Problem 95 - Thomas Calculus

Question

The Cantor set To construct this set, we begin with the closed interval \([0,1]\) . From that interval, remove the middle open interval \((1 / 3,2 / 3),\) leaving the two closed intervals \([0,1 / 3]\) and \([2 / 3,1] .\) At the second step we remove the open middle third interval from each of those remaining. From \([0,1 / 3]\) we remove the open interval \((1 / 9,2 / 9),\) and from \([2 / 3,1]\) we remove \((7 / 9,8 / 9),\) leaving behind the four closed intervals \([0,1 / 9]\) \([2 / 9,1 / 3],[2 / 3,7 / 9],\) and \([8 / 9,1] .\) At the next step, we remove the middle open third interval from each closed interval left behind, so \((1 / 27,2 / 27)\) is removed from \([0,1 / 9],\) leaving the closed intervals \([0,1 / 27]\) and \([2 / 27,1 / 9] ;(7 / 27,8 / 27)\) is removed from \([2 / 9,1 / 3],\) leaving behind \([2 / 9,7 / 27]\) and \([8 / 27,1 / 3],\) and so forth. We continue this process repeatedly without stopping, at each step removing the open third interval from every closed interval remaining behind from the preceding step. The numbers remaining in the interval \([0,1],\) after all open middle third intervals have been removed, are the points in the Cantor set (named after Georg Cantor, \(1845-1918\) . The set has some interesting properties. a. The Cantor set contains infinitely many numbers in \([0,1]\) . List 12 numbers that belong to the Cantor set. b. Show, by summing an appropriate geometric series, that the total length of all the open middle third intervals that have been removed from \([0,1]\) is equal to \(1 .\)

Step-by-Step Solution

Answer

a. 0, 1/9, 2/9, 1/3, 2/3, 7/9, 8/9, 1, 1/27, 2/27, 7/27, 8/27. b. Sum of removed lengths = 1.

Step 1: Introduction to Cantor Set

The Cantor set is constructed by repeatedly removing the middle third of each interval. We start with the interval \([0, 1]\) and continue this process indefinitely.

Step 2: Identify the pattern of removal

In the first step, remove \((1/3, 2/3)\) from \([0, 1]\), leaving \([0, 1/3] \) and \([2/3, 1]\). In the second step, remove \((1/9, 2/9)\) from \([0, 1/3]\) and \((7/9, 8/9)\) from \([2/3, 1]\). Follow this pattern: remove the middle third from each current interval, leading to subsequent subdivisions.

Step 3: Determine 12 numbers in the Cantor set

The Cantor set includes endpoints of the removed intervals like 0, 1/9, 2/9, 1/3, 2/3, 7/9, 8/9, and 1. Additionally, numbers like 1/27, 2/27, and 7/27 from deeper steps (not endpoints of any removed interval) belong to the set. Thus some elements are 0, 1/9, 2/9, 1/3, 2/3, 7/9, 8/9, 1, 1/27, 2/27, 7/27, and 8/27.

Step 4: Calculate the length of removed intervals

The length of the intervals removed follows a geometric series. Initially, the length removed is 1/3. In each subsequent step, the length of each removed interval is one-third of the preceding removed interval, which forms a geometric series.

Step 5: Sum the geometric series

To find the sum, consider the geometric series where \( a = \frac{1}{3} \) and the common ratio is \(r = \frac{2}{3} \). The sum of an infinite geometric series is given by: \ \[ S = \frac{a}{1 - r} = \frac{1/3}{1 - 2/3} = 1 \]. Therefore, the total length of all removed intervals sums up to 1.

Key Concepts

Geometric Series

A geometric series is a sequence 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. It's a fascinating concept because it helps us understand patterns and behavior in many mathematical contexts.

In the context of the Cantor set, we explore a geometric series to determine how much of the total length of the interval [0,1] is removed when forming the Cantor set. Initially, we remove \(\frac{1}{3}\) of the original interval in the first step.

• The first term of our series is \(a = \frac{1}{3}\).
• The common ratio \(r\) is \(\frac{2}{3}\) because each subsequent removal corresponds to two thirds of the previous middle third.
• The formula for an infinite geometric series is \(S = \frac{a}{1 - r}\).
• Plugging in the values, we find \(S = \frac{1/3}{1-2/3} = 1\).

This calculation shows that the total length of all removed intervals adds up to 1, perfectly balancing the entire length of our initial interval.

Infinite Series

An infinite series is essentially a sum of an infinite sequence of numbers. Unlike finite series, they go on endlessly, presenting unique challenges and fascinating results.

In the Cantor set, we look at an infinite series to represent the removal of parts of an interval. This process, carried out infinitely, results in a set with unique features:
- Each step involves removing an open middle third from previous intervals, increasing the count of intervals but reducing their size - As a result, the total length of all these small segments removed is 1, although each individual piece becomes infinitesimally small
- Usefulness and beauty lie in this surprising characteristic: infinitely removing portions still leads to a finite total length This exercise showcases how infinite processes can sometimes produce quite finite and meaningful results. It tells us that some infinite series converge to a real number, in this case 1, which may initially seem counterintuitive.

Fractal Geometry

Fractal geometry explores figures that display self-similarity across different scales, and the Cantor set is a classic example of this fascinating concept. As the process of constructing the set continues indefinitely by removing the middle third of each interval, what remains keeps forming a pattern repeated at every scale.

- **Self-similarity:** Each step in the Cantor set construction is a smaller yet identical version of the whole. Remove the middle third at each stage endlessly and what remains is still a form of the Cantor set. - **Infinite Detail and Complexity:** Though it is simple to see how to build the set, when complete, it shows complex structure because the pattern never actually ends. - **Measure Zero:** Surprise lies in the fact that despite containing infinitely many points, the total "volume" or "measure" of the Cantor set is zero due to unlimited removals. These features exemplify the concept of fractal geometry where infinitely repeating detail and self-similar patterns blend into astonishing simplicity and complexity. This blend makes the Cantor set more than just a mathematical curiosity; it's a gateway to understanding how simple processes can yield complex structures.