Cantor's Comb

What is Cantor's Comb?

This activity allows the user to step through the process of building Cantor's comb.

The Cantor Comb is a way of visualizing the famous Cantor Set. Start with a line segment like:

Then remove the middle third:

Next, remove the middle thirds of each remaining line segment. The next stage would be:

Now repeat this process indefinitely. The Cantor comb is the stuff left after removing the middle thirds of the line segments "infinitely many times."

Other Cantor combs can be created by removing different sized middle pieces. For example, we can remove the middle quarter (1/4) or the middle half (1/2) to get different Cantor combs. To distinguish between these we sometimes call the original version the "Cantor middle-thirds comb" using a similar name for the others.

Georg Cantor (1845-1918) was very interested in infinite sets, especially ones with unusual properties. He built what is now called the Cantor set to illustrate an unusual infinite set. This set has only points left, no line segments - so no length. If you stop the process at any stage, there are line segments, very short ones. The idea is that at any stage you have length, but in the limit you don't. This makes the Cantor comb a very curious set.

How Do I Use This Activity?

This activity allows the user to work step by step through the process of building Cantor's comb.

Controls and Output

  • The rectangular area at the top of the applet shows the fractal, which consists of a number of red lines.
  • The Previous Step and Next Step buttons move you forward and backward one step in the drawing process.
  • This label tells you how long each of the red lines that make up the fractal are (as a fraction of the total width) and what step of the fractal iteration you are on.
  • In the boxes marked Fraction Removed, enter the fraction of each line segment that you want removed at each step. Enter the numerator in the first box and the denominator in the second box. 1/3 is the default. Press the Set button to restart with a new value for Fraction Removed.

Description

This activity allows the user to work step by step through the process of building Cantor's comb. This activity would work well in mixed ability groups of two or three for about thirty to forty-five minutes if you use the exploration questions and ten minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • practice students' fraction manipulation skills
  • practice students' pattern finding skills
  • demonstrate the complexity of infinity
  • motivate the ideas of self-similarity and recursion
  • demonstrate fractal objects
  • motivate the idea of a limit

Standards Addressed

Fifth Grade

  • Operations and Algebraic Thinking

    • Analyze patterns and relationships.

Grades 3-5

  • Algebra

    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grade 5

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with non-negative rational numbers.

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.
  • Geometry and Measurement

    • COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts to solve problems.

Technical Mathematics I

  • Number and Operations

    • Competency Goal 1: The learner will apply various strategies to solve problems.

5th Grade

  • Algebra

    • The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

4th Grade

  • Algebra

    • The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

Textbooks Aligned

Grade Six

  • Bits and Pieces I

    • Investigation One: Fund-Raising Fractions
    • Investigation Two: Comparing Fractions
    • Investigation Four: From Fractions to Decimals
    • Investigation Five: Moving Between Fractions and Decimals

Grade Seven

  • Accentuate the Negative

    • Investigation Two: Adding Integers
    • Investigation Three: Subtracting Integers

Book 1

  • Module 2 - Patterns and Designs

    • Section 2: Fractions
    • Section 2: Mixed Numbers
    • Section 3: Equivalent Fractions
  • Module 3 - Statistical Safari

    • Section 2: Fractions, Percents
  • Module 5 - Creating Things

    • Section 1: Fraction Number Sense
    • Section 1: Common Denominators
    • Section 3: Adding Fractions, Subtracting Fractions, Adding Mixed Numbers, Subtracting Mixed Numbers

Book 2

  • Module 2 - Search and Rescue

    • Section 2: Comparing Integers
    • Section 3: Adding Integers
    • Section 3: Subtracting Integers
  • Module 4 - The Art of Motion

    • Section 1: Multiplying Fractions, Dividing Fractions
  • Module 5 - Recreation

    • Section 4: Fractions to Decimals to Percents

Book 3

  • Module 2 - At the Mall

    • Section 5: Working with Negative Fractions
  • Module 4 - Patterns and Discoveries

    • Section 2: Rational and Irrational Numbers
    • Section 3: Equations with Fractions

6th

  • Module 3 - Mind Games

    • Section 5: Fraction and Mixed Number Multiplication

Book 1

  • From Zero to One and Beyond

    • Lesson 1: Folding Fractions
    • Lesson 4: Out of One Hundred
    • Lesson 5: Percents That Make Sense
    • Lesson 9: All Three at Once
  • Number Powerhouse

    • Lesson 6: Multiplication Made Easy
    • Lesson 7: The Great Fraction Divide

Book 2

  • Buyer Beware

    • Lesson 6: Which Brand Has the Most Chocolate?
  • Making Mathematical Arguments

    • Lesson 6: Counterexamples and Special Cases

Grade 8

  • Reflections on Number

    • Divisibility and Prime Factorization
    • Multiplication and Division
    • Operations with Inverses

Grade 5

  • Measure for Measure

    • Equivalent Decimals
    • Fraction/Decimal Equivalence
    • Adding and Subtracting Decimals
  • Per Sense

    • Using Percents
    • Fraction/Percent/Ratio Equivalence
    • Estimating Percents
  • Some of the Parts

    • Fractions
    • Relationships between Fractions
    • Operations with Fractions

Grade 6

  • Fraction Times

    • Operations with Fractions
    • Fraction/Percent/Decimal/Ration Relationships
  • More or Less

    • Fraction/Decimal/Percent Relationships
    • Operations with Percents
    • Decimal Multiplication
  • Ratios and Rates

    • Ratio/Fraction/Decimal/Percent Relationships
    • Part-Part Ratios
    • Part-Whole Ratios
    • Scale Factor
    • Linear Functions

Grade 7

  • Cereal Numbers

    • Volume
    • Surface Area
    • Relationship between Volume and Surface Area
    • Comparisons with Ratios
    • Fractions
    • Decimals and Percents
    • Multiplying and Dividing Fractions

Be Prepared to

  • work through the first table on the exploration questions with the students
  • discuss the concepts of infinity and limit
  • review fraction multiplication