Computer graphics can be tricky, and you often need a lot of experience with programming before you can get started. However, for over fifty years, the LOGO Turtle has allowed elementary-school students to draw cool and interesting shapes on their computer. And if they can do it, so can you.

The Turtle is a cursor on the screen that uses a Turtle for its icon. To draw a line, you push the Turtle forward. The screen draws a line of the same color as the Turtle (which can be changed at any time) along its path of movement. To draw more interesting shapes, you simply change the direction of the Turtle. With a few more commands you can even draw solid shapes.

While programming languages have come a long way since the early days of LOGO, the graphics Turtle lives on. Every major programming language, from C/C++ to Java to Python, has a version of the Turtle. For today’s assignment you get to participate in this 50 year tradition, and hopefully have some fun in the process.

This assignment is a little harder than the previous one, but you have two full weeks to work on it. In the past we have found this to be more than enough time for this assignment. But get started early! If you do not know where to start, or if you are completely lost, please see someone immediately. A little in-person help can do wonders.

As before, remember to fill out the survey telling us how long you worked on this assignment.

Authors: D. Gries, W. White, L. Lee, and S. Marschner

Learning Objectives

This assignment is designed to help you understand the following concepts.

• It introduces you to the famous Turtle, allowing you to draw basic shapes.
• It gives you practice with writing simple for-loops
• It gives you practice writing recursive functions from an explicit definition.
• It gives you practice with complex objects having both attributes and methods.
• It gives you practice with using asserts to enforce your preconditions.

Table of Contents

• Academic Integrity and Collaboration
• Introduction to Turtle Graphics
  • Using a Window
  • Using a Turtle
  • Using a Pen
• Assignment Source Code
• Assignment Instructions
  • Precondition Enforcement
  • Task 1. Triangles
  • Task 2. Hexagons
  • Task 3. Radial Shapes
  • Task 4. Recursive Fractals
  • Task 5. Minkowski Island
• Finishing Touches

Academic Integrity and Collaboration

Academic Integrity

This is a classic assignment that is continuing because students really like it. But when we say classic, we mean really, really old. While we do make changes to the assignment every year (the fractals this year are brand new), there are guaranteed to be solutions to many of these problems in the wild. Please avoid looking at solutions online, or looking code from other students this semester, or semesters previous.

In this assignment, it is highly unlikely that your code for this assignment will look exactly like someone else’s. We will be using Moss to check for instances of plagiarism. We also ask that you do not enable violations of academic policy. Do not post your code to Pastebin, GitHub, or any other publicly accessible site.

Collaboration Policy

You may do this assignment with one other person. If you are going to work together, form your group on CMS as soon as possible. If you do this assignment with another person, you must work together. It is against the rules for one person to do some programming on this assignment without the other person sitting nearby and helping.

With the exception of your CMS-registered partner, we ask that you do not look at anyone else’s code or show your code to anyone else (except a CS1110 staff member) in any form whatsoever. This includes posting your code on Ed Discussions to ask for help. It is okay to post error messages there, but not code. If we need to see your code, we will ask for it.

Introduction to Turtle Graphics

Python actually has a built-in Turtle provided by the turtle module. However, we find this module a bit confusing to use, particularly for a beginner. In addition, it appears that Python 3 broke the Turtle on Windows. That is why we provide an alternative Turtle, which we call the Introcs Turtle. This Turtle is provided by the module introcs, which you should be well familiar with now.

However, you cannot access the Turtle by importing introcs. Why not? Because the Turtle uses a lot of memory and processing power and we did not want it to start up every time you import introcs. Instead, it is contained in a module inside of introcs (yes, modules can contain other modules) called introcs.turtle. So to use this module, you would instead type

import introcs.turtle

This is a bit of a mouthful, because you will have to write introcs.turtle before all the functions in this module as well. That is why, for this assignment, we prefer the from syntax. You only need three things from this module: Window, Turtle, and Pen. So you can import all of these as follows:

from introcs.turtle import Window, Turtle, Pen

While we describe these in detail below, you can get even more information from the official documentation. We wrote this version of the Turtle from scratch, and it is quite powerful.

Using a Window

To create a window, you use the constructor Window() and assign the result to a variable. Try this command at the interactive prompt:

>>> from introcs.turtle import Window
>>> w = Window()

This will display a window on your screen. The window object has several attributes that you can change.

