4.3. What about Reflections?
i
“bookmt” — 2006/8/8 — 12:58 — page 224 — #236
i
i
224
4. SYMMETRIES OF POLYHEDRA
Show that
2
0
3
f
0
4
p
5g [ A [ B ˙ 5=2 is the set of vertices of an icosahedron.
4.2.3. Each vertex of the icosahedron lies on a 5–fold axis, each midpoint of an edge on a 2–fold axis, and each centroid of a face on a 3–fold axis.
Using the data of the previous exercise and the method of Exercises 4.1.2, and 4.1.3, you can compute the matrices for rotations of the icosahedron. (I have only done this numerically and I don’t know if the matrices have a nice closed form.)
4.3. What about Reflections?
When you thought about the nature of symmetry when you first began reading this text, you might have focused especially on reflection symmetry. (People are particularly attuned to reflection symmetry since human faces and bodies are important to us.)
A reflection in R3 through a plane P is the transformation that leaves the points of P fixed and sends a point x 62 P to the point on the line through x and perpendicular to P , which is equidistant from P with x and on the opposite side of P .
Figure 4.3.1. A reflection.
For a plane P through the origin in R3, the reflection through P is given by the following formula. Let ˛ be a unit vector perpendicular to P .
For any x 2 R3, the reflection j˛ of x through P is given by j˛ .x/ D
x
2hx; ˛i˛, where h; i denotes the inner product in R3.
In the Exercises, you are asked to verify this formula and to compute the matrix of a reflection, with respect to the standard basis of R3. You are
i
i
i
i
i
“bookmt” — 2006/8/8 — 12:58 — page 225 — #237
i
i
4.3. WHAT ABOUT REFLECTIONS?
225
also asked to find a formula for the reflection through a plane that does not pass through the origin.
A reflection that sends a geometric figure onto itself is a type of sym metry of the figure. It is not an actual motion that you could perform on a physical model of the figure, but it is an ideal motion.
Let’s see how we can bring reflection symmetry into our account of the symmetries of some simple geometric figures. Consider a thickened version of our rectangular card: a rectangular brick. Place the brick with its faces parallel to the coordinate planes and with its centroid at the origin of coordinates. See Figure 4.3.2.
r3
r2
r1
Figure 4.3.2. Rotations and reflections of a brick.
The rotational symmetries of the brick are the same as those of the rectangular card. There are four rotational symmetries: the nonmotion e, and the rotations r1; r2, and r3 through an angle of about the x–, y–, and z–axes. The same matrices E, R1, R2, and R3 listed in Section 1.5 implement these rotations.
In addition, the reflections in each of the coordinate planes are symme tries; write ji for jOe , the reflection in the plane orthogonal to the standard i
unit vector Oei . See Figure 4.3.2.
The symmetry ji is implemented by the diagonal matrix Ji with
1
in the i th diagonal position and 1’s in the other diagonal positions. For example, 21
0 03
J2 D 0 1 0
4
5 :
0
0 1
There is one more symmetry that must be considered along with these, which is neither a reflection nor a rotation but is a product of a rotation i
i
i
i
i
“bookmt” — 2006/8/8 — 12:58 — page 226 — #238
i
i
226
4. SYMMETRIES OF POLYHEDRA and a reflection in several different ways. This is the inversion, which sends each corner of the rectangular solid to its opposite corner. This is implemented by the matrix E. Note that E D J1R1 D J2R2 D J3R3, so the inversion is equal to ji ri for each i .
Having included the inversion as well as the three reflections, we again have a group. It is very easy to check closure under multiplication and inverse and to compute the multiplication table. The eight symmetries are represented by the eight 3—by—3 diagonal matrices with 1’s and 1’s on the diagonal; this set of matrices is clearly closed under matrix multiplication and inverse, and products of symmetries can be obtained immediately by multiplication of matrices. The product of symmetries (or of matrices) is a priori associative.
