Question

The Sierpinski carpet is a fractal created using squares. The process involves removing smaller squares from larger squares. First, divide a large square into nine congruent squares. Then remove the center square. Repeat these steps for each smaller square, as shown below. Assume that each side of the initial square is 1 unit long. a. Let \(a_n\) be the total number of squares removed at the \(n\)th stage. Write a rule for \(a_n\). Then find the total number of squares removed through Stage 8 . b. Let \(b_n\) be the remaining area of the original square after the \(n\)th stage. Write a rule for \(b_n\). Then find the remaining area of the original square after Stage 12 .

Short answer

The total number of squares removed through Stage 8 is 16777216. The remaining area of the original square after Stage 12 is approximately 0.07937 units square.

Step-by-step solution

Determine Rule for \(a_n\)

In the Sierpinski carpet, at each stage \(n\), we are essentially dividing our squares into nine congruent squares and removing one, which means we are removing \(8^n\) squares in each stage. So the rule for \(a_n\) is \(8^n\).

Find \(a_8\) Using the Rule

We can substitute the value of '8' into the rule we've determined to find the total number of squares removed through Stage 8. That gives us \(a_8 = 8^8\). The total number of squares removed at Stage 8 is \(16777216\).

Determine Rule for \(b_n\)

The original square has an area of 1 unit square. In the first stage, we remove 1/9 of the total area. In each subsequent stage, we are removing from each of the remaining squares, which are 1/9 of the area of the previous squares. This gives us the rule for \(b_n\) as \(b_n = 1 * (8/9)^n\).

Find \(b_{12}\) Using the Rule

We can substitute the value of 12 into the rule we've derived to find the remaining area of the original square after Stage 12. This gives us \(b_{12} = 1 * (8/9)^{12}\). The remaining area of the original square after Stage 12 is approximately 0.07937 units square.

Key concepts

Fractal Geometry

Imagine you are walking through a never-ending pattern that repeats itself at different scales, endlessly complex and incredibly detailed. This is the essence of fractal geometry—a branch of mathematics that describes irregular and fragmented shapes occurring naturally in our universe.

Take, for example, the Sierpinski carpet. It is a classic illustration of fractal geometry and is formed through a deceptively simple process of iteration. Starting with a square, we repeatedly remove smaller squares, creating a repeating pattern that never looks the same as you zoom closer and closer into it. This pattern's beauty lies in its self-similarity; each part resembles the whole, no matter how deeply you delve into the carpet's intricacies.

Exponential Functions

In the realm of mathematics, exponential functions represent growth or decay at an increasingly rapid rate. They are vital for understanding phenomena where changes occur multiplicatively rather than additively.

In the steps involved in forming the Sierpinski carpet, we see exponential functions come to life. With each stage, the number of squares removed is represented as an exponential function where the base number is 8, reflected in the rule for the number of squares, \(a_n = 8^n\). This means that with each additional stage, the count of squares removed multiplies exponentially, illustrating an explosive growth that underlines the very concept of exponential increase.

Sequence and Series

Sequences and series are fundamental in understanding patterns and summations in mathematics. A sequence is an ordered list of numbers following a particular rule, while a series represents the sum of the elements of a sequence.

In the formation of the Sierpinski carpet, we encounter a geometric sequence when we consider the remaining area of the carpet after each stage. This is given by a rule, \(b_n = (8/9)^n\), with each term representing the carpet's area after successive stages, demonstrating how sequences can define iterative processes in fractal geometry. As we observe the series formed by these areas, we understand the convergence of the leftover area, which intriguingly becomes smaller and smaller, almost whispering the story of the infinite within the finite confines of the initial square.