Geometry Explorer Help

Exploring Turtle Geometry:

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Turtle geometry was created as part of the development of the LOGO programming language. LOGO was designed in part to give children a relatively easy way to program a computer. In turtle geometry one imagines a small turtle on the computer screen. (The little green guy at point F in the picture above) The turtle understands commands like move forward, turn left, turn right, change color, among others.

Turtle Geometry is extremely useful in drawing simple shapes quickly, and also in creating fractals of almost any type. Look here for an example on how to use turtle geometry and grammar re-writing to generate the Koch Curve. (Note: This example is at a fairly advanced level and assumes knowledge of the turtle geometry basics found on this page.)

Turtles in Geometry Explorer can understand these basic commands:

One can also change the drawing color of the turtle and use color tables to create multi-colored turtle designs.  Look here for an example of a multi-colored regular 17-gon.

The turtle starts out with a specified heading and length. The heading is the direction in which the turtle will move. The length  is how far the turtle should move when told to go forward or backward. The heading and length are given by a vector, that is a pair of points.  The vector's length is just the distance between the points and the vector's  heading is given by an arrow from the first point towards the second.

A turtle must also know what angle to turn by when it turns. This is specified by a set of three points: the initial, vertex, and terminal points of an angle.

As an example, let's use a turtle to construct a regular pentagon.  First, we need to define a turtle. As described above we need to define a vector and angle. The angle that will be needed for our pentagon is one of 72 degrees. To construct a 72 degree angle first construct a segment AB. Then select A and set it as a center of rotation by choosing "Center'' under the Mark menu in the Transform Panel. Then, click on "Rotation'' under the Custom menu in the Transform Panel and type in 72 for the angle and hit the Okay button. Then, select point B and hit the rotation button in the Transform Panel. We now have a 72 degree angle BAC.

We need to store this angle for future use.  To store the angle select B, A, and C (in that order)  and choose "Turtle Turn Angle'' under the Turtle menu in the main window.  We next construct a segment DE that will serve as a vector for our turtle.  Create a segment DE near angle BAC and select D and E (in that order) and then choose "Turtle Heading Vector'' under the Turtle menu.

We have now stored an angle and vector for a turtle. To create the turtle, however, we need to specify a position, i.e. a point, at which the turtle will start at. Create a point F anywhere on the canvas and select it. At this point the "Create Turtle At Point'' menu under the Turtle menu will be active.  Once we choose this menu item a small green turtle will be created at point F.

Also, a Turtle Controller window will pop up. We will use just the section of this window labeled "Simple Turtle''. (For more information on turtles and the use of the Controller window see the chapter on Turtle Geometry in the pdf User Guide.)  The "Simple Turtle" section has a set of seven buttons that we can use to control the movement of our turtle.

turtle control

We are now ready to construct our pentagon. We do this by carrying out a sequence of five Draw Forward -- Turn Left ("<--'') actions. Carry out these actions by pressing the Draw Forward button followed by the Turn Left button five times, getting the pentagon shown below.

Suppose that we move point D. By lengthening the distance from D to E we lengthen all of the turtle movements. Also, by changing the heading of DE we change the orientation of the pentagon drawn by the turtle.  However, moving point A does not change angle BAC and thus has no effect on the pentagon.

The regular pentagon has many fascinating properties. One of these has to do with the ratio of a diagonal of the pentagon to a side. Select points I,G, and H as shown and choose "Ratio'' from the Measure menu. This ratio is called the golden ratio. This ratio pops up in many surprising places in nature.