## Description

Checkpoint 1

Attempts to define the formal foundations of mathematics at the start of the 20th century

depended heavily on the mathematical notion of recursion. While ultimately completing

this formalization was shown to be impossible, the theory that was developed contributed

heavily to the formal basis of computer science.

The idea is to start with primitive, axiomatic definitions and build from there. We will build

basic arithmetic starting with just one value — 0 — and three operations: (1) adding 1 to a

value (the successor operation), (2) subtracing 1 from a value (the predecessor operation),

and (3) testing a value to see if it is 0. In this case, the value 1 is the successor of 0, the

value 2 is the successor of the successfor of 0, etc.

Using just these operations, we can implement addition of positive integers using a Python

recursive function:

def add(m,n):

if n == 0:

return m

else:

return add(m,n-1) + 1

To make sure you understand this, show the sequence of calls made by

print add(5,3)

You may add print statements to show the results.

Building on this idea, please write a recursive function to multiply two non-negative integers

using only the add function we’ve just defined, together with +1, -1 and the equality with 0

test. Call this function mult. Demonstrate the result by multiplying 8 and 3. As a recursive

function, mult call itself of course.

Now, define the integer power function, power(x, n) = x

n

, in terms of the mult function

you just wrote, together with +1, -1, and equality. Demonstrate the result by computing

power(6,4).

To complete Checkpoint 1: Show:

1. the calls from add(5,3),

2. a working version of mult, and

3. a working version of power.

Checkpoint 2

In this section, you will write a recursive function to draw a self repeating plus sign. You

are given the code — lab12_check2_start.py — to draw the first plus sign in the middle

of the canvas. First, remember that (0,0) is the upper left corner, and (600,600) is the

lower right corner.

At each iteration, your code will draw the same pattern at the four end points of the current

plus sign and then reduce the length of the sign to half of its current value. The expected

figure at iterations 0, 1, 2 and a much higher level is shown below.

When you start, your origin is at location (300,300) and original length is given as 150 on

each side. You first draw a plus sign by drawing two lines, between:

(150,300)-(450,300) (horizontal line, total length 300)

and

(300,150)-(300,450) (vertical line, total length 300)

Now, you must start from the end points of this plus sign and draw four new plus signs

(recursively) of length 75 at on each side (the center of these four new plus signs will be:

(150,300), (450,300), (300,150), and (300,450).)

Each time you call the function recursively, the length will decrease by half. Think of a

good stopping condition for your recursion. May be a lower limit on the length would be a

good choice.

To solve this, you are going to follow the same pattern we have used for the Sierpinski

triangle in class.

To complete Checkpoint 2: Show your working program.

Congratulations! You have completed all the labs in the course!

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Some extra things to try for fun!

You can change a few things in your program for Checkpoint 2 and get different designs.

First, try drawing the lines not at the end of the plus sign, but 1/4 way in from the end

points. For example, the second lines will have center locations at follows:

(300,375), (375,300), (225,300), (300,225)

(basically length/2 away from the center in all four dimensions.) If you repeat this process,

you will get a different pattern, as you can see below:

You can use the same code for the original checkpoint, but simply change the center locations. You may want to keep two versions of the code or include an if statement to toggle

between the two. We will call this new version the rectange version.

For both the rectangle and the plus version, the new lines don’t overlap with the old ones

because we keep dividing the length by half and moving the center by the same amount

at least. You can change this as well and change the length by a different fraction at each

step. Try setting the length at each iteration to:

length = 2*length/3

length = 3*length/5

instead of length/2. You will see that as patterns overlap, you start seeing some very

interesting Moir´e patterns:

http://en.wikipedia.org/wiki/Moire_pattern

Here are some example patterns you might get (for 2*length/3 for plus and rectangle versions):

Now, try changing the length to 3/2 of its current value every third iteration and halving it

at other iterations. Try shifting the starting values a little every time and see new patterns

emerge.

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