Question

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller equilateral triangles. Leave the middle small triangle blank, but for each of the other 3 small triangles, draw lines connecting the midpoints of the sides to create 4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to divide the others into 4 parts. Find the infinite series for the total area left blank if this process is continued indefinitely. (Suggestion: Let the area of the original triangle be 1 ; then the area of the first blank triangle is \(1 / 4 .\) ) Sum the series to find the total area left blank. Is the answer what you expect? Hint: What is the "area" of a straight line? (Comment: You have constructed a fractal called the Sierpi?ski gasket. A fractal has the property that a magnified view of a small part of it looks very much like the original.)

Short answer

The total area left blank is 1.

Step-by-step solution

Identify the initial blank area

Given that the area of the original equilateral triangle is 1, the area of the first blank triangle, which is formed by connecting the midpoints of the sides of the original triangle, is \(\frac{1}{4}\).

Determine the total area of subsequent blank triangles

Notice that each time the process is repeated, the side length of the new triangles is half of the previous triangle's side length. The area of each new blank triangle is \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\) of the area of the larger triangle it is within. Moreover, there are 3 triangles left blank each iteration.

Formulate the infinite series

Sum the areas of the blank triangles to form the infinite series. The total blank area after infinite iterations is given by the sum: \[ \frac{1}{4} + 3 \times \frac{1}{4^2} + 9 \times \frac{1}{4^3} + \text{...} \]

Recognize the series as geometric

The series can be identified as a geometric series with the first term \(a = \frac{1}{4}\) and common ratio \(r = \frac{3}{4} \): \[ S = \frac{a}{1 - r} = \frac{\frac{1}{4}}{1 - \frac{3}{4}} = 1 \]

Interpret the sum

The sum of the infinite series is 1, implying that as the process continues indefinitely, the total area left blank approaches the total area of the original triangle.

Key concepts

fractal geometry

In the exercise, you encountered a structure known as the Sierpinski gasket. This is a famous example of fractal geometry. Fractals are shapes that can be split into parts, each of which is a reduced-scale copy of the whole. They have a property called self-similarity, meaning that a magnified view of a small part of the fractal looks very much like the original.

The Sierpinski gasket is constructed by repeatedly removing smaller and smaller triangles from an equilateral triangle. Despite the fractal's seemingly complex structure, it follows simple iterative rules. This makes it an excellent example of how complex shapes can emerge from simple recursive processes.

Because fractals like the Sierpinski gasket exhibit self-similarity, they are not easily described by traditional Euclidean geometry. Instead, special mathematical methods are used to explore their properties, such as their dimension, which can be fractional (hence the term 'fractal').

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 fixed, non-zero number called the common ratio. The series in the exercise arises from repeatedly removing smaller triangles, making it a perfect fit for this kind of mathematical sequence.

For the Sierpinski gasket, the original triangle's area is defined as 1. The first blank triangle has an area of \(\frac{1}{4}\). Each subsequent blank triangle is smaller by a factor related to the process of removing triangles and the halving of sides. The total blank area can be expressed as an infinite series:

\[ \frac{1}{4} + 3 \times \frac{1}{4^2} + 9 \times \frac{1}{4^3} + ... \]

This is a geometric series because each term is a fixed fraction of the previous one. The sequence continues indefinitely, making it an infinite series.

infinite series

An infinite series is a sum of infinitely many terms. In the Sierpinski gasket exercise, we encountered an infinite series representing the total blank area of the triangles.

The series starts with \(\frac{1}{4}\) and each term is smaller than the previous one, following the pattern given by the geometry of the gasket. This pattern can continue indefinitely, adding terms that grow progressively smaller.

One of the important aspects of infinite series is determining their sum. For an infinite geometric series with first term \(\frac{1}{4}\) and common ratio \(\frac{3}{4}\), the sum can be calculated using the formula:

\[ S = \frac{a}{1 - r} \]

Substituting the values, we get:

\[ S = \frac{\frac{1}{4}}{1 - \frac{3}{4}} = 1 \]

This demonstrates an intriguing property: although we are adding an infinite number of decreasing areas, their total sum equals the area of the original triangle, which is 1. This is a key concept when dealing with fractals and infinite series in mathematics.

mathematical methods

Mathematical methods provide the tools and techniques necessary to understand complex problems like the Sierpinski gasket. In this exercise, we applied several key methods to find the solution.

Initially, we identified the iterative process and calculated the areas of the blank triangles using basic geometry. Then, we recognized the emerging pattern as a geometric series. Finally, we used the formula for summing an infinite geometric series to find the total blank area left by the fractal process.

These steps illustrate the broader application of mathematical methods:

• Recognizing patterns in iterative processes
• Employing geometric measurements
• Formulating and simplifying series expressions
• Calculating sums of infinite series

Mastering these mathematical methods allows students to tackle a wide range of problems beyond the scope of this particular exercise. They form a foundational part of higher mathematics, enabling the study of complex and beautiful structures like fractals.