Try changing the values of these attributes (with assignment statements). For example, what happens when you type the following command:

>>> w.width = 100

In addition, there are two important methods:

Pixels inside of the window follow a rather intuitive coordinate system. The point (0,0) is the center of the window. The x-coordinates get larger to the east and the y-coordinates get larger going up.

Using a Turtle

The Turtle class is used to draw on your Window. Each Turtle object t has the following important attributes:

To create a Turtle, you use the constructor Turtle() which takes a single argument: the Window that you want to draw on. Assuming that you made a Window object w in the previous section, try the following at the interactive prompt:

>>> from introcs.turtle import Turtle
>>> t = Turtle(w)

You should now see a (red) Turtle on your Window. The Turtle will always start at coordinate (0,0), which means it is centered in the window. It will also face east.

The fact that Turtle and Window are separate allows you to have as many Turtles as you like so that you can draw different things with them. If at any time you have too many Turtles, use the method w.clear(). This removes all Turtles from the Window (which also means that attempts to do anything with any old Turtles will fail). If you want to start drawing again, you will need to add a brand new Turtle.

Position and Orientation

The position and heading of the Turtle are maintained using floating point numbers. This is needed for accuracy. If integers were used, errors would be introduced after only a few calculations. However, whenever a point is to be drawn in the window, its x- and y-coordinates are rounded to the nearest integer because the pixel coordinates are represented as integers.

The direction of the Turtle is called its heading. It is a number representing the angle in degrees counterclockwise from east (to the right). Thus east is 0 degrees, north is 90 degrees, west is 180 degrees, and south is 270 degrees. Negative angles and angles greater than 360 are allowed; the remainder modulo 360 is used.

Measurement of the Turtle heading

While the heading attribute can be modified, the x and yattributes cannot. You can only control the Turtle’s position via the methods listed below.

Important Methods

In addition to its attributes, a Turtle object t has several important methods:

Note that most of these methods are used to move the Turtle about the screen. This is why the attributes x and y cannot be altered directly (e.g. you cannot assign values to them). You should use these methods instead. All of these methods autoflush when the speed is not 0 (see below), and so you will see the Turtle draw as soon as they are called.

Colors

To change the Turtle color, you assign a new value to the color attribute. You can use the RGB and HSV objects from the last assignment. You cannot use a CMYK object with a Turtle; that color model is designed for printing, and not for displaying on your screen.

The Turtle also supports strings as colors. Just put the name of the color that you want in quotes; make sure the name is all lower case. For example, to make your Turtle blue, try

>>> t.color = 'blue'

If you are familiar with web colors, those are also supported. Just remember to start the string with a hashtag (#), like this:

>>> t.color = '#0099CC'

Speed

As you will discover with this assignment, the turtle can be quite slow. You can control the speed of the Turtle by setting is speed attribute. It is a number in the range 1 ≤ speed ≤ 10, with 1 slowest and 10 fastest. But even speed 10 can be quite slow. Speed 10 will draw a single line instantaneously. However, if the Turtle is drawing multiple lines, it will draw each one separately. For the radial shapes, this can take a while.

If you really want to draw quickly, you should set the speed to 0. This speed is exactly what it sounds like: the Turtle does not draw at all. This seems counter-intuitive; of course you want the Turtle to draw. But you can always tell the Turtle to draw later by calling the flush method, like this:

>>> t.flush()

When you call flush on a Turtle with speed 0, it instantaneously draws all the lines that you asked the Turtle to draw. Hence, speed 0 remembers all the drawing commands, but draws them only when you ask it to finish. This is really handy for fast drawing (and we will rely on it for grading).

For this reason, you will need to remember to flush in many of your functions in this assignment. But you should be careful how you use flush. If you call it after each Turtle step, it will update the screen before you are ready. In this case, your Turtle will be no faster than speed 10. You should only call flush at the end of a function, and only in the main functions (i.e. not the helpers).

If the speed is 1 through 10, it is unnecessary to call flush. All of the drawing methods above will autoflush (meaning they call flush for you). But it is still safe to call flush for these speeds anyway.