Now consider a thickened version of the square card: a square tile, which we place with its centroid at the origin of coordinates, its square faces parallel with the .x; y/–plane, and its other faces parallel with the other coordinate planes. This figure has the same rotational symmetries as does the square card, and these are implemented by the matrices given in Section 1.5. See Figure 4.3.3.
r
c
b
a
d
Figure 4.3.3. Rotations of the square tile.
In addition, we can readily detect five reflection symmetries: For each of the five axes of symmetry, the plane perpendicular to the axis and passing through the origin is a plane of symmetry. See Figure 4.3.4 on the next page
Let us label the reflection through the plane perpendicular to the axis of the rotation a by ja, and similarly for the other four rotation axes. The reflections ja; jb; jc; jd , and jr are implemented by the following matrices:
i
i
i
i
i
“bookmt” — 2006/8/8 — 12:58 — page 227 — #239
i
i
4.3. WHAT ABOUT REFLECTIONS?
227
2
1 0 03
21
0 03
21 0
03
Ja D
0 1 0
0
1 0
0 1
0
4
5
Jb D 4
5
Jr D 4
5
0 0 1
0
0 1
0 0
1
2
0
1 03 20 1 03
Jc D
1
0 0 1 0 0
4
5
Jd D 4
5 :
0
0 1 0 0 1 Figure 4.3.4. Reflections of the square tile.
I claim that there are three additional symmetries that we must con sider along with the five reflections and eight rotations; these symmetries are neither rotations nor reflections, but are products of a rotation and a reflection. One of these we can guess from our experience with the brick, the inversion, which is obtained, for example, as the product aja, and which is implemented by the matrix
E.
If we can’t find the other two by insight, we can find them by computa tion: If 1 and 2 are any two symmetries, then their composition product 12 is also a symmetry; and if the symmetries 1 and 2 are implemented by matrices F1 and F2, then 12 is implemented by the matrix product
F1F2. So we can look for other symmetries by examining products of the matrices implementing the known symmetries.
We have 14 matrices, so we can compute a lot of products before find ing something new. If you are lucky, after a bit of trial and error you will discover that the combinations to try are powers of R multiplied by the reflection matrix Jr :
i
i
i
i
i
“bookmt” — 2006/8/8 — 12:58 — page 228 — #240
i
i
228
4. SYMMETRIES OF POLYHEDRA
20
1
03
Jr R D RJr D 1
0
0
4
5
0
0
1
2
0 1
03
Jr R3 D R3Jr D 1 0
0
4
5 :
0 0
1
These are new matrices. What symmetries do they implement? Both are reflection-rotations, reflections in a plane followed by a rotation about an axis perpendicular to the plane.
(Here are all the combinations that, as we find by experimentation, will not give something new: We already know that the product of two rotations of the square tile is again a rotation. The product of two reflections appears always to be a rotation. The product of a reflection and a rotation by about the axis perpendicular to the plane of the reflection is always the inversion.)
Now we have sixteen symmetries of the square tile, eight rotations (in cluding the nonmotion), five reflections, one inversion, and two reflectionrotations. This set of sixteen symmetries is a group. It seems a bit daunting to work this out by computing 256 matrix products and recording the multiplication table, but we could do it in an hour or two, or we could get a computer to work out the multiplication table in no time at all.
However, there is a more thoughtful method to work this out that se riously reduces the necessary computation; this method is outlined in the Exercises.
In the next section, we will develop a more general conceptual frame work in which to place this exploration, which will allow us to understand the experimental observation that for both the brick and the square tile, the total number of symmetries is twice the number of rotations. We will also see, for example, that the product of two rotations matrices is again a rotation matrix, and the product of two matrices, each of which is a reflection or a rotation–reflection, is a rotation.
Exercises 4.3 4.3.1. Verify the formula for the reflection J˛ through the plane perpendicular to ˛.
4.3.2. J˛ is linear. Find its matrix with respect to the standard basis of R3.
(Of course, the matrix involves the coordinates of ˛.)
i
i
i
i