Coursehero >> Illinois >> University of Illinois, Urbana Champaign >> MATH 231

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

231 Math Project Option #6 Space-Filling Curves: Covering a Whole Lot of Ground Did you ever have an Etch a Sketch? It s a drawing toy by the Ohio Art Company. In it, you twist two knobs one for up/down, one for left/right to move a point around a screen, tracing out a picture.1 One thing that some people like to do with an Etch a Sketch is move the point so that it traces over every point on the whole screen, turning it all dark. It s very time-consuming (and you have to be careful not to jostle the Etch a Sketch in the meantime, so that nothing s erased), but it seems possible. Is the same thing possible theoretically? Say you had a pencil with an in nitesimal point. Would it be possible, in theory, to trace over every point in a square of side length 1 without lifting up your pencil? And to do it in a nite amount of time? How would you do it? It should probably come as a surprise to you that such a thing is possible, in theory. It s accomplished by what s called a space- lling curve. A space- lling curve is de ned as a curve (a continuous function) whose domain is [0, 1], and whose range is the entire unit square with vertices (0, 0), (0, 1), (1, 0), and (1, 1). Let s see a picture. The thing about space- lling curves is that it s not very helpful to graph the space lling curve itself. Why not? Sketch what the graph of every space- lling curve will look like. What s more, many space- lling curves are de ned as the limits of sequences other curves, so it s more instructive to show some of the curves in a sequence converging to the space- lling curve. Shown below2 are the rst several graphs in the sequence of curves which converges to the Hilbert Curve, a space- lling curve designed by the mathematician David Hilbert (1862 1943): You ll notice that each curve appears to ll up much more of the square than the curve before it. This sequence converges to a curve that lls up the entire square, with no gaps at all left. We re going to examine a particular space- lling curve. 1 If 2 Image you re nostalgic, or have never played with one before, you might visit http://www.etchy.org. Enjoy! found at http://www.idynamix.org/ pmath370FinalProj/PMATH370.hilbertCurve.iterations1-6.jpg. A space- lling curve The curve we will describe was constructed by I. J. Schoenberg in 1938. To start with, we need to de ne a function f (t) on the real line by letting 0, 0 t < 1/3, 3t 1, 1/3 t < 2/3, f (t) = 1, 2/3 t < 4/3, 3t + 5, 4/3 t < 5/3, 0, 5/3 t < 2, and then making f periodic. In other words, the piecewise de ned function above traces out the boxed portion of the graph below, and we repeat it over and over again to get the entire graph: t 2 1 0 1 2 3 4 As you can see, the graph repeats itself every 2 units and oscillates between 0 and 1. The function f (t) is not itself the space- lling curve, but now we can de ne the Schoenberg space- lling curve parametrically as x(t) = f (32n 2 t) , 2n n=1 y(t) = f (32n 1 t) , 2n n=1 0 t 1. Let s look at where the curve is at a few speci c times. Find where the curve starts, i.e., nd x(0) and y(0). Find where the curve ends, i.e., nd x(1) and y(1). (Hint: remember that f (t) repeats itself every 2 units. What values does f (T take ) on when T is an odd integer? What does this mean for the sums for x(1) and y(1)?) Find x(1/3) and y(1/3). Find x(2/3) and y(2/3). And now let s look at a few general questions. How do we know that the series for x(t) and y(t) will always converge as t ranges between 0 and 1? Use an appropriate convergence test to show that it does. Going a little bit farther than the last question, explain why we always get 0 x(t) 1 and 0 y(t) 1), no matter what t is. We saw above the start of a collection of curves which converges to the Hilbert curve. We can get a sequence of curves which converge to the Schoenberg curve if we take partial sums of the series. Graph the rst partial sum, which is the parametric curve x(t) = f (t) , 2 y(t) = f (3t) , 2 0 t 1. Suggestion: It will probably be easiest just to see where the corners in the graphs of x(t) and y(t) occur, and connect the dots from there. For example, we know that the graph f (t) changes its behavior its direction, anyway at integer multiples of 1/3 (e.g., at x = 1/3, 2/3, 4/3, 5/3, etc.). In the rst curve above, y(t) involves f (3t). Since f changes when its inside is a multiple of 1/3, we need to pay attention to when 3t is a multiple of 1/3, which happens when t is a multiple of 1/9. We might make a table: t 0 1/9 2/9 3/9 4/9 . . . x(t) = f (t)/2 y(t) = f (3t)/2 Complete the table for t equal to all multiples of 1/9 up through 1. Then plot the points (x(t), y(t)) and connect consecutive dots. Do the same type of thing for the second curve above, but only on the interval 0 t 1/3. (Beyond t = 1/3, things get pretty messy pretty quickly.) Notice that there the function y(t) involves f (27t), and since f (t) changes its behavior of multiples of 1/3, f (27t) will make some changes when 27t equals certain multiples of 1/3 which means that t will be a multiple of 1/81. Use enough points to get an accurate picture of the second curve, and then draw it. The third, fourth, and fth partial sums of the Schoenberg curve are shown below. 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Based on your sketch of the rst partial sum and the graphs of the partial sums above, what properties of symmetry would you guess that the partial sums of the Schoenberg curve have? Use the de nition and/or graph of f (t) in part to explain why there is symmetry in the partial sum x(t) = you sketched above. Let s nish the project by talking about the speed at which the Schoenberg curve is traced out. Intuitively, you might think that since the curve has to trace out all points in the time between t = 0 and t = 1, it has to move pretty fast in some places (maybe even in nitely fast?). Finding the velocity at which the complete Schoenberg curve is traced out gets a little messy, so as our last project problem we ll just take a quick look at a partial sum. Write a piecewise-de ned formula for the velocity of the point tracing out the partial sum x(t) = f (t) , 2 y(t) = f (3t) , 2 0 t 1. f (t) , 2 y(t) = f (3t) , 2 0 t 1. Describe the motion of the point as it traces out the curve (which, remember, you sketched above). In particular, does it trace out the curve at a constant speed, or does the speed vary as t changes?

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: