In Lesson 3, you were introduced to functions as units of computational abstraction. When given a predefined function such as Math.sqrt, you were able to call and use that function in computations, without worrying about how it was implemented. In this lab, you will work within a simple graphics environment, making function calls to control drawings on the screen.
In 1967, Seymour Papert and other researchers at MIT developed the programming language LOGO. LOGO was specifically designed to teach children how to program and solve problems on the computer. Central to the LOGO approach is the use of "turtle graphics", where the child/programmer draws pictures by moving an object (called a turtle) around on the screen. Simple commands such as "move forward 100 steps" and "turn right 90 degrees" are used to control the turtle, which leaves a trail (i.e., a line) in its wake.
Research by Papert and others has shown that the graphical LOGO environment has numerous advantages over textual environments. Beginners can see the effect of their actions, and are often drawn to the more creative aspects of drawing. In this following exercises, you will be introduced to some of the basic turtle graphics functions, and will experiment with calling these functions. Hopefully, this more visual approach will assist you in understanding how function calls behave, and appeal to your creative side as well.
The Web page Turtle.html contains a simple Turtle Graphics environment. In the top half of the page there is a box in which commands can be entered for controlling the movement of the turtle. When the button to the right is clicked, these commands are then executed and the turtle moves accordingly in the graphics screen below. The graphics screen is 400 units wide and 250 units tall, where each unit is a single pixel (dot) on the screen. The center of the screen is considered to be coordinate (0,0), so the x-axis (horizontal) ranges from -200 to 200 while the y-axis (vertical) ranges from -125 to 125. When you first load the page, the turtle is located at the center of the screen (0,0) facing straight up. You can't see the turtle itself, but you can follow its trail as it moves around on the screen. The coordinates of the turtle, along with its current heading (0 = up, 90 = right, 180 = down, etc.) and distance from the origin, are also displayed to the right. Functions for controlling the turtle's movement include:
forward: moves the turtle the specified number of steps in the direction it is currently facing.
For example, forward(20) moves 20 steps forward.
left: turns the turtle the specified number of degrees to the left, relative to the current heading.
For example, left(90) makes a right angle turn to the left (counter-clockwise).
right: turns the turtle the specified number of degrees to the right, relative to the current heading.
For example, right(90) makes a right angle turn to the right (clockwise).
Each time the "Execute code" button is clicked, the turtle will move according to the function calls in the large box at the top of the page. By default, each new drawing will begin where the previous one ended, although you can force all drawings to start at the center by checking the box labeled "Start each drawing at (0,0)". The lines drawn by the turtle will persist between executions, until the "Reset graphics screen" button is clicked. Note: If the browser window containing the turtle graphics environment becomes obscured (i.e., covered by another window), the turtle lines will NOT be redrawn when the page is brought to the forefront again.
EXERCISE 1: Sketch what is drawn by executing each of the following sequences of function calls. Be sure to reset the graphics screen between each sequence by clicking the button labeled "Reset graphics screen". Note: it is perfectly legal to put more than one function call on a line -- calls on one line are evaluated in a left-to-right order before moving on to the next line.
forward(50); left(90); forward(25); right(180); forward(50); ________________________ left(45); forward(50); left(45); forward(80); left(45); forward(50); left(90); forward(50); left(45); forward(80); left(45); forward(50); ________________________ forward(60); right(90); forward(60); right(90); forward(60); right(90); forward(60); right(90);
You may have noticed that the last sequence of function calls is repetitive. In particular, it consists of the same two function calls repeated four times:
For repetitive drawing tasks such as this, the Turtle Graphics page provides a simpler mechanism for execution. To the right of the "Execute code" button is a box that specifies the number of times the code is to be executed. By entering the above two function calls and specifying 4 repetitions in this box, the same effect as before is obtained. That is, the two function calls are executed four times in succession.
EXERCISE 2: Execute the following sequence of function calls 4 times, and verify that it yields the same result as in the previous exercise.
Execute the following sequence of function calls 4 times and sketch the results:
forward(40); right(90); forward(20); left(90); forward(10); left(90);
Execute the following sequence of function calls 5 times and sketch the results:
Execute the following sequence of function calls 5 times and sketch the results:
forward(10); right(90); forward(10); left(90);
EXERCISE 3: As you verified above, the function calls
when executed 4 times in succession, draw a square on the screen. By changing the turning angle and the number of executions, other regular polygons can likewise be drawn. For example, changing the angle to 120 and then executing 3 times results in a triangle. Likewise, changing the angle to 72 and executing 5 times results in a pentagon.
In general, what must the turning angle be in order to draw an N-sided polygon? Give your answer as a formula involving arbitrary variable N, such that if you assign N=3 and execute 3 times, a triangle is drawn; assigning N=4 and executing 4 times draws a square, etc. Verify your answer in the page.
