[ S = \frac{a}{1-r} \ \]
where \ \( a \ \) is the first term and \ \( r \ \) is the common ratio. In our series, \ \( a \ \) is \ \( \frac{4}{9} \ \) and \ \( r \ \) is \ \( \frac{4}{9} \ \). By applying this formula, we can solve the problem.

fractals

Fractals are complex geometric shapes that can be split into parts, each of which is a reduced-scale copy of the whole. The Koch snowflake is an example of a fractal. It is created by starting with an equilateral triangle and recursively adding smaller triangles to its sides. This pattern creates a shape that has an infinitely long boundary but encloses a finite area. This self-similarity property makes fractals fascinating and useful in various fields such as computer graphics and nature modeling. Fractals always contain patterns that recur at progressively smaller scales, which is the hallmark of their intricate and beautiful structure.

infinite series

An infinite series is a sum of an infinite sequence of terms. The summation continues indefinitely. The Koch snowflake is a prime example where the area is represented by an infinite series. Each additional triangle adds an infinite number of progressively smaller areas to the total. The formula for the area of the Koch snowflake was broken down into an infinite series, showing how we keep adding smaller triangular areas forever. Despite the infinite number of terms, the series converges to a finite number; in the Koch snowflake's case, 6.8 square meters. This shows how infinite processes can yield finite results, a key concept in mathematics.

mathematical sequences

A mathematical sequence is an ordered list of numbers following a particular pattern or rule. In the Koch snowflake's series, the sequence involves adding triangles of diminishing sizes. The coefficients of each term (3, 12, 48, 192, ...) follow a specific sequence where each number is four times the previous one, except for the starting point. This sequence helps derive the series' pattern and eventually leads to understanding the overall shape and area. Understanding sequences is crucial for breaking down complex problems into manageable parts and finding comprehensive solutions, such as calculating the Koch snowflake's area accurately.