Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

Sierpinski sieve The Sierpinski sieve, designed in 1915 , is an example of a fractal. It can be constructed by starting with a solid black equilateral triangle. This triangle is divided into four congruent equilateral triangles, and the middle triangle is removed. On the next step, each of the three remaining equilateral triangles is divided into four congruent equilateral triangles, and the middle triangle in each of these triangles is removed, as shown in the first figure. On the third step, nine triangles are removed. If the process is continued indefinitely, the Sierpinski sieve results (see the second figure). (a) Find a geometric sequence \(a_{k}\) that gives the number of triangles removed on the \(k\) th step. (b) Calculate the number of triangles removed on the fifteenth step. (c) Suppose the initial triangle has an area of 1 unit. Find a geometric sequence \(b_{k}\) that gives the area removed on the \(k\) th step. (d) Determine the area removed on the seventh step.

Step-by-Step Solution

Answer

(a) \(a_k = 3^{k-1}\). (b) 4782969 triangles. (c) \(b_k = \left(\frac{3}{4}\right)^{k-1}\times\frac{1}{3}\). (d) Area removed on step 7 is \(\frac{243}{4096}\).

Step 1: Understand the Process

The Sierpinski sieve is created by removing triangles from a larger triangle. In each step, the remaining triangles are divided, and the middle triangle of each group is removed. This removal process follows a pattern, forming a geometric sequence.

Step 2: Define the Sequence for Part (a)

At each step, more triangles are removed as the pattern grows. The first step removes 1 triangle, the second step removes 3 triangles, the third removes 9, and so forth. This forms a sequence where the number of triangles removed at step \(k\), \(a_k\), is given by \(a_k = 3^{k-1}\), because each step triples the number of triangles removed from the previous step.

Step 3: Calculate Number of Removed Triangles - Step 15

Using the sequence from Step 2, calculate the number of triangles removed on the 15th step. Substitute \(k=15\) into the formula: \(a_{15} = 3^{15-1} = 3^{14}\). Calculate \(3^{14} = 4782969\).

Step 4: Define the Sequence for Part (c)

The area removed at each step also forms a geometric sequence. On step 1, the area removed is \(\frac{1}{4}\), and each successive step removes an area \(\frac{3}{4} \) of the previous step. This means the area removed at each step is \(b_k = \left(\frac{3}{4}\right)^{k-1}\times\frac{1}{3}\).

Step 5: Calculate Area Removed - Step 7

Using the formula for \(b_k\), calculate the area removed on the 7th step. Substitute \(k=7\) into the formula: \(b_7 =\left(\frac{3}{4}\right)^{6}\times\frac{1}{3}\). Calculate it: \(b_7 = \left(\frac{3}{4}\right)^6 \times \frac{1}{3} = \frac{729}{4096} \times \frac{1}{3} = \frac{243}{4096}\).

Key Concepts

Fractal Geometry

Fractal geometry is an exciting field that explores shapes that self-repeat at different scales. The Sierpinski sieve, also known as the Sierpinski triangle, is a classic example of a fractal. This intriguing shape begins with a solid equilateral triangle. Through a series of steps, smaller triangles are removed from it.
The removal of triangles follows a recursive pattern, creating smaller and smaller triangles that replicate the overall design.
Fractals like the Sierpinski sieve are not only beautiful; they are used in various fields such as computer graphics, nature modeling, and even art.

• Self-similar: Fractals are known for their repeating patterns at different sizes.
• Infinite detail: As you look closer, more complexity emerges.
• Non-linear scaling: Fractals don't scale in a regular geometric way, making them fascinating to study.

Geometric Sequences

Geometric sequences play a key role in understanding the Sierpinski sieve. In simple terms, a geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio."
In the context of the Sierpinski sieve, we see two main geometric sequences. One relates to the number of triangles removed at each step, and the other to the area removed.
For the number of triangles removed, the sequence grows by a factor of three each time. It starts with 1, then 3, 9, and so on. The general term for the sequence is given by the formula: \( a_k = 3^{k-1} \). For calculating steps in this sequence, knowing the common ratio helps predict the number of triangles removed at any given step.

• Consistent ratio: Each term is generated by multiplying the previous term by the common ratio.
• Predictable growth: This allows easy calculation of later terms, as seen with the Sierpinski sieve.

Area Calculation

In the process of creating a Sierpinski sieve, not only are triangles removed, but their areas are also systematically decreased in a predictable manner. Initially, the triangle has a total area of 1 unit.
With each step in the fractal creation, a fraction of each sub-triangle's area is removed. This remains connected to geometric sequences, where each step sees an area reduced by a factor of \(\frac{3}{4}\) of the previous step's total removed area. The area removed at each step can be represented as:\[ b_k = \left(\frac{3}{4}\right)^{k-1} \times \frac{1}{3} \] where \(b_k\) is the area removed at step \(k\). This forms a predictable pattern in how much area gets subtracted as the sieve is constructed.

• Progressive reduction: The removal of area follows a systematic formula.
• Fractional change: Each step sees the area decrease in a recurring fractional pattern.

Triangle Removal Process

The Sierpinski sieve's striking appearance comes from its recursive triangle removal process. Starting with a single solid triangle, each step involves dividing remaining triangles into four smaller congruent equilateral triangles and then removing the central triangle of each set.
This gets repeated for every smaller triangle, increasing the complexity while revealing the same fundamental pattern at each level. The removal process is an essential feature of the fractal, responsible for its characteristic look.
At each progressive step, the removal follows a consistent geometric sequence. The process begins by removing one triangle, followed by three, then nine, and so on, continuing until the pattern is infinitely replicated.

• Recursive division: Each iteration sees triangles divided into more triangles.
• Central cutoff: The key removal strategy involves taking away the central triangle in each set.
• Growing complexity: With each step, the sieve gains more intricate layers.