1. Suppose one has an equilateral triangle with sides of length 1. If one modifies each of the line segments composing the triangle by cutting out the middle third of the segment, and then inserting an outward-pointing "tooth," both of whose sides are as long as the removed third, one gets a six-pointed star. Suppose one repeats this process for each of the line segments making up the star, then to each of the line segments making up the resulting figure, and so on, as in the diagram:

.

(Note that the lengths of the line segments at each stage are a third of the length of the segments at the preceding stage.) The curve which is the limit of this process (if one takes infinitely many steps) is often called the snowflake curve.

a. Find the area of the region enclosed by the snowflake curve. [3]

b. Find the length of the snowflake curve. [3]

2. Suppose one has a cube which has sides of length 1. Modify this cube by cutting out the centre one of the 27 equal-sized sub-cubes, plus the six sub-cubes that share a face with the centre sub-cube. Proceed further by modifying each of the remaining sub-cubes in the same way, then each of the remaining sub-sub-cubes, and so on. (You draw the pictures!) Consider the object consisting of the points in the cube remaining after infinitely many steps, call it the Cheese.#

a. Find the volume of the Cheese. [2]

b. Find the surface area of the Cheese. [2]

Hint: Computing the surface area by working out a formula for the surface area at each stage is probably quite difficult, so you should look to find a sequence or series which gives a suitable lower bound. (You might be able to guess from that what the answer is...)

# I have no idea if anyone has studied this object before... Hail, O pioneers!.

Department of Mathematics

Trent University