Math 103 Finite Math with a Special Emphasis on Math & Art by Lun-Yi Tsai, Spring 2010, University of Miami

2 Symmetry notes (continued)

2.1 Direct and Opposite Symmetries

In our exploration of symmetry, we have thus far encountered rotations and reflections. They fall into two classes known as direct and opposite symmetries.1

Definition 2.1. A direct symmetry is a symmetric transformation that preserves orientation. An opposite symmetry is one that reverses orientation.

Let’s see what we mean by a couple examples.

Example 2.2. Consider once more the symmetries of an equilateral triangle. Let’s see the effect of the rotation R = R120◦ on the “numbering” of the triangle.

1 3 Before the transformation we count the numbers “1 − 2 − 3” in a counterclockwise direction. Ro- R120◦ tation by 120◦ doesn’t affect this orientation: al- P P though the positions of the numbers change, we still count “1 − 2 − 3” in a counterclockwise direc- tion. 2 3 1 2

As a rotation symmetry doesn’t change the orientation, it is a direct symmetry.

Example 2.3. Now let’s consider the transformation of reflection F = F1 about the vertical axis through the center P .

1 1 Before the transformation we count the numbers F “1 − 2 − 3” in a counterclockwise direction. Reflec- tion about the vertical axis switches 2 and 3. Now P P when we count “1−2−3” we do so in a clockwise direction. Thus, reflection reverses orientation. 2 3 3 2

A reflection results in a reversal of orientation, therefore it’s an opposite symmetry. Remark 2.4. All rotations are direct symmetries, while all reflections are opposite symmetries.

1In fact, it turns out that the set of all direct symmetries of an object together with the set of all its opposite symmetries form the group of all its symmetries.

1 Before we complete our list of symmetries, let’s list the direct and opposite symmetries of the following symbols.

fig.1 Which of these have the same direct and opposite symmetries?

2.2 Translation Symmetry

2.2.1 General Definition

Definition 2.5. An object, shape, design has translational symmetry if we displace it in certain direction, a certain distance, its appearance doesn’t change.

Remark 2.6. It’s clear that for an object to have a non-identity translational symmetry, the object must be infinite in at least one dimension. Thus, the objects we’ve been looking at do not have this kind of symmetry2.

Let’s take a look at some examples of this kind of symmetry. Here are typical Greek band ornaments.

fig.2 If we assume that the patterns extend infinitely to the right and left, then these two band ornaments have translational symmetry.

2Aside from the identity translation that displaces the object a distance 0, i.e., doesn’t move it at all.

2 Now, let us turn to describing the natural mathematical elements that describe translation.

2.2.2 The Language of Vectors

Definition 2.7. A vector is a mathematical quantity with a direction and a length.

A vector3 describes displacement, i.e., a direction and a distance that an object is to be moved; as such we usually think of it as an arrow (pointing in the appropriate direction) whose length indicates the distance. As vectors describe displacement, they are not fixed to any particular point and can be thought to be floating about freely.

Example 2.8. Here are a few vectors.

−→v −→v 2 The top vector, , represents a displacement to the right by two units; while the bottom vector, −→u , represents a displacement to the left by three −→u units. 3

Furthermore, we can multiply a vector by a number, which can be thought of as scaling the vector up (or down) by that number. If the number is negative, the direction of the vector is reversed.

Example 2.9. Assuming −→v and −→u to be as in the above example, we have:

2−→v The top vector, 2−→v , now represents a displace- 4 ment to the right by four units; while the bottom vector, 1 −→u , now represents a displacement to the 1 −→ 3 3 u left by one unit.

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−−→v If we multiply by a negative number, the direc- tion of the vector is reversed and its length scaled 2 by the absolute value of the number. Thus, the top vector, −−→v , now represents a displacement 1 −→ to the left by two units; while the bottom vector, − 3 u 1 −→ − 3 u , now represents a displacement to the right 1 by one unit.

3Here we barely touch upon some aspects of this very rich and important area of mathematics that studies linear transformations, which are transformations that take lines to lines. Essentially, one side of the calculus story tells us that we can approximate things by a linear model, i.e., we can look at a linear approximation of an object to get a better idea of what’s happening. This is known as the Differential Calculus.

3 We can add vectors by putting the tail of one vector −→v to the head of another vector −→u , then the vector that has the head of the one vector −→v and the tail of the other −→u is the sum of the vectors −→u + −→v .

