SNOWFLAKE FRACTALS

Those who contemplate the beauty of the earth find reserves of strength that will endure as long as life lasts. –Rachel Carson

 

Once is an instance. Twice may be an accident. But three or more times makes a pattern. –Diane Ackerman

 

Fractals represent a new geometry that mirrors the universe. – Benoit Mandelbrot

 

 

Grade level: upper middle school

 

Topic: Geometric sense, FRACTALS

 

Purpose: The simple forms of traditional Euclidean geometry don’t match the shapes of the natural landscape and elements in nature. Benoit B. Mandelbrot, the creator of fractal geometry, wrote: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight lines.” Mandelbrot developed the geometry that could analyze and quantify some unique patterns in nature. “Fractals represent a new geometry that mirrors the universe.” Not every irregular shape however, is a fractal. To fit into this category, a shape must have what Mandelbrot called self-similarity—the details must look much like the larger picture. (By Nature’s Design)

 

Snowflakes demonstrate this fractal pattern. Through the Koch snowflake activity students will step through the generation of a fractal made from deforming a line by bending it. The Koch Curve was studied by Helge von Koch in 1904. When considered in its snowflake form the curve is infinitely long but surrounds finite area. This lesson is designed to get students to think about several of the concepts from fractals, including recursion and self-similarity. The mathematical concepts of line segments and infinity are used, and skill at pattern recognition is practiced. Later lessons focus on mathematical concepts of perimeter, area, and ratios, particularly Fibonacci and golden ratio.

Objectives: Students will be able to:

·        Recognize patterns

·        Construct repeating patterns

·        Predict repeating pattern consequences

·        Define fractals

·        Work successfully in groups to produce a Koch Snowflake

 

 

Materials:

·        triangle graph paper

·        pencils

·        erasers

·        colored pencils

·        scissors

·        construction paper

·        straight edge (rulers) are optional

 

Vocabulary:

  • fractal
  • self similarity
  • iterate
  • Koch snowflake
  • recursive relationship

 

Hook:

Have you ever noticed patterns in snow?

 

Instruction:

We are going to work with some unique patterns today. We will use some materials and models to help us identify and think through some specific pattern qualities. What do you know about patterns? What is a pattern? (brain drain with students what they know about patterns listing words on the board. You may come back to some of those words they initially identified to solidify your summary discussion of fractals) What patterns do you see in snow?

 

“You will need to take out a pencil and clear your desk space”

Distribute triangle graph paper, and supplies to each student.

“Quietly look for patterns. Do you see any patterns here?”

There are many possibilities. We are going to create one of those options right now and then practice repeating this pattern.

 

Draw a line segment on your paper that is 18 units long. One unit equals the length of the base of the smallest already scribed triangle. You may have to draw your segment on a diagonal.

Divide the segment into thirds. How many units in each third? (6).

Now, imagine bending that middle third into an equilateral triangle. What will it look like?

Erase the middle third of your line segment. And draw that figure with two lines the same length as the section you removed.

Notice the figure is more complicated than before.

How many line segments are there now?

What if we repeat what we did to the first segment to each of the four segments?

What was the process we performed?

  1. divide the segment into thirds.
  2. erase the middle third
  3. imagine bending that middle third into an equilateral triangle
  4. Replace the middle third with two lines the same length as the section you removed.

Go ahead and perform that function./draw that pattern.

What do you notice?

The figure is now more complicated.

What if we did it again?

Go ahead and do it this third time.

What if we did it again? J What would you predict would happen?

Would there be a limit to the number of times we could perform this function on a line segment? Could we do it infinitely many times?

Okay, can someone summarize or reiterate what we’ve done to this figure?

What we’ve done is an important quality of patterns. Iterate. How many iterations did we perform? Think of this also as a recursive relationship. What might that mean?

What if we started with an equilateral triangle and performed the same iterations to each segment there? What might the figure become? What might it look like?

 

Find out! Here’s what I want you to do.

Darken the outer lines of your figure. (show example)

Each person will carefully cut their figure away from their graph paper. (show example)

Find two other people in the class to team up with.

Arrange your three figures so that the initial bases will form that original triangle.

Glue your new shape to a piece of construction paper.

(And color in the center area.) *save for later instruction and lesson on finite area.

 

Now as a group look at the figure you’ve created. Talk about what do you notice? What similarities do you observe?

If the red image is the original figure, how many similar copies of it are contained in the blue figure?

 

This is the pattern of self similarity.

Notice also that another feature that results from the iterative process it that of self-similarity, i.e., if we magnify or "zoom in on" part of the figure, we see copies of itself.

What does your figure look like?

A snowflake?

We’ve constructed what is known as a Koch snowflake. The Koch Curve is the most recognized "line bender" fractal. Helge von Koch in studied the iterate process and self similarity in this shape in 1904. Self similarity and recursive or iterate process are the backbones of fractals. 

What is a fractal?

Geometric figures, just like lines, circles, rectangles, and squares, but fractals also have two special properties—they are self similar and created through a recursive process.

 

Welcome to the fantastic and fabulously fun world of fractals. Wahoo!

They are not just found in this snowflake or in the room. They are found everywhere and quite often in nature.

(Show examples; ferns…)

Next time, we’ll look at these Koch snowflakes and search for patterns and changes in perimeter and area.

