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
· triangle graph paper
· colored pencils
· construction paper
· straight edge (rulers) are optional
Have you ever noticed patterns in snow?
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?
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?
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.
(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
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.
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)
the fifth, ? ___________________________(3072)
the sixth, ? ______________________ (12288)
The seventh, ? _______________ (49152)
The eight, ? __________________(196608)
Do you see any patterns in these relationships?
In reverse order:
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.
Number of Segments per Side
Length of a Segment
Total Length of Curve
Answer the following questions:
Murphy, P. (1993). By
Nature’s Design: An Exploratorium Book.
triangle grid paper
infinite perimeter in the koch snowflake
finite area of koch snowflake
excellent defination of fractals
koch snowflake perimeter worksheet
self similarity discussion
java program to create iterate snowflakes and fractal curves
introduction to fractals lesson
fractals and spirituality, hindu temples, cosmos, unity