1.4. SYMMETRIES AND MATRICES
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“bookmt” — 2006/8/8 — 12:58 — page 11 — #23
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1.4. SYMMETRIES AND MATRICES
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Define r k D .r 1/k for any positive integer k. Show that r k D r3k D rm, where m is the unique element of f0; 1; 2; 3g such that m C k is divisible by 4.
1.3.3. Here is another way to list the symmetries of the square card that makes it easy to compute the products of symmetries quickly.
(a) Verify that the four symmetries a; b; c; and d that exchange the top and bottom faces of the card are a; ra; r 2a; and r 3a, in some order. Which is which? Thus a complete list of the symmetries is fe; r; r2; r3; a; ra; r2a; r3ag:
(b) Verify that ar D r 1a D r3a: (c) Conclude that ar k D r ka for all integers k.
(d) Show that these relations suffice to compute any product.
1.4. Symmetries and Matrices
While looking at some examples, we have also been gradually refining our notion of a symmetry of a geometric figure. In fact, we are developing a mathematical model for a physical phenomenon — the symmetry of a physical object such as a ball or a brick or a card. So far, we have decided to pay attention only to the final position of the parts of an object, and to ignore the path by which they arrived at this position. This means that a symmetry of a figure R is a transformation or map from R to R. We have also implicitly assumed that the symmetries are rigid motions; that is, we don’t allow our objects to be distorted by a symmetry.
We can formalize the idea that a transformation is rigid or nondis torting by the requirement that it be distance preserving or isometric. A transformation W R ! R is called an isometry if for all points a; b 2 R, we have d. .a/; .b// D d.a; b/, where d denotes the usual Euclidean distance function.
We can show that an isometry W R ! R3 defined on a subset R of R3 always extends to an affine isometry of R3. That is, there is a vector b and a linear isometry T W R3 ! R3 such that .x/ D b C T .x/ for all x 2 R. Moreover, if R is not contained in any two–dimensional plane, then the affine extension is uniquely determined by . (Note that if 0 2 R and .0/ D 0, then we must have b D 0, so extends to a linear isometry of R3.) These facts are established in Section 11.1; for now we will just assume them.
Now suppose that R is a (square or a nonsquare) rectangle, which we suppose lies in the .x; y/–plane, in three–dimensional space. Consider an isometry W R ! R. We can show that must map the set of vertices of R to itself. (It would certainly be surprising if a symmetry did not
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“bookmt” — 2006/8/8 — 12:58 — page 12 — #24
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1. ALGEBRAIC THEMES map vertices to vertices!) Now there is exactly one point in R that is equidistant from the four vertices; this is the centroid of the figure, which is the intersection of the two diagonals of R. Denote the centroid C . What is .C /? Since is an isometry and maps the set of vertices to itself, .C / is still equidistant from the four vertices, so .C / D C . We can assume without loss of generality that the figure is located with its centroid at 0, the origin of coordinates. It follows from the results quoted in the previous paragraph that extends to a linear isometry of R3.
The same argument and the same conclusion are valid for many other geometric figures (for example, polygons in the plane, or polyhedra in space). For such figures, there is (at least) one point that is mapped to itself by every symmetry of the figure. If we place such a point at the origin of coordinates, then every symmetry of the figure extends to a linear isometry of R3.
Let’s summarize with a proposition:
Proposition 1.4.1. Let R denote a polygon or a polyhedron in three– dimensional space, located with its centroid at the origin of coordinates.
Then every symmetry of R is the restriction to R of a linear isometry of R3.
Since our symmetries extend to linear transformations of space, they are implemented by 3-by-3 matrices. That is, for each symmetry of one of our figures, there is an (invertible) matrix A such that for all points x in our figure, .x/ D Ax.5 Here is an important observation: Let 1 and 2 be two symmetries of a three-dimensional object R. Let T1 and T2 be the (uniquely determined) linear transformations of R3, extending 1 and 2. The composed linear transformation T1T2 is then the unique linear extension of the composed symmetry 12. Moreover, if A1 and A2 are the matrices implementing T1 and T2, then the matrix product A1A2 implements T1T2. Consequently, we can compute the composition of symmetries by computing the product of the corresponding matrices.
This observation gives us an alternative, and more or less automatic, way to do the bookkeeping for composing symmetries.
Let us proceed to find, for each symmetry of the square or rectangle, the matrix that implements the symmetry.
5A brief review of elementary linear algebra is provided in Appendix We still have a slight problem with nonuniqueness of the linear transformation im plementing a symmetry of a two–dimensional object such as the rectangle or the square.
However, if we insist on implementing our symmetries by rotational transformations of space, then the linear transformation implementing each symmetry is unique.