Command Sequences

To draw shapes with the Turtle, you string together a sequence of drawing commands – method calls and attribute assignments. These commands move the turtle and change its colors. As a demonstration, start up the Python interactive shell and try these commands:

>>> from introcs.turtle import Window, Turtle
>>> w = Window()
>>> t = Turtle(w)
>>> t.color = 'green' # This will flush and show the turtle
>>> t.forward(100)
>>> t.color = 'red'
>>> t.right(90)
>>> t.forward(150)

As you type the lines up to and including t.color = 'green', you will see a window appear with a Turtle at the center facing east. As you type the other commands, the Turtle will change color, move, and draw lines.

Using a Pen

Objects of type Pen are very similar to Turtle objects, except that they draw a bit differently. You create a Pen just as you would a Turtle. At the interactive prompt try

>>> from introcs.turtle import Pen
>>> w = Window()
>>> p = Pen(w)

The pen icon does not look like a turtle. Instead, it looks like a diamond on its side with two different colors. With that said, this Pen object has a lot of attributes in common with a Turtle object. It draws from the tip on the left, and the left-side color is the drawing color.

However, the Pen does not have a heading attribute. Instead, for a Pen object p, you draw with the following methods.

Solid Shapes

The Pen also does not have a drawmode attribute. The four methods listed above either always draw (drawLine, drawTo, drawOval) or never draw (move). What the Pen does have is a solid attribute. When this attribute is True, the Pen will enter into a “solid mode”. Anything that is drawn between now and when the attribute becomes False (or when a call to move is made) will result in a solid shape.

For example, to draw a solid square, try the following sequence of commands with your Pen.

>>> p.fillcolor = 'blue'
>>> p.solid = True
>>> p.drawLine(0,50)
>>> p.drawLine(50,0)
>>> p.drawLine(0,-50)
>>> p.drawLine(-50,0)
>>> p.solid = False

When you finish, the pen will fill the insides of the square with the color blue.

Because the pen can draw solid shapes, it actually has two color attributes: fillcolor and edgecolor (there is no simple color attribute in Pen). The fillcolor is the color it uses inside a solid shape, and edgecolor is the color for hollow shapes as well as the border of solid shapes. When you look at the pen icon, the edge color is the color on the left and the fill color is the color on the right.

Assignment Source Code

Download the source code to this assignment before you do anything else. This time there are only two files. There is a module a4.py and a test script a4test.py. This time you only need to complete the file a4.py. The test script is provided to help you, but you do not need to add anything to it. You will not submit the test script and you will not be graded on your tests.

The module a4.py

The module a4.py contains all of the functions that you are to implement for this assignment. You will see that there are a lot of functions. That is because some are completed for you already and some are optional. See the instructions for more information on what is required.

All of the functions that you must implement take a Window object as an input and draw to that window. To test out one of these functions, navigate to the directory containing the file a4.py and start up the interactive shell. Then type:

>>> from introcs.turtle import Window
>>> w = Window()
>>> import a4
>>> a4.draw_two_lines(w,2)

This will draw two lines in the window w, at speed 2. Study the body of draw_two_lines, as it will help you with all of the tasks in this assignment.

Throughout this assignment, you will be writing procedures that draw shapes, much like draw_two_lines. As you write a procedure, refer constantly to the specification. Follow it carefully. If you have to call another procedure, look at its specification and make sure you follow it. A huge number of programming errors arise from not following specifications carefully.

The module a4test.py

Testing a graphical program is hard. You cannot automate tests with introcs.assert_equals. You have to let the turtle draw, look at the picture, and see if it is what you were expecting. This is why we have written the test script for you.

When you run the test script, it will pop up a window and ask you for a speed (we always like to use 0 or 10 to keep the tests moving). It will then procedure to draw pictures. After each picture it will wait until you type return in the Terminal window, giving you time to look at the picture.

This test script contains a test procedure for every function in the assignment (though function helpers are grouped together with their main functions). Each test procedure will draw a picture. If you are unsure if your picture is correct, post a screenshot on Ed Discussions and we will tell you. If your picture is right, you pass the test.

This test script is not guaranteed to be complete. Passing this script will not guarantee you a perfect on the assignment. However, you are not required to complete the script and you will not be graded on this file. This test script is simply provided as a convenience to make the assignment easier.

The turtle can take a long time to draw, so you may get tired of drawing the same pictures over and over. All of the test procedures are called by the master test procedure test_all at the bottom of the script. To disable a test, comment out the call to the relevant test procedure.