In 1828, a biologist Robert Brown used a microscope to discover that small pollen grains in a drop of water seemed to quiver in perpetual motion. Brown showed that this motion was not biological but physical and the phenomenon became known as Brownian motion in his honor. In 1905, the 25 year old Albert Einstein wrote a brilliant paper that developed a theory of Brownian motion, explaining quantitatively how the perpetual motion was a natural consequence of many ongoing collisions of tiny invisible atoms with the pollen grain. This paper created much excitement since it provided, for the first time, an experimental way to prove the existence of atoms, which was a very controversial issue at the turn of the century. In 1926, a French experimental scientist, Jean Perrin, won the Nobel prize in Physics for verifying Einstein's predictions and for making the first quantitative measurements of Brownian motion.
One simplified model of Brownian motion involves a particle that takes successive steps of a fixed size but in randomly chosen directions. This approximates the effects of collisions of a big object with many small atoms. The path taken by such a particle is often referred to as a "random walk", with the common analogy being that of an inebriated person staggering in random directions.
EXERCISE 4: Enter this code into the Turtle Graphics page and set the number of repetitions to be 100. Execute the code and describe the general look of the walks below.
Increase the number of repetitions to 1000 and conduct several more random walks (resetting the screen between each walk). Do walks ever extend beyond the boundaries of the graphics screen? Do walks ever extend off the screen but then later return? Describe the patterns you see.
Since random walks are, by definition, random in nature, you might think that there is no way to predict the result of a random walk. Statistically speaking, this is not true. Think back to the random darts thrown in the Monte Carlo PI lab. Despite the fact that location of each dart was random, a large number of random darts yielded a predictable distribution. This same sort of predictability holds for random walks. While it is possible for a walker to head off in one direction and continue in a straight line, this is very unlikely. More often, the path taken will loop back upon itself and the walker will not venture too far off from its starting point. In fact, it has been proven that after N random steps, the expected distance of the walker is sqrt(N) steps from its starting place. Or, perhaps a better way to say it is that the square of the final distance will be equal on average to the number of steps taken.
EXERCISE 5: Set the step size to 1 and the number of repetitions to 100. Then, conduct 10 random walks by clicking the "Execute code" button repeatedly. Be sure to check the box labeled "Start each drawing at (0,0)" to ensure that each random walk begins at the center of the screen.
Record the squares of the final distances below. Is the average of these 10 numbers close to the expected value of 100?
EXERCISE 6: Set the step size to 4 and the number of repetitions to 100. Then, conduct 10 random walks by clicking the "Execute code" button repeatedly. As before, be sure to check the box labeled "Start each drawing at (0,0)" to ensure that each random walk begins at the center of the screen.
Record the squares of the final distances below. Is the average of these 10 numbers close to the expected value of 1600?
Since 10 random walks are not very many, your averages from the last two exercises may not have been very close to the expected values. The page Walk.html makes it easier for you to conduct a large number of random walks and gather statistics on those walks. You can enter the number of desired random walks and details for each walk (the number of steps, the step size, and the turning angle). To save time, only the final position of the turtle is marked for each walk, and averages over all of the walks are displayed.
EXERCISE 7: Using this page, conduct 100 random walks, each consisting of 100 steps. What is the average step distance2 of your 100 walks? Is it close to the expected value of 100? Note: you can change the step size for the walks without affecting the average, since the box labeled average step distance2 normalizes the distance with respect to the step size.
Now, conduct 1000 random walks, each consisting of 100 steps. What is the average step distance2 of your 1000 walks? Is this number closer to the expected value than the average over 100 walks? Should it be? Explain.
With each step, a random walk chooses one of 360 directions in which to take a step. Sometimes, a walk is more constrained. Imagine walking in a city such as Manhattan, where each block constitutes a step. Since most streets meet at right angles, you generally have four choices in direction (including turning around) at each corner. Because of this analogy, a random walk where the direction is constrained to right angles only is known as a Manhattan walk.
EXERCISE 8: In the Turtle.html page, modify your random walk code to simulate a Manhattan walk, randomly choosing between four directions (0, 90, 180 or 270 degrees) for each step. List the modified code below. Hint: to randomly select from 0, 90, 180, and 270, first generate a random integer in the range 0 to 3, then multiply by 90.
Conduct several Manhattan walks using your code and describe the general look of the walks below:
EXERCISE 9: How would you expect Manhattan walks to compare with unconstrained 360-degree random walks? That is, would you predict that the expected distance for a Manhattan walk would be greater or smaller than the corresponding unconstrained random walk with the same number of steps and step size? State a hypothesis and a possible rationale for that hypothesis.
Using the Walk.html page, test your hypothesis. That is, conduct 1000 Manhattan walks, each of 100 steps, and compare the average distance2 with that obtained in EXERCISE 7. Do the results support or contradict your hypothesis? Is there enough of a difference to be considered significant? You may need to conduct further experiments to be more convincing.