Transformations: Motion Geometry
We have seen that geometric figures have symmetry properties such as rotational and reflection symmetry. Now we define these properties in terms of mappings or functions called transformations .
Previously we investigated symmetry properties of geometric figures by means of motions that make the figure coincide with itself. Now we investigate correspondences between points in the plane. A transformation is a one-to-one correspondence between points in the plane such that each point P is associated with a unique point P', called the image of P.
Transformations that preserve size and shape are called isometries or rigid motions . We will study three isometries: translations, reflections and rotations.
Translations : Informally, a translation is the movement of a point or figure by a sliding motion. More rigorously, a translation is a motion of the plane that moves every point of the plane a specified distance in a specified direction along a straight line.
There are two ways to specify a translation. One is by a directed line segment or vector .
In the diagram the dotted line has both distance and direction and is called a vector. The original triangle and its image is shown translated by this vector.
Alternatively, we can describe a translation on a coordinate grid. The triangle in the first figure is displaced 5 units horizontally and 3 units vertically. This would be the vector (5,
3) and a mapping that would describe this translation would be ( , )

( x

5, y

3) .
The quadrilateral in the second figure is displaced -5 units horizontally, and -4 units vertically. This would be the vector (-5, -4) and a mapping that would describe this translation would be: ( , )

( x

5, y

4) .

Rotations : Informally, a rotation is turning a figure about a fixed point. More rigorously, a rotation is a transformation of the plane determined by holding one point called the center fixed, and rotating the palne about this point by a fixed amount in a certain direction. Positive rotations are anti-clockwise. The figure shows three objects being rotated 180 degrees about point O.
Reflections : Informally, a reflection is a mirror image. More rigorously, A reflection in a line l is a transformation of the plane that pairs each point P of the plane with a point P' in such a way that l is the perpendicular bisector of PP ' , as long as P is not on l . If P is on l , the P = P'.
The figure shows the image of P on l and the image of triangle ABC on l .
Tessellations:

An arrangement of shapes that can completely cover a plane without gaps or overlaps is called a tessellation. For shapes to fill the plane without gaps or overlaps, their angles, when arranged around a point, must have measures that add up to 360 degrees. If the sum is less than 360 degrees there will be a gap. Regular pentagons do not tessellate the plane since 3(108) = 324 degrees and 4(108) = 432 degrees.
Investigation: Do all triangles tessellate?
1.
Make 12 congruent scalene triangles and use them to try to create a tessellation.
2.
Look at the angles about each vertex point. What do you notice?
3.
What is the sum of the measures of the three angles of a triangle? What is the sum of the measures of the angles that fit around a point?
4.
Compare your results with others and state a conjecture.
Investigation: Do all quadrilaterals tessellate?
1.
Cut out 12 congruent quadrilaterals. Label the corresponding angles in each quadrilateral a, b, c and d.
2.
Using you 12 congruent quadrilaterals, try to create a tessellation.
3.
Note the angles about each vertex point. How many times does each angle of your quadrilateral fit at each point? What is the sum of the measures of the angles of a quadrilateral? Compare your results with others and state a conjecture.
Tessellations Using Only Translations:
The four steps below show how Escher may have created his Pegasus tessellation. Notice how it uses only translations and starts with a square.

You can also use the translation technique with regular hexagons. The only difference is that there are three sets of opposite sides on a hexagon. So you will need to draw three sets of curves and translate them to the opposite sides. The six steps below show how student Mark Purcell created his tessellation, Monster Mix .

Tessellations Using Only Reflections:
In this Escher print each reptile is made by rotating three different curves about three alternating vertices of a regular hexagon.

Another method used by Escher utilizes rotations on an equilateral triangle grid. Two sides of each equilateral triangle have the same curve, rotated about their common point.
The third side is a curve with a point of symmetry. The following steps show how you might create a tessellating flying fish like that created by Escher.

Or with a slight variation in the curves, the resulting shape will appear more like a bird than a flying fish.
Tessellations Using Glide Reflections:
In this drawing by Escher some birds are facing in one direction and others in the opposite direction. This is accomplished by a combination of translation (gliding) and reflection.

Notice how in step 2 the curve is first translated to the opposite side and then reflected.