The procedure assert_error

If you look at the test procedures, you will notice that they actually do more than just draw a picture. They also have some calls to assert_equals. If you read the specifications to the drawing methods, you see that they are supposed to restore certain attributes (turtle position, heading, color) when they are done. These test cases check that this is happening properly.

There is also a call to a new procedure, assert_error. This is a tool to check whether or not a precondition is being enforced. For example, draw_two_lines must have a speed that is an int in the range 0..10. Therefore the call

a4.draw_two_lines(w,-1)

should crash, since the function enforces all preconditions. Since it is supposed to crash, this makes it a little difficult to test (because we do not want the test script to crash). Instead of calling the function ourselves, we get assert_error to call it for us. The first argument of assert_error is the function name we want to test and the remaining arguments are the arguments to use in that function.

To test that the code above crashes, we write

introcs.assert_error(a4.draw_two_lines,w,-1)

This procedure now does the opposite of draw_two_lines. It crashes if the function call does not crash and does nothing if it crashes. It will also crash if the function call does not crash correctly, meaning that it crashes with an error other than an AssertError (enforcing a precondition).

With this procedure, you can test that all of the preconditions are enforced. If you look at the test procedures in a4test.py, you can see that we have done this for all of the assignment functions. However, part of this assignment is to make sure that all preconditions are enforced. If we omitted an assertion in a4.py, then we also omitted its test in the script a4test.py. It is up to you if you want add these tests to a4test.py. Once again, you will not be graded on your tests.

Assignment Instructions

This assignment is broken up into five tasks. Each task corresponds to a procedure stub (or collection of stubs) in a4.py. You will find this assignment to be a lot easier if you complete and fully test one task before moving on to the next.

Once again, we do not require you to modify or even submit the file a4test.py. This test script is provided merely as a convenience.

Precondition Enforcement

As we saw in class, it is very helpful to assert your preconditions when you are using recursion or iteration. This keeps you from being caught in an (effectively) infinite loop.

Throughout the code in a4.py, we have placed assert statements in the various function stubs. However, we do not guarantee that they are enough. When you complete a function, we expect that you fully enforce your precondition with assert statements. If the provided assert statements do not fully enforce your precondition, then you must add more.

To help you with this process, we have provided you with several helper functions at the very top of a4.py. All of these helper functions return a boolean value: True or False. These helper functions are to be used inside of an assert to check part of a precondition, as shown throughout the code.

One of the main reasons we have provided you with all these helper functions is because the preconditions in this assignment can be quite complex.vIn particular, look at the function for is_valid_color(). There are many ways for a color to be valid. Using these functions allows us to simplify our assert statements a lot.

You will also notice that we have a helper function called report_error. In the past, we discovered that students are quite prone to make coding mistakes in their assert error messages (particularly adding a non-string to a string). This function is a nice way to make error messages that is fairly foolproof.

Task 1. Triangles

Complete the procedure draw_triangle. This procedure is given a Turtle as a parameter. In implementing the function do not need to make a new Turtle, nor a new Window.

This procedure should draw an equilateral triangle of side length s and color c using Turtle t. It should draw the triangle using the current position, orientation, and speed of t. The Turtle should end its drawing at the same position and orientation as when it started. Do not save the Turtle’s position and orientation at the beginning and then restore them at the end. If you draw the triangle correctly, following the instructions in the procedure specification, then this should happen automatically.

Remember to flush the Turtle at the very end of the procedure. If you do not do this, your Turtle will not draw anything when the speed is 0. However, only flush once. Do not flush after each line drawn.

To try out the procedure, type the following in interactive mode.

>>> from introcs.turtle import Window, Turtle
>>> import a4
>>> w = Window()
>>> t = Turtle(w)
>>> a4.draw_triangle(t,200,'green')

Task 2. Hexagons

Complete the procedure draw_hex. This method should draw six equilateral triangles using color 'cyan' with side lengths s. This triangles should form a hexagon, as illustrated to the right. Follow the specification and hints carefully.

A cyan Hexagon

In particular, be sure to use the helper function suggested. Do not try to repeat code already written. However, you should still remember to flush in this procedure after you restore the Turtle attributes, even though you already flushed in draw_triangle.

For both draw_triangle and draw_hex, it is very important that you follow the specifications. If you do not follow the specifications exactly, we will deduct points. Pay close attention what we say about the state of the Turtle. Did you make any changes to Turtle attributes that need to be changed back to what they were orginally?