If we add the vectors −→u and −→v we get the resul- −→ −→ u + v tant vector −→u + −→v . −→v

−→u

Example 2.10. Let −→u be the vector pointing up of length 1, and −→v be the vector pointing right of length 3. Draw −→u + −→v ; 2−→u + −→v ; and −→u − −→v .

2.2.3 Vector Definition of Translation

Definition 2.11. A translation by a vector −→u is a transformation that results by displacing each point of an object in the direction of −→u a distance equal to the length of −→u . −→ Notation 2.12. We use τ−→u to denote a translation by u . Thus τ−→u (P ) indicates the position of the point P after translation by −→u . −→ Definition 2.13. An object, shape, design X has translation symmetry if there exists a vector u such that τ−→u (X) is indistinguishable from X. In other words, if all the points of object X have been translated by −→u and its appearance remains unchanged, then it has translation symmetry.

The band ornaments considered earlier are now accompaneid with vectors that give rise to the translation sym- metries. Of course, there are infinitely-many translation symmetries, but these vectors give the shortest possible translations. All other translations are in fact gotten by multiplying these vectors by integers.

fig.3 Here each vector gives a translation symmetry of the band ornament below it.

For example, translation of the first band to the right two is gotten by multiplying −→u by 2 to get 2−→u , which is the −→ −→ transformation τ2−→u . Translation to the left one is gotten by multiplying u by −1, i.e., − u has the translation symmetry τ−−→u .

Note that the two bands have other symmetries besides translation. Can you find reflections or rotations?

Finally we introduce our last symmetry.

4 2.3 Glide Reflection Symmetry

Definition 2.14. An object has glide reflection symmetry if a translation by −→u followed by a reflection along a line parallel to −→u doesn’t change its appearance.

Assume again that the pattern below extends infinitely to the right and left. Then it has glide reflection symmetry. If we shift the pattern to the right a certain distance (as indicated by −→u ) and then reflect across the green line (which is parallel to −→u ), our pattern’s appearance is unchanged.

fig.4 Glide reflection symmetry. Here two petals are colored red to demonstrate more clearly the transformation. We assume the pattern to be of one color.

In the following drawing by the Dutch artist M.C. Escher, notice how the first band appears at first sight to have glide reflection symmetry. But then you realize that the two colors would get switched by the reflection. Furthermore, the green fish have eyes, while the purple ones don’t. However, the other Escher band with the birds and fish does have glide reflection symmetry.

fig.5 The role of color.

Now, see if you can detect all the symmetries in this band. (Hint: There may be more than just translations and glide reflections.)

fig.6 List the symmetries of this band ornament.

5 2.4 Notation for Transformations

In the last class, we finished up our presentation of symmetry. In discussing some questions regarding the home- 4 work , I introduced some more mathematical notation (similar to the one we use for translation, τ−→u ) for the other transformations.

2.4.1 Rotation notation

◦ When we talked about rotation previously, we just used R = R90◦ to indicate a rotation of 90 . It was assumed that the rotation of the given figure (whether a letter or an equilateral triangle) was about the center of the figure. In general, we can rotate about any point, so our notation has to provide this information in addition to the angle of rotation.

The standard is to use the Greek letter rho, ρ, which sounds like the “ro” in rotation. Thus, we write ρC,90◦ when we mean rotate about the point C an angle of 90◦.5

2.4.2 Reflection notation

We use the Greek letter mu, written µ and pronounced “mew,” for reflection. I think the reason for this is because it sort of sounds like mirror, which makes it easy to remember what it’s for.

In any case, in order to perform a reflection, we need to have a line across which we do the reflection. And so the notation needs to have this information. Thus, we may write µl2 to indicate a reflection across some line l2.

2.4.3 Glide reflection notation

And finally, we introduce the notation for the last of our symmetric transformations, the glide reflection. In order to define it, we need a vector −→v for the “glide” or translation and a line l (parallel to −→v ) across which to reflect.

We use the Greek letter gamma, γ, to suggest “glide.” Thus, we write γl,−→v = µl · τ−→v to mean translate by a vector −→v and reflect across the line l.

4These were problems taken from a book by my former professor, the late Dr. Martin M. Guterman, entitled “Symmetry Groups of the Plane.” 5As mentioned before, when the angle is a positive number, we mean rotation in the counterclockwise direction.

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