Point Reflection in Transformational Geometry
Let A be a given point in the plane, and let P be another point in the plane. We construct the point P the point reflection of P with center A, as follows: let P be the point on line AP distinct from P with |AP| = |AP|. (Special case: if P=A, then define P= A also.) In other words, A is the midpoint of PP'.
This defines a function or transformation H_A that takes P to P. This transformation is also sometimes called point reflection or point symmetry. This transformation is the same as a half-turn, a rotation by 180 degrees.
1.1 Point Reflection Images using Sketchpad
In this first Sketchpad exercise we explore several important properties of point reflections and then solve a problem using these properties. The properties are:
Experiment 1A. Image of 2 points.
Experiment 1B. Image of lines, circles, etc.
Suppose S is a set of points in the plane, and F is a transformation of the plane. The F-image S of set S (also written as F(S)) is the set of points P for which P = F(P) for a point P in S. Working with the image of a set rather than a point at a time is a very powerful way to understand and use a transformation.
In this experiment we observe that when a point moves along a geometrical object, how its images moves along the image of the object.
What happens if you animate D along the segment B'C' instead of BC?
A figure F has A as a center of Point Symmetry if the half-turn H_A is a symmetry of F. This means that the H_A image of F is F itself. Consider and explain these examples:
1.2 Solving a Problem with a Point reflection
Here is a curious geometry problem.
Version 1. Given two lines m and n and a point A, find a point M on m and a point N on n so that A is the midpoint of MN.
It may not be clear how to attack this problem, but point reflections give an approach to the problem. It can be reformulated thus:
Version 2. Given two lines m and n and a point A, find a point M on m and a point N on n so that N is the point reflection of M with center A.
Finally, there is a third version:
Version 3. Given two lines m and n and a point A, find a point M on m and on the image of n by the point reflection with center A.
2. Composition of Point reflections
One of the powerful concepts about transformations is the concept of composition. We can get some interesting geometric figures by composing point reflections.
This experiment explains composition of point reflections in terms of elementary geometry.
This figure would usually be constructed from a triangle and its midpoints, but instead, construct the figure in a special way.
Start with segment AB and point C.
If H_A is the point reflection with center A and H_B is the point reflection with center B, then C'' = H_B(C') = H_B(H_A(C)). If we ignore the intermediate step and move straight from C to C'', we can think of this as a single transformation. This transformation T is the composition of the two point reflections. We denote this as a product. This T = H_B H_A, and T(C) = C''.
Experiment 2C. Translations as compositions
Sketchpad can create transformations that are compositions.
The Moral of this Story is that we can get exactly the same transformation 3 different ways.
3. Exploring Dynamic Quadrilateral Midpoint figures
One can learn a lot by finding relationships in figures, but some relationships that may be difficult to see in a static figure jump out when the figure moves. Such figures are possible with dynamic software for geometry such as The Geometer's Sketchpad.
Experiment 3A. From quadrilateral to midpoints
We have proved before that this midpoint figure is a parallelogram. Now let's construct ABCD from the midpoint figure.
Experiment 3B. From midpoints to quadrilateral
If you see the connection between the quadrilateral figure and the triangle midpoint figure, it may not surprise you that there is a way to think about this figure using point reflections.
Related Problem: Construct a quadrilateral with vertex P given the midpoint parallelogram ABCD. Explain why there is a solution on if ABCD is a parallelogram and only one solution.
4. Quadrilateral Tessellations and Midpoints
We say a shape tessellates the plane, if congruent copies of the shape can be laid down edge to edge as tiles cover a floor. We know some common shapes tile the plane. For example, equilateral triangles tile the plane; also squares or rectangles can be used. In fact general parallelograms tile the plane like this.
Experiment 4A: Sketchpad Construction of Parallelogram Tessellation
Experiment 4B: Hands-on Experiment with General Quadrilateral Tessellations
This shows that special quadrilaterals can tile the plane. But what if we take as a tile a "random" quadrilateral with no special shape. Can we lay down tiles of this shape to cover the plane?
A geometrical fact that may seem surprising is that given any quadrilateral, the plane can be tessellated, or tiled, by congruent copies of that quadrilateral.
Cut out a set of congruent quadrilaterals like this one, or one of your own choosing.
Assemble them like a puzzle to cover (tessellate) the plane. Do this! Then note carefully that adjoining shapes are can be moved one onto the other by an isometry. What isometry is this? (Save the figure by taping it to paper or tracing around the shapes onto paper.
Try another example. This time take a shape that is not convex and tile the plane with it. Answer the same question about how neighboring tiles are related by isometries.
Experiment 4C: Sketchpad Tessellation by General Quadrilaterals
In a new sketch, draw a quadrilateral ABCD (no special shape).
Create a tessellation with Sketchpad like the one you did by hand. An important feature of the Sketchpad construction is that if you drag any one of A, B, C or D to change the shape of the original quadrilateral, the shape of ALL the quadrilaterals will change so that all are congruent. And the tiles will still cover the plane. To do this you will need to use transformations. Save this figure!
Experiment 4C continued: The Midpoint Connection.
Form the midpoint parallelograms of the quadrilaterals and observe how they fit together. How are these parallelograms related to the translations that carry certain quadrilaterals to others? Discuss everything in the light of what you know about composition of point reflections.
When the quadrilaterals are put together to tile the plane, the midpoint parallelograms fit together to give a tiling by parallelograms. The sides of the parallelograms are half the length and direction of the translations that take the tiling to itself.
Experiment 4D: From parallelograms to general quadrilaterals
Return to your sketch with tessellation by parallelograms. Place a point P inside one of the parallelograms with vertices EFGH. Reflect P in each of the vertices E, F, G, and H and connect P to each reflection image point by a segment.
Now each of the images P' of P are also in a parallelogram. Reflect each of these points in the vertices of their parallelograms and each point to its images.
If you keep repeating this constructing and connecting points and their images, you will soon find you have formed a the same figure as in 4C, but this time starting with the parallelograms rather than with the general quadrilateral!
Question: If you imagine the quadrilateral tessellation pattern in 4D or 4C continuing on over the whole plane, what are the symmetries of this infinite figure?