Creating a Sierpirfski Gasket Fractal Name(s):

By using the Iterate command r f ou can repeat an action again and agai:.on the same figure. Repeatedly taking the result of an action and applr-i:,sthe action to that result again is called iteration in mathematics and iscentral to the creation of fractals. In this activity, you'll create the firstfew stages of a fractal called the Sierpiiski gasket. A true Sierpiriski gaske:created in infinitely many stages, has dimension between 1 and2 and it-.perimeter and area have strange properties.

Using the Text tool,click once on a point

to show its label.Double-click the

label to change it,

Sketch and lnvestigate

1. In a new sketch, construct a triangle ABC,with point A at the bottom left corner.

2. Construct the midpoints of the three sides.

3. If necessary, change labels to match thediagram.

"",,,18!l",l,f.l ft +. Co"struct the interior of the triangle.in the Construct I

;";e,;1";; I As shown at right, the three midpoints helpTriansfe lnterior'l define four "subtriangles" of. A,ABC, namely

LAFD, LFBE, LDEC, and AEDF. (The dottedlines are shown for illustration only-theyshouldn't be in your sketch.)

Take a moment to think about what you've " done"

A

to triangle ABC: You've constructed its midpoints and its interior. Nowimagine doing the same thing (iternting the construction) to L,AFD, then toLFBE and ADEC (butnot to AEDfl, and then hiding the original interior.Make a drawing below of what the resulting figure (the first iteration)would look like.

Can you imagine what the second iteration would look like (in otherwords, if you now applied the iteration rule to the three outer subtrianglesin your drawing above)? You'll now use Sketchpad's Iterate command toconstruct an iterated image that can show several stages of the iteration.

Select, in order, points A, B, and C. Choose Iterate from the Transformmenu. The Iterate dialog box appears.

Choose Final Iteration Only from the Display pop-up menu. This tellsSketchpad to hide triangle interiors or,." tih"y getiterated on. \

5.

6.

204 . Chapter 10: Trigonometry and Fractals

J

Exploring Geometry with The Geometer's Sketchpad

@ 2002 Key Curriculum Press

Drao rhp rriarnn hnv, Creating a Sierpiriski Gasket Fractal (continued)bv its title bar if vou I

c'an't see a::::ll F 7. Click on point A first, then on point F, and then on point D. This tellsl^1 .1 t.//1Sketchpad to "do to triangle AFD what was done to triangle ABC."

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IteratEffiI

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Pre-lnagq To Firsl lmtgq :

A = l-A--lB +l F I l

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Number of ii8rstiom: 3-

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9.

After step 7

8. Choose Add New Map fromthe Structure menu.

Repeat stepT for points F, B,

and E, choose Add New Map,then repeat step 7 for points D,E, and C. When you're done, theIterate dialog box should looklike the one at right. Click Iterate.

Deselect all objects. Now click inthe center of LABC to select its interior, then hide it.

Hide the midpoints.

Select the iterated image and press the - key until nothing changes.This is the stage 1 iteration. How does it compare with your drawingon the previous page?

With the iterated image still selected, press the + key once to show thestage 2 iteration. Sketch the resulting figure below.

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i,,,: Qllphy ;i j structure-l tJ t_SEIlllJ ltlgl3lgt

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12.

10.

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Exploring Geometry with The Geomeler's Sketchpad@2002 Key Curriculum Press

Chapter 10: Trigonometry and Fractals . 205

Creating a Sierpiriski Gasket Fractal (continued)

13. Use the + and - keys to explore other stages of the Sierpiriski iteration-

Q2 What stage gasket is shownin the figure at right?

Q3 Suppose a stage 0 gasket (a single triangle)has area 1 square unit. Fill in the middlerow of the chart with the areas of stage 1,2,3, and 4 gaskets. As you increase thenumber of stages, what's happening tothe area of the figure (the shaded part)? Enter an expression in thechart for the area of a stage n gasket.

Qa Suppose the stage 0 gasket started with perimeter 3 units. Whathappens to the perimeter, including the perimeters around all thelittle shaded triangles inside, as the stages increase? Complete thebottom row of the chart for perimeter.

\Atrhat would the area of aSierpiriski gasket be at stage infinity?

\tVhat would the perimeter of aSierpiriski gasket be at stage infinity?

Q5

Q6

Stage 0 1 2 J 4 n

Area ,l

Perimeter J

206 . Chapter 10: Trigonometry and Fractals Exploring Geometry with The Geometer's Sketchpad

@2002 Key Curriculum Press