Part 1: Creating snowflakes with paper and pencil

Fractals deal with "self similar" objects. If you were to look at a coastline from an airplane it would probably appear as a jagged edge. As the plane flew closer to the ground features of the coastline that were not visible from higher elevations would now become apparent. There may even be portions of the coastline that appear "self similar" to portions of the larger coastline. Koch’s Snowflake is one such example of a "self similar" object.

• With paper and pencil construct an equilateral triangle of side length S, equal to three inches. This is a level one, L=1, "Snowflake."

• Trisect each side of your equilateral triangle and construct another equilateral triangle on the exterior of each side using the middle third of each side as the base. Erase the base line. This is a level two "Snowflake." Note: The sides of this snowflake will each have length of one inch.

• Repeat this "trisect and construct process" for each of the 12 sides of your level two snowflake to create a level three snowflake.

• Discuss how these snowflakes are self-similar.

• Calculate the area and perimeter for each level snowflake. Create a chart to record the snowflake level, area and perimeter.

• Hypothesize what the area and perimeter would be if this procedure continued indefinitely.

Part 2: Creating Snowflakes with MW

This part explores a MW code that will construct Koch’s Snowflake at different fractal levels.

• Open MW and go to the Procedures page. Write a procedure to construct Koch’s snowflake. Good Luck! See below and enter the following procedure. (Code taken from Serra, M., (1989), Discovering Geometry: An Inductive Approach, Key Curriculum Press, p. 745.)

Note: L is the fractal level and S is the length of the side. L values are integer values greater than or equal to one. S values should exceed 50 turtle steps for clarity of the image. If S is too large the image will not fit on the viewing screen and will wrap around the screen.

to snowflake :L :S
repeat 3 [divside :L :S rt 120]
end
to divside :L :S
if :L = 1 [fd :S stop]
divside :L – 1 :S / 3 lt 60
divside :L – 1 :S / 3 rt 120
divside :L – 1 :S / 3 lt 60
divside :L – 1 :S / 3
end

• Return to page1 and put the pen down. Run the code for a level one snowflake, L=1 and S=60.

• Drag the turtle to a new location and continue to run this program changing levels of L, L={2, 3, 4…}. For each successive level drag the turtle to a new screen location.

• Observe the resulting constructions and discuss how the areas and perimeters are changing and how they might or might not be bounded.

Part 3: Algebraic derivations of series for area and perimeter

This activity steps away from the programming and investigates numerically the previous conjectures stated with respect to area and perimeter.

• Derive an expression for the area of Koch’s Snowflake as a function of L. (Hint: Assume area of a level one snowflake is equal to one).

• Derive an expression for the perimeter of Koch’s Snowflake as a function of L.

• Discuss the convergence or divergence of the above expressions as L goes to infinity and relate this to your observations from Part 2.

Part 4: Analyzing the code

to snowflake :L :S
repeat 3 [div.side :L :S rt 120]
end

to div.side :L :S
if :L = 1 [FD :S stop]
div.side :L – 1 :S / 3 lt 60
div.side :L – 1 :S / 3 rt 120
div.side :L – 1 :S / 3 lt 60
div.side :L – 1 :S / 3
end

• Work through this code, one level at a time, beginning with L = 1 through L = 3 and explain the code in your own words.