Symmetries and Isometries

The concepts of symmetry and isometry are central to the study of geometry. An isometry is a distance preserving map from some space it itself: a rigid motion. For example, f(x)=x+5 is a isometry of the real line; the whole line is shifted by 5 and distances between points remain unchanged. A symmetry of a figure in some space is an isometry of that space which maps the figure to itself. If this seems unclear, do not despair! This whole section is devoted to understanding symmetries and isometries.

Activity 1: Isometries of the line revealed
  1. Shifting the real line by any constant, as in the f(x)=x+5 example above is a perfectly fine isometry: it maps the real line to itself and preserves distances. Is stretching by a constant factor, say f(x)=2x an isometry? (Hint: for any two real numbers x and y, is f(x)-f(y)=x-y?)
  2. What about a reflection? That is, f(x)=-x. Is this an isometry?
It's a theorem that the only isometries of the line are translations and reflection. Let's prove it together!
  1. On the real line below 1 (red) is mapped by some unknown isometry f to 4 (green). What can f map 2 (purple) to? (Hint: recall that an isometry must preserve distances.)
  2. Suppose that f(1)=4 and f(2)=3. What is f(3) and why? Write down an equation for this isometry. (Hint: the form is f(x)=ax+b for some a and b.)
  3. What if instead, while f(1)=4, we have f(2)=5. What is f(3)? What is the isometry?
QED! real line

Set aside the isometries of a line for a moment and consider the plane. We can still move it along by a translation of the form f(x,y)=(x+a,y+b) for any two numbers a and b. A reflection across a line is also possible, for instance the reflection f(x,y)=(-x,y) across the y-axis. But unlike in the case of a line, we now have an additional type of isometry, a rotation about a point.

Activity 2: A square, a plane, a symmetry: yrtemmysaenalpaerauqsa!

Consider a square lying in the plane. We can rotate the plane about the center of the square (denoted p) by 90 degrees. This rotation is an isometry of the plane and at the same time it maps the square to itself. Thus it's a symmetry of the square in the plane.

square
  1. What other symmetries of the square can you find?
  2. We can also rotate the square by 450 degrees about the point p. Is this a different symmetry from a rotation by 90 degrees or the same? (Hint: does it map all points of the plane to the same points as a 90 degree rotation?)
  3. A reflection about the orange line is also a symmetry. What other lines can you use to reflect the plane such that the square is mapped to itself?
  4. Are any (non-zero) translations of the plane symmetries of the square?

In the above exercise you have (hopefully) found eight different symmetries: the three rotations illustrated below, the four reflections about the orange lines, and the identity symmetry, which we always count as one of the symmetries of a figure (you can think of it as a 0 degree rotation if you wish).

square symmetries

How do we know these are all the symmetries? By using the theorem that all isometries of the plane are translations, rotations, reflections, and compositions of translations and reflections (that is, a translation followed by a reflection). Since no translation takes the square to itself, only reflections and rotations are possible as symmetries of the square. The square must be mapped to itself by a symmetry, so only the rotations and reflections indicted above actually work.

Activity 3: More symmetries in the plane
  1. Is the composition of two symmetries, that is, applying one symmetry and then another, itself a symmetry? (Hint: use the definition.)
  2. Suppose we follow a reflection about the vertical orange line by a 90 degree rotation. Which of the above 8 symmetries is this? (Hint: see where different points are mapped to.)
  3. Find all symmetries of an equilateral triangle in the plane.
Activity 4: Symmetries of sets

We can also look at symmetries of more unusual sets. Here by set we just mean some, possibly infinite, collection of points.

  1. Consider the function y=cos(x). What are the symmetries of the graph of this function? (Hint: unlike for the square above, there are infinitely many different symmetries, but they are easy to describe.)
  2. Consider the set S consisting of five points in the plane: (0,0), (1,0), (0,1), (0.5,1), and (1,1). What are the symmetries of S?
  3. Let R be the set S without the point (0.5,1). What are the symmetries of R?
  4. What is the smallest number of points you need to place on the plane such that this set of points will only have the trivial (0 degree rotation) symmetry?
two points

As you might suspect, the collection of symmetries of a set depends on the space in which it lies. For instance, the two black points to the right have three different symmetries if placed on a plane, namely the identity symmetry, rotation by 180 degree about point p, and reflection about the orange line. If the space in which the points lie is a line however (so now a symmetry is an isometry of the line which preserves the two points), then there are only two symmetries, the identity and a reflection about the midpoint between them. After all, on a line a 180 degree rotation is the same as a reflection.

Activity 5: Space, the final frontier cube
  1. What are the symmetries of a cube in 3 dimensional space (i.e. our everyday world)? Draw a picture to help you. (Hint: they are rotations about some lines and reflections across some planes. But which ones?)
  2. What are the symmetries of a line (say the x-axis) in the plane? In 3-space?

Another important concept is that of a fixed point, a point which is mapped to itself by a symmetry or an isometry. In case of a symmetry, there are two types of fixed points involved: the fixed points of the ambient space and the fixed points of the figure itself. For instance, the identity symmetry fixes all points of the space and hence also of the figure. A rotation in a plane fixes only the center point of rotation, while a rotation about a line in 3-spaces fixes only the axis of rotation.

Activity 6: Idée fixe

What are the fixed points of the four types of isometries of the plane mentioned before (rotation, reflection, translation, and translation followed by a reflection)?


Next: Groups of symmetries