MIRROR, MIRROR, ON THE WALL
An Activity Using Geometer’s Sketchpad Software
(You will get an overview of the activity but Geometer’s Sketchpad software
is needed to complete the assignment)
Felix Klein (look him up!) was appointed to a professorship at the German university of Erlangen in 1872. As was customary at the time, the newly appointed professor delivered an address to the faculty, in which he discussed the state of the art of his discipline, and his indications for the future. Klein’s address contained the first formal formulation of what is considered today the true essence of geometry.
Here are Klein's words:
As a generalization of Geometry, the following comprehensive problem arises:
Given a variety and a group of transformations on it, study the shapes belonging to the variety for what concerns those properties which are not altered by the transformations of the given group.
What Klein is saying is essentially the following.
- Choose the place (variety) where you want to do geometry: the plane, the sphere, three-dimensional space…
- Choose the rules of the game. Choose what kinds of transformations are allowed in your variety to move shapes around. For example, you might decide you want your transformations not to alter distances, or you might want your transformations just not to alter angles, or you might want your transformation just to be continuous…
- You are ready to play geometry. Pick different shapes, move them around your variety, and look for properties of your shapes that do NOT change by the action of the transformations.
We are now going to play Euclidean Geometry in the plane. Our variety is the plane and our transformations are the so-called isometries.
An isometry is a transformation from the plane to the plane that preserves distances.
Basic examples of isometries are: reflections, rotations, and translations.
Reflection
A reflection across a line l is a transformation that takes a point P to a point P’, so that l is the perpendicular bisector of the segment PP’.
Reflection with Sketchpad:
STEP 1: Construct a line (in Sketchpad this can be a line segment, an infinite line, or a ray; it does not matter.) To tell Sketchpad that you want this line to be your mirror of reflection, select it and pull down the Transform menu. Highlight the Mark Mirror option. You can achieve the same by double clicking on the line. Sketchpad lets you know that it understood your intentions by flashing the selection squares on the line.
STEP 2: Now create an object, any object, in your sketch. Select it, go to the Transform menu and select Reflect. Sketchpad will reflect all the items that were selected in your sketch.
STEP 3: Take a minute to observe how Sketchpad keeps track of labels in a reflection. As usual, if you move either the mirror or the object you reflected, Sketchpad will adjust the whole sketch accordingly.
Rotation.
A rotation around a point C (called the center of rotation), by an oriented angle a, is a transformation that takes a point P to a point P’ so that CP is the same length as CP’ and the oriented angle Ð PCP’ has the same size and orientation as a.
Rotation with Sketchpad
STEP 1: Create a point. Label it C. To tell Sketchpad that you want this point to be your center of rotation, select it and pull down the Transform menu. Highlight the Mark Center option.
There are two ways to tell Sketchpad which oriented angle you want to use as your angle of rotation:
i) Through a dialog box.
ii) Creating an angle in your sketch.
STEP 2 shows you how to use the dialog box. STEP 3 shows you the more dynamic way of creating a modifiable angle.
STEP 2: Create an object in your sketch and select it. In the Transform menu select Rotate. You will be prompted with a dialog box in which you can specify the angle of rotation. Sketchpad follows the usual trigonometry convention of considering positive angles as oriented counterclockwise and negative angles as oriented clockwise. When you click on OK, Sketchpad will perform the rotation for you.
STEP 3: Create an angle with three points either free in the sketch or on objects, it does not matter. Select the angle by selecting the three points in the following order: point on the initial side, vertex, point on the terminal side. Be careful that the order in which you select the points changes the angle and the angle orientation. If you want to use the angle Ð PQR, oriented counterclockwise, you MUST select the three points in the order P, Q, R. Selecting R, Q, P will give you the same angle but oriented clockwise.
STEP 4: Under the Transform menu select Mark Angle. Notice that Sketchpad lets you know that it understood your intentions by flashing an arc through the angle, showing you which angle and with which orientation it is looking at.
STEP 5: Create an object and select it. Under the Transform menu, choose Rotate. In the dialog box now you will see the option by Marked Angle automatically selected. Choose OK (or hit return) and watch Sketchpad rotating your object.
STEP 6: Exploit fully the dynamic nature of Sketchpad by playing with your angle.
Translation
A translation given by a vector v is a transformation that takes a point P to a point P’ so that the vector with tail at P and tip at P’ is equal to v (it has the same length, slope, and orientation).
Translation with Sketchpad
STEP 1: Create a vector by creating a line segment or just two points. Label the vertices of the segment or your two points. To tell Sketchpad that this is the vector you want to use to translate, select the vertices of your segment or your two points. Be careful that the order in which you select the points is read by Sketchpad as tail, tip for your vector, so a different order changes the direction of the vector. Go under the Transform menu and select Mark Vector. Sketchpad will flash the vector, from tail to tip, to confirm your intentions.
STEP 2: Create an object and select it. Go under the Transform menu and select Translate. The dialog box will propose by default to translate by the vector you have marked in STEP 1. Click on OK or hit return.
STEP 3: Exploit fully the dynamic nature of Sketchpad by playing with your vector.
PROBLEM 1: What kind of isometry do I get if I perform two consecutive reflections?
Think about the possible mutual positions of two lines in the plane.
Experiment with Sketchpad. Talk to your neighbor.
We suggest that you use triangles or quadrilaterals as objects to reflect. Concave quadrilaterals are particularly effective since they easily reveal changes in orientation. Reflecting a single point or two points will NOT be sufficient.
When you have a clear conjecture, write it down as concisely as you can.
PROBLEM 2: Suppose you are reflecting consecutively across two mirrors that meet at a point. From Problem 1, you believe that the resulting isometry is a rotation. Can you find the angle of rotation and the center of rotation knowing only where the mirrors are?
PROBLEM 3: What happens if you pivot your two mirrors in Problem 2 around their point of intersection, keeping the angle between them fixed? Does the final resulting isometry change? Why ?
To explore problem 3 with Sketchpad follow the steps below:
STEP 1: Mark the point of intersection of the two mirrors as center by double clicking on it.
STEP 2: Hold down the mouse button while you are clicking on the Select and Translate (Arrow) tool on the tool palette. A sub-menu will spring out to the right. Select the Rotate tool.
STEP 3: Select both mirrors in your sketch and drag them around their point of intersection.
PROBLEM 4: Suppose you are reflecting across two parallel mirrors. You believe, from Problem 1, that the resulting isometry is a translation. Can you find the vector (its slope, direction and length) knowing the position of the mirrors ?
PROBLEM 5: What happens if you move your two mirrors in Problem 4 around the plane, keeping the distance between them fixed? Does the final resulting isometry change? Why?
To explore Problem 5 with Sketchpad follow the steps below:
STEP 1: Make sure that the Select and Translate tool is selected at the top of the tool palette.
STEP 2: Select both mirrors in your sketch and drag them around the plane.