Geometric Models of Infinite Series

Geometric Models of Infinite Series1. The Sierpinski Carpet is constructed by removing the center one-ninth of a square of side 1, thenremoving the centers of the eight smaller remaining squares, and so on. (The figure shows the firstthree steps of the construction.) Show that the sum of the areas of the removed squares is 1. Whatdoes this tell you about the area of the Sierpinski Carpet?

2. A right triangle ABC is given with ∠A = θ and |AC| = b. CD is drawn perpendicular to AB, DE isdrawn perpendicular to BC, EF ⊥ AB, and this process is continued indefinitely as in the figure.Find the total length of all the perpendiculars, |CD| + |DE| + |EF| + |FG| + … , in terms of b and θ.

3. To construct thesnowflake curve startwith an equilateraltriangle with sides oflength 1.Step 1 in theconstruction is todivide each side intothree equal parts,construct anequilateral triangleon the middle part,and then delete themiddle part.Step 2 is to repeatstep 1 for each sideof the resultingpolygon.This process isrepeated at eachsucceeding step.The snowflakecurve is the curvethat results fromrepeating thisprocess indefinitely.a) Complete the following table.Steps n , thenumber ofsidesl n , the length of asidep n , the total length ofthe nth approximatingcurveA n , the area enclosed bythe snowflake curve01234Find formulas for s n , l n , and p n .nb) Show that p n , → ∞ as n → ∞.c) Sum an infinite series to find the area enclosed by the snowflake curve.d) What "strange" relationship do parts (b) and (c) show between the snowflake curve's perimeterand area.

4. Square ABCD has sides of length 1. Square EFGH is formed by connecting the midpoints of thesides of the first square, as shown in the figure. Assume that the pattern of shaded regions in thesquare is continued indefinitely. What is the total area of the shaded regions?It might help to complete the following table and then look for a pattern.Stepthe length ofa sidethe area of thesquare1 AB = 1 1the area of the triangle182EF = 2234

5. In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, eachcircle touching other circles and sides of the triangle. If the triangle has sides of length 1, find thetotal area occupied by the circles.It might help to complete the following table and then look for a pattern.Steph, height of thetriangles, length of the sideof the triangles = 2 3 hr, radius of the A, area of the circlecircler = 3A = π r 26 s1 31 32 6π ⎜ ⎛ 3⎝ ⎠ ⎟⎞26 234