Of course to answer that question for a given Sierpinski Gasket that has been made by cutting and gluing black pieces of paper and white ones, one must know exactly how many black triangles were used, or to what "level" the Sierpinski Gasket was made. Each time you make a set of smaller black triangles to glue onto the white ones, the area of the black increases, and the area of the white decreases.
Theoretically, the addition of the black triangles goes on forever. Does the black ever completely cover the white? Answers of yes or no are justifiable. It is interesting to discuss and debate them.
Instead of being concerned with the actual area of the equilateral triangles, it is possible to assign an arbitrary value to the area of the large white one and compute the area of the black in terms of that value. In the example below, a value of 1 will be assigned to the area of the large triangle. All of the other (black) triangles are fractional sizes of that larger triangle, so determining the area of each, and adding up the total area will involve arithmetic with fractions. If students are not comfortable working with fractions, this can be avoided by choosing a larger value---such as 16---for the area of that larger triangle. Whatever value you choose, be sure to work out the example fully before presenting it to students so that the results don't surprise or confuse you in the midst of helping your students do it!
To make the first black triangle that will be glued on the large white triangle (the one that connects the midpoints of its sides), a black triangle the same size as the white one is folded in a way that actually makes 4 smaller black triangles. So the area of the first black triangle is 1/4.
A black triangle whose size is 1/4 will be cut into 4 smaller triangles. Each of these triangles has an area of (1/4) X (1/4) = 1/16. Three of them will be placed inside the white triangles. Then the total black area will be:
The next triangles to place on the Sierpinski Gasket will be a quarter the size of the 1/16 triangles, or 1/64, and there will be 3 X 3 = 9 of them, making the total area:
There will be 9 X 3 = 27 of the next size smaller triangles. Since they are a quarter of the size of the previous ones, their size will be 1/256. When those triangles are places, the total area covered by black triangles is:
The purpose of this exercise isn't to go through the agonzing process of adding fractions with different denominators, but to notice the patterns that are developing and begin to predict how the computation will progress as more and more triangles are added.
Each time a new (smaller) size of triangle is added to the Sierpinski Gasket, another term is added to the series. The numerators of each term represent the number of triangles of that size that are used. Notice that they increase by a factor of three each time. Look at a Sierpinski Gasket and verify why this would be so. The fraction 1/d, (where d is the denominator of each term) is the measure of the area of one triangle in relation to the large white triangle whose area was taken to be 1. Notice that the denominator increases by a factor of 4 as each term is added. That is because the area of the triangles in each set is is 1/4 the size of the area of the triangles in the preceding set.
What is interesting about the sums? So far they are all less than 1. It is helpful to plot them on a number line so that you can see clearly just where they are located between 0 and 1. Continue adding terms to the series and computing the sums. Each sum represents the area covered by black triangles as more and more of them are added.
Recall that the value of 1 was chosen somewhat arbitrarily to represent the size of the large white triangle with which we began (before any black ones were glued on). If the area covered by black triangles was ever to become greater than 1, this would mean that the black triangles somehow covered an area larger than the original white triangle. Is there a way that this could happen?
We find ourselves in an interesting and bizarre situation. Taking information from the way that the Sierpinski Gasket is made (which is relatively simple), we designed a scheme for coming up with a long list of numbers to be added together. No matter how long that list gets, we can always add more numbers to it, according to our scheme. And no matter how many numbers we add together, if we keep to the scheme, it doesn't seem like we can ever get the grand total to be more than 1. How can this be? Will the sum ever actually be equal to 1? It seems like it will get closer and closer. In fact, it seems that no matter what value you pick that is very close to 1, you can keep adding terms to the series so that the sum of all the terms is greater than that value, but, of course, still less than 1. But will the sum ever actually be 1? This question is very similar to questions posed by the ancient Greek mathematician Zeno who is credited with setting forth some famous paradoxes . This kind of thinking is also very important in the study of Calculus.
Don't be disappointed or frustrated if this discussion puzzles you, or if you can't find some simple explanation that resolves all possible contradictions. From the time of Zeno to the present, mathematicians and philosophers have wrestled with the notion of infinity. Always, there has been one loose end or another which cannot be tied up neatly with the rest. The story of the Hotel Infifnity reflects the state of the debate over infinity at the beginning of the 20th century.
