GEOMETRY AND TRANSFORMATION
Geometry in the art of M.C. Escher
1. TRANSLATION

A transformation is the movement of the "position" of a shape on the plane. To be more specific, a transformation maintains the form, size and dimensions (distances between points and angle measurements) of the shape exactly. There are three different types of transformation: translation, rotation and symmetry (or reflection)
1.- We can think of a translation as being the movement of a shape from one position to another. Think of different situations in everyday life where you might see translations, e.g. opening a drawer or a car driving along a straight road.

2.- Try and deduce what exactly determines a translation. In other words, what information do we need to know to work out the final position of a shape if we know its initial position.

A vector is a segment which indicates direction and magnitude (length).

3.- Work out the final position of the yellow triangle after it is translated according to the information given by the red and blue vectors.

The show parameter (increase its value to one) allows you to see the final shape after the translation (referred to as homologous).

Change the arrow by dragging either the head or the end point or both.

4.- What is the change in position of any of the points on the shape, in this case the yellow triangle, after the translation?

If you alter the option parameter (increase its value to one) you will see what happens to the three vertices.

5.- If you know the coordinates of the vector that represents the translation you can work out the equations governing the translation. Another way of doing it is to see how point  P, with known coordinates (p0, p1) undergoes a translation represented by the vector v with known coordinates (a,b). What would the coordinates of the translated point P' be whose unknown coordinates are (x,y)?
2. ROTATION

Our intuitive concept of rotation is present in many everyday activities, e.g. the movement made by the hands of a clock or by a door handle when we open the door.

6.- Think of other everyday situations where we can find examples of rotation.

7.- Develop this idea to deduce the basic characteristics that determine rotation.

8.- Look carefully at what happens to a shape (the yellow triangle) when it is rotated.

Rotation is a transformation which requires a centre to rotate around and an angle of rotation. The orange point is the centre of rotation, which can be moved by dragging it. The angle of rotation can be altered either by using the red and blue arrows or by writing in the new angle directly and pressing the Enter key.

9.- Explain why a point and its corresponding point after it has undergone the rotation are found at an equal distance from the centre of rotation and why when they are joined to the centre form an angle which is equal to the angle of rotation.

Change the option parameter (increase it to 1) to see the results.
3. SYMMETRY OR REFLECTION

A reflection about the straight line e transforms point P on the plane to point P' on the plane so that:

  1. Segment PP' is perpendicular to line e
  2. The distance from point P to axis e is equal to the distance from point P' to the same axis.

10.- Work out where the reflection of the yellow triangle would be in the window.

Increase the show parameter to 1 to see the solution.

11.- Try it a few times with different axes.

If the line isn't vertical you can change it by entering the gradient value and the y-coordinate value when the x-coordinate is 0.

If you want to make the axis vertical then change the vertical line parameter to 1.

The k parameter determines the vertical line x=k and only works if the vertical line is activated

12.- Give examples of situations and physical facts which are governed by symmetry, e.g. think about different anatomical shapes, buildings etc.
Enrique Martínez Arcos
© Spanish Ministry of Education and Science. Year 2001