Task 3. Radial Shapes

Choose two (and only two!) from the following three activities: spirals, polygons, and radiating petals. Once you have done two of these, you are free (but not required) to do the remaining one. These are pretty fun assignments. If you decide to do all three, we will grade you on the best two (though there is no extra credit beyond that).

Each of these tasks involves creating a helper procedure. In each case, the main procedure does not have a Turtle as parameter, but its helper procedure does. The main procedure clears the Window, creates a Turtle, calls the helper procedure to do the work, then hides the Turtle. Note that some of these procedures are very particular about which way that the newly created Turtle should start out facing. Remember that you can control the facing of your Turtle via the heading attribute.

When one of these procedures completes, you should flush the Turtle to handle speed 0 properly. However, you should only flush in the main procedure, and at the end. Never flush in the helper functions.

When writing these procedures, write the main procedure first, then the helper, and finally test both by calling the first one in python. If the main procedure is foo, its associated helper is called foo_helper. We have created stubs for all of these procedures in a4.py. Do not change the headers (either the names or the parameters), as our grading software will be calling them by those names. Just fill in the bodies.

Once again, it is very important that you follow the specifications for all procedures below. If you do not follow the specifications exactly, we will deduct points. This includes the helper procedures as well. We are not just grading the main procedures. For each problem we grade the main procedure and the helper procedure.

Spirals

The procedure draw_spiral draws a spiral of alternating colors. The pictures below show two different calls to draw_spiral. Both of them draw 10 lines with lengths 10, 20, 30, … In the first picture, the Turtle turns left 90 degrees after drawing each line. In the second picture the Turtle turns left 75 degrees after each line.

Turning 90 degrees	Turning 75 degrees

Complete the procedures draw_spiral and draw_spiral_helper. Pay close attention to how the lines grow at each step. Also pay close attention to how these change color. These are all import parts of the specification.

While there is a test in a4test.py, these tests are not complete. If you want to add your own tests, we recommend that you use 10 for the initial side length. Try different angles, like 90 degrees, 92 degrees, 88 degrees, and so on. You will be amazed at what these procedures do.

Here are some particular good tests to try out (after creating the Window w):

draw_spiral(w,  8,  90, 300, 10)
draw_spiral(w,  8, 135, 400, 10)
draw_spiral(w,  9,  60, 100, 10)
draw_spiral(w,  9, 121, 500, 10)
draw_spiral(w, 10,  89, 400, 10)
draw_spiral(w, 10, 150, 300, 10)
draw_spiral(w, 10,-144, 500, 10)

Polygons

The procedure multi_polygons draws one or more polygons on the screen, in alternating colors. Each polygon is drawn starting at the same place (within roundoff errors), but the turtle turns left 360.0/k degrees after each polygon.

The pictures below show two different calls to multi_polygons. The first is a single 10-sided polygon. The second image is series of 100 10-sided polygons, the first starting at angle -90, the second at an angle of 360.0/100-90, the third at an angle of 2*360.0/100-90, and so on. This demonstrates the kind of cool pictures you can draw just with polygons (for those of a certain age, this is very reminiscent of a Spirograph).

One 10-sided polygon	100 10-sided polygons

Complete the procedures multi_polygons and multi_polygons_helper so that your program can draw such designs. You should use the procedure draw_polygon, which we have provided, to draw the individual polygons (do not modify this procedure).

You should also pay attention to the color alternation. As you can see in the 100 polygon picture, we alternate the colors blue and orange. When your are finished, experiment to see what neat designs come out. Once again, relying on a4test.py is not enough.

Here are some particular good tests to try out (after creating the Window w):

multi_polygons(w, 45,  3, 100, 10)
multi_polygons(w, 60, 30,  20, 10)

Radiating Petals

The procedure radiate_petals draws a wheel of colorful petals. A petal is either open (a diamond) or closed (a straight line). These petals are arranged with equal angles between then, alternating between open and closed. If n petals are drawn, the angle between them is 360.0/n.

The pictures below show two different calls to radiate_petals. The first contains eight petals of the same length. The second contains 60 petals. Note that the color of each petal depends on the angle (i.e. the direction) of each petal. The angle of an open petal is the angle of the line that goes through its center.