 

EALRs:

(1.2    Understand and apply concepts and procedures from measurement

understand the concepts of and the relationships among perimeter, area, and volume and how changes in one dimensions affect perimeter, area, and/or volume * note this applicable for the lesson extensions.)

1.3    Understand and apply concepts and procedures from geometric sense

            identify, describe, or draw objects in the surrounding environment in geometric terms

            perform geometric constructions using a variety of tools

Students will draw iterations of a line segment, producing a Koch Curve. Students will come together to “draw” and create a Koch Snowflake from their individual curves.

(1.5    Understand and apply concepts and procedures from algebraic sense

            recognize, extend, and create patterns and sequences)

2.1    Investigate situations

            search systematically for patterns in simple situations

Students will look critically at the iteration process in the construction of the Koch Curve and identify patterns of self similarity.

3.1    Analyze information

            validate thinking and mathematical ideas using models, known facts, patterns,     relationships…

Students will use triangle graph paper and the Koch Curve to validate their understanding of fractal patterns.

3.2    Predict results

            make conjectures based on analysis of new problem situations

Students will predict the infinite possible iterations of the Koch curve fractal.

3.3        Draw conclusions and verify results

            Check for reasonableness of results

Students will be shown a further iterated fractal.
4.3    Represent and share information

            Explain or represent mathematical ideas and information in ways appropriate for            audience and purpose

Students, in groups, will represent their knowledge of fractals by creating and displaying a Koch Snowflake.
5.1    Relate concepts and procedures within mathematics

In recognizing patterns, students will relate algebraic sense with geometric sense. Students will also relate fractions and measurement to the iteration steps in this geometry focused lesson.

5.2        Relate mathematics concepts and procedures to other disciplines

Students will see the fractal properties of a snowflake help one understand the science behind the crystal structure. Math is found in nature.        
5.3    Relate mathematics concepts and procedures to real life situations

Students will see fractal relationships in common snowflakes.

 

Assessment:

Excellence in this activity will be clear if students are following steps to create individual iterations. Students will make connections and practice patterns. Students will also be engaged and active and working both independently and with a group. Students will create a Koch snowflake. Finally, further success will be evident when students expand upon the concept to see fractals in other objects and models.


 

line segment extension/challenge questions (moving into defining and constructing an equation of the recursive relationship):

if you start with a triangle, how many line segments? ____________ (3)

after the first iteration, how many line segments? _______________ (12)

after the second, how many line segments? ________________ (48)

after the third, how many line segments? ________________ (192)

the fourth, how many line segments? _______________ (768)

keep going?

the fifth, ? ___________________________(3072)

the sixth, ? ______________________ (12288)

The seventh, ? _______________ (49152)

The eight, ? __________________(196608)

Do you see any patterns in these relationships?

In reverse order:

1/6561

1/2187

1/796

1/243

1/81

1/27

1/9

1/3

 

3 to the x power.

 

The Koch Curve is the most recognized "line bender" fractal.

Directions: Draw several iterations of the Koch Curve, filling in the table below.

Iteration

1

2

3

4

5

Number of Segments per Side

  
  
  
  
  

Length of a Segment

  
  
  
  
  

Total Length of Curve

  
  
  
  
  

Answer the following questions:

  • What would the total length of the N-th iteration be? Look at the patterns made by the numbers both before and after simplifying.
  • What do you expect the Koch Curve to look like? In other words, what would you expect to happen if you repeated this infinitely many times?
  • What is the length of the Koch Curve?
  • Can you estimate the area at each stage? What is the area of the final snowflake?

Resources:

Murphy, P. (1993). By Nature’s Design: An Exploratorium Book. San Francisco:     Chronicle Books.

 

http://math.rice.edu/~lanius/fractals/selfpr.html

self similarity

 

http://math.rice.edu/~lanius/fractals/selfpr.html

iterative formation

 

http://math.rice.edu/~lanius/fractals/selfpr.html

triangle grid paper

 

http://math.rice.edu/~lanius/frac/ko2pr.html

infinite perimeter in the koch snowflake

 

http://math.rice.edu/~lanius/frac/ko3pr.html

finite area of koch snowflake

 

http://www.tacoma.washington.edu/education/intel/Projects/lammpics/define.gif

excellent defination of fractals

 

http://www.tacoma.washington.edu/education/intel/Projects/malinda_lamm.htm

relative EARLS

 

http://occawlonline.pearsoned.com/bookbind/pubbooks/billstein_awl/medialib/shodor_site/activities/koch/worksheet.html

koch snowflake perimeter worksheet

 

http://occawlonline.pearsoned.com/bookbind/pubbooks/billstein_awl/medialib/shodor_site/discussions/recursion.html

recursion discussion

 

http://occawlonline.pearsoned.com/bookbind/pubbooks/billstein_awl/medialib/shodor_site/discussions/selfsim.html

self similarity discussion

 

http://www.shodor.org/MASTER/fractal/software/Snowflake.html

java program to create iterate snowflakes and fractal curves

 

http://occawlonline.pearsoned.com/bookbind/pubbooks/billstein_awl/medialib/shodor_site/lessons/frac1.html

introduction to fractals lesson

 

http://php.iupui.edu/~wijackso/

fractals and spirituality, hindu temples, cosmos, unity