10.2. Teacher Note: Turtle Geometry

The turtle is actually useful for exploring a wide variety of ideas in geometry. The book Turtle Geometry does a wonderful job of showing how turtles can be used to explore a wide variety of geometric, mathematical, and scientific ideas (e.g., using turtles to model insect behavior). The example pattern below is drawn from that book.

Figure 1: A scan of Mark’s copy of the book

10.3. Total Turtle Trip Theorem

That last piece of code is actually a pattern for a wide variety of geometric shapes. Here’s a triangle. It may not be obvious why we turn 120 in this program, but it will be soon.

from turtle import * # use the turtle library space = Screen() # create a turtle space avery = Turtle() # create a turtle named avery avery.setheading(90) # point due north for sides in range(3): # repeat the indented lines 3 times avery.forward(100) # move forward by 100 units avery.right(120) # turn by 120 degrees

And here’s a pentagon.

from turtle import * # use the turtle library space = Screen() # create a turtle space will = Turtle() # create a turtle named will will.setheading(90) # point due north for sides in range(5): # repeat the indented lines 5 times will.forward(100) # move forward by 100 units will.right(72) # turn by 72 degrees

The Total Turtle Trip Theorem states that the turtle will draw a closed figure with n sides when the sum of the angles turned is a multiple of 360. In the triangle example 3 * 120 = 360 and in the pentagon example 5 * 72 = 360.

Change the ?? in line 7 below to the amount to turn each time to draw a 12 sided polygon, which is called a dodecagon. If you get it right the turtle will draw a 12 sided closed polygon.

from turtle import * # use the turtle library space = Screen() # create a turtle space mia = Turtle() # create a turtle named maria mia.setheading(90) # point due north for sides in range(12): # repeat the indented lines 12 times mia.forward(40) # move forward by 40 units mia.right(??) # change ?? to the amount to turn

How much does mia need turn in the program above to create a closed dodecagon (12-sided figure)? Only one of these works.

• 15
• This one will not close
• 30
• Exactly! 12 * 30 = 360
• 12
• No, 12 * 12 is 144, which is not a multiple of 360
• 90
• This one will generate a square, three times. 12 * 90 = 1080 = 360 * 3

The following program uses a turtle to draw a triangle as shown below, but the lines are mixed up. The program should do all necessary set-up and create the turtle. After that, iterate (loop) 3 times, and each time through the loop the turtle should go forward 100 pixels, and then turn left 120 degrees. Drag the needed blocks of statements from the left column to the right column and put them in the right order with the correct indention. There may be additional blocks that are not needed in a correct solution. Click on Check Me to see if you are right. You will be told if any of the lines are in the wrong order or are the wrong blocks.

        from turtle import *
---
space = Screen()
---
space = screen() #paired
---
marie = Turtle()
---
# repeat 3 times
for i in range(3):
---
# repeat 3 times
for i in range(3) #paired
---
    marie.forward(100)
---
    marie.forward(100 #paired
---
    marie.left(120)
---
    marie.turn(120) #paired
        

Write a procedure which takes in a turtle and the number of sides as parameters to draw a polygon. Call the function to test it.

# DEFINE THE PROCEDURE def polygon(turtle, sides): degree = 360 / sides for side in range(sides): mia.forward(40) mia.right(degree) # CREATE THE TURTLE WORLD from turtle import * space = Screen() mia = Turtle() mia.setheading(90) # CALL THE PROCEDURE polygon(mia,5)

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