8 petals	60 petals

Note that the Turtle color attribute will accept HSV objects (a variation of HSL from assignment 3. A petal drawn at angle ang uses the color HSV(ang, 1.0, 1.0). Just assign the object to the attribute and start drawing. This should make this part of the assignment fairly straightforward. Remember the invariants for an HSV object when you are drawing.

Complete the procedures radiate and radiate_helper. When finished, test them with small values of n, like 4 or 8. After the procedures are completely tested, try them with 360 petals of length 200. Also try 2000 lines with width 1 and Turtle speed 0 (which should be almost instantaneous if you wrote the procedure correctly), and notice how much more filled in the disk becomes.

Task 4. Recursive Fractals

In the next two tasks you will draw some fractals. A fractal is a shape that has parts which (when you zoom in) look like the whole shape. This suggests that you will need to use recursion to draw them. The number of recursive steps (or depth) determines the resolution of the fractal drawn. Wikipedia has a wealth of information about these and other fractals.

This time you are to choose one (and only one) from the following two shapes: the Sierpinski triangle or the Cantor stool. Once again you are free (but not required) to complete them both. If you do both of them, we will grade the best one.

Throughout both of these tasks, we ask that you use a Pen instead of a Turtle because (1) there is no need to maintain the direction and (2) Pen methods can draw solid shapes. See the overview of the Pen above for more information.

As with the radial shapes, for each of these recursive tasks, you will implement two procedures, a main procedure and a helper. The main procedure clears the Window and creates a new Pen. It also calls the helper to do the drawing, then cleans up afterward. The main procedure does not have a Pen as a parameter (though it does have the Window as a parameter), while the helper does. You should flush at the end of the main procedure, but not in the helper.

The helper is the function that does all the real drawing. It is the function that is supposed to call itself recursively. The main procedure is not recursive. This is why flushing in the helper is particularly bad.

Once again, ** it is very important that you follow the specifications for all three procedures below**. If you do not follow the specifications exactly, we will deduct points. Pay attention to when the Pen should and should not be visible.

Sierpinski Triangle

There are a lot of fractals named after Sierpinski. Some of you may have seen examples of this in your Calculus class, like the Sierpinski Carpet. The Sierpinski triangle is the same idea (removing a shape from the middle), except that it is built of triangles and not squares.

Below are Sierpinski triangles for depths 0, 1, 2, and 3 respectively. The width and height of the fractal is size. In depth 0, there is no recursive call and so the fractal is a simple square. In depth 1, there is a single recursive call, creating a cross out of five squares (the center and each arm). Each of these squares is one-third of the size of the fractal. In depth 2, the center and arms of the fractal are themselves fractals.

Depth 0	Depth 1	Depth 2	Depth 3

In general, we construct a Sierpinski triangle as shown below. In depth 0, there is no recursive call and so it is just an equilateral triangle whose sides are length side. In depth 1, there is a single recursive call and so break up that triangle into four smaller triangles and remove the middle one. At depth 2, we break up the remaining triangles, and so on.

When drawing your shape, do not “remove” the center triangle (e.g. draw a white triangle to erase it). Instead, you should just refuse to draw it. Except in the base case, only draw in the outer three triangles. Those triangles should be Sierpinski triangles of one less depth. This is illustrated below.

This fractal is slightly tricky because you have to place the subtriangles correctly. To do that, you need to understand how to compute the height of a equilateral triangle from its side. In particular, for an equilaterial triangle of length side, the height is side√3/2. Look at the image below for more information.

Height of an Equilateral Triangle

Note that when we talk about the center in this problem, we are talking about the middle of this line indicating this height. This is different from the rotational center of a equilateral triangle (e.g. the center of the circle inscribed inside the triangle). That is a much more complicated definition of center. Do not use that.

Rotational Center (DO NOT USE)

Instead, you should make sure that when you break your triangle into three parts, the top triangle should be side√3/4from the bottom as shown below.

Dividing up an Equilateral Triangle

We have stubbed in the procedures triangle and triangle_helper for you to complete. We have also provided a procedure fill_triangle, which you can use to draw a solid square. You should not modify fill_triangle (even though it is missing asserts).

Cantor Stool

The Cantor stool is a 2-dimensional version of the Cantor set. The legs of the stool are formed by the recursive calls beyond the base case. Below are the Cantor stools for depths 0, 1, 2, and 3 respectively. The width of each stool stool is side and the height of the stool is hght. In depth 0, there is no recursive call and so the stool is a simple rectangle. In depth 1, there is a single recursive call, and the stool was cut in half vertically. The horizontally, the legs take up the first and last third of the width. In depth 2, the legs of the stool are themselves stools.

Depth 0	Depth 1	Depth 2	Depth 3

In general, we construct a Cantor stool as shown below. In the base case, we draw a simple rectangle. For later depths, divide the rectangle in half vertically, drawing a rectangle with width size and height hght/2. For the legs, draw stools of one less depth whose width is size/3. You will need to reposition the stools so that they line up with the edges of the top rectangle.

We have stubbed in the procedures cantor and cantor_helper for you to complete. We also have provided a procedure fill_rect, which you can use to draw a solid hexagon. You should not modify fill_rect (even though it is missing asserts).

Task 5. Minkowski Island

The Minkowski island is different from the shapes in Task 4 in that it is a line drawing and not a solid shape. The basic shape is a simple square. Later shapes embellish the edges to add cool crystalline effects.

Depth 0	Depth 1	Depth 2	Depth 3

The important thing to understand about the Minkowski island is that the recursion is not applied to the square. It is applied to the edges. The basic edge is a straight line. Later shapes break the edge into four parts and put a “bump” in the middle half (using six edges instead of three). These eight edges are themselves Minkowski edges of one less depth.

Depth 0	Depth 1	Depth 2	Depth 3

Because this is a line drawing, you will be using the Turtle once again. Using the Turtle to draw a recursive shape can be quite tricky because you have to handle the orientation correctly. In the diagram below, we assume that the Turtle is drawing the edge west-to-east. However, if you code the turns correctly, it will work no matter what the initial orientation of the turtle is.

At each step, the Turtle draws an edge of one less depth. The Turtle first draws an edge for one quarter of the side length (Step 1). Once it reaches a quarter of the way, it turns right to start the bump and draws another edge of a quarter of the side length (Step 2). It turns back to the left 90 degrees and draws another edge of one quarter the side length (Step 3). It then turns left and draws two edges in a row of one quarter length (Steps 4 and 5). It then turns right to draw one quarter length (Step 6) and left again, repeating the process (Step 7). Finally, it turns left and draws one last edge of one quarter length (Step 8).

To combine the edges together, we start the Turtle in the bottom right corner of the square. We draw an edge and turn left four times, creating the shape seen below.

This composition step is not recursive, so it belongs in the main procedure and not the helper function. We have stubbed in the procedures island and island_edge for you to complete. You will notice that, while island assumes the Turtle starts at the bottom facing up, island_edge does not. That is because island_edge must be able to draw the edge in several different orientations.

If you are unsure of how to approach this function, look at the Hilbert curve example from class. This example uses a Turtle to recursively draw a shape in much the same way.

One of the interesting features of Minkowski islands is that they are particularly stable. At some point the shapes all look the same, because the changes are at smaller and smaller steps – eventually too small to show on your screen. For example, the pictures below are for depth 5 and depth 7, and they look very similar. However, Depth 5 only took a few seconds, while Depth 7 took well over 15 minutes to draw at speed 0 (Each depth that you add more than doubles the drawing time for the Turtle).

Depth 5	Depth 7

Finishing Touches

Before you submit this assignment, you should be sure that everything is working and polished. Unlike the first assignment, you only get one submission for this assignment. If you make a mistake, you will not get an opportunity to correct it. With that said, you may submit multiple times before the due date. We will grade the most recent version submitted.

Once you have everything working you should go back and make sure that your program meets the class coding conventions. In particular, you should check that the following are all true:

• You have indented with spaces, not tabs (Pulsar handles this automatically).
• Functions are each separated by two blank lines.
• Lines are short enough (~80 characters) that horizontal scrolling is not necessary.
• Docstrings are only used for specifications, not general comments.
• Your name(s) and netid(s) are in the comments at the top of the modules.

Upload only the file a4.py to CMS by the due date: Thursday, October 31. We do not need any other files. In particular, we do not want the file a4test.py.

Completing the Survey

In addition to turning in the assignment, we ask that you complete the survey posted in CMS. Once again, the surveys will ask about things such as how long you spent on the assignment, your impression of the difficulty, and what could be done to improve it. Please try to complete the survey within a day of turning in this assignment. Remember that participation in surveys is 1% of your final grade.