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“bookmt” — 2006/8/8 — 12:58 — page 13 — #25
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1.4. SYMMETRIES AND MATRICES
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We can arrange that the figure (square or rectangle) lies in the .x; y/– plane with sides parallel to the coordinate axes and centroid at the origin of coordinates. Then certain axes of symmetry will coincide with the coordinate axes. For example, we can orient the rectangle in the plane so that the axis of rotation for r1 coincides with the x–axis, the axis of rotation for r2 coincides with the y–axis, and the axis of rotation for r3 coincides with the z–axis.
The rotation r1 leaves the x–coordinate of a point in space unchanged and changes the sign of the y– and z–coordinates. We want to compute the matrix that implements the rotation r1, so let us recall how the standard matrix of a linear transformation is determined. Consider the standard basis of R3:
213
203
203
Oe1 D 0
1
0
4
5
Oe2 D 4 5 Oe3 D 4 5 :
0
0
1
If T is any linear transformation of R3, then the 3-by-3 matrix MT with columns T . Oe1/; T . Oe2/, and T . Oe3/ satisfies MT x D T .x/ for all x 2 R3.
Now we have r1. Oe1/ D Oe1; r1. Oe2/ D Oe2; and r1. Oe3/ D Oe3; so the matrix R1 implementing the rotation r1 is
21
0
03
R1 D 0
1
0
4
5 :
0
0
1
Similarly, we can trace through what the rotations r2 and r3 do in terms of coordinates. The result is that the matrices
2
1 0
03
2
1
0 03
R2 D 0 1
0
0
1 0
4
5
and R3 D 4
5
0 0
1
0
0 1 implement the rotations r2 and r3. Of course, the identity matrix 21 0 03
E D 0 1 0
4
5
0 0 1 implements the nonmotion. Now you can check that the square of any of the Ri ’s is E and the product of any two of the Ri ’s is the third. Thus the matrices R1; R2; R3, and E have the same multiplication table (using matrix multiplication) as do the symmetries r1; r2; r3, and e of the rectangle, as expected.
Let us similarly work out the matrices for the symmetries of the square:
Choose the orientation of the square in space so that the axes of symmetry for the rotations a, b, and r coincide with the x–, y–, and z–axes, respectively.
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“bookmt” — 2006/8/8 — 12:58 — page 14 — #26
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1. ALGEBRAIC THEMES
Then the symmetries a and b are implemented by the matrices
21
0
03
2
1 0
03
A D 0
1
0
0 1
0
4
5
B D 4
5 :
0
0
1
0 0
1
The rotation r is implemented by the matrix
20
1 03
R D 1 0 0
4
5 ;
0
0 1 and powers of r by powers of this matrix
2
1
0 03
2
0 1 03
R2 D
0
1 0 1 0 0
4
5
and
R3 D 4
5 :
0
0 1 0 0 1
The symmetries c and d are implemented by matrices
2
0
1
03
20 1
03
C D
1
0
0
1 0
0
4
5
and
D D 4
5 :
0
0
1
0 0
1
Therefore, the set of matrices fE; R; R2; R3; A; B; C; Dg necessarily has the same multiplication table (under matrix multiplication) as does the corresponding set of symmetries fe; r; r2; r3; a; b; c; d g. So we could have worked out the multiplication table for the symmetries of the square by computing products of the corresponding matrices. For example, we compute that CD D R2 and can conclude that cd D r2.
We can now return to the question of whether we have found all the symmetries of the rectangle and the square. We suppose, as before, that the figure (square or rectangle) lies in the .x; y/–plane with sides parallel to the coordinate axes and centroid at the origin of coordinates. Any symmetry takes vertices to vertices, and line segments to line segments (see Exercise 1.4.4), and so takes edges to edges. Since a symmetry is an isometry, it must take each edge to an edge of the same length, and it must take the midpoints of edges to midpoints of edges. Let 2` and 2w denote the lengths of the edges; for the rectangle ` ¤ w, and for the square ` D w.
The midpoints of the edges are at ˙` Oe1 and ˙w Oe2. A symmetry is determined by .` Oe1/ and .w Oe2/, since the symmetry is linear and these two vectors are a basis of the plane, which contains the figure R.
For the rectangle, .` Oe1/ must be ˙` Oe1, since these are the only two midpoints of edges length 2w. Likewise, .w Oe2/ must be ˙w Oe2, since these are the only two midpoints of edges length 2`. Thus there are at most four possible symmetries of the rectangle. Since we have already found four distinct symmetries, there are exactly four.
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“bookmt” — 2006/8/8 — 12:58 — page 15 — #27
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1.4. SYMMETRIES AND MATRICES
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For the square (with sides of length 2w), .w Oe1/ and .w Oe2/ must be contained in the set f˙w Oe1; ˙w Oe2g. Furthermore if .w Oe1/ is ˙w Oe1, then .w Oe2/ is ˙w Oe2; and if .w Oe1/ is ˙w Oe2, then .w Oe2/ is ˙w Oe1.
Thus there are at most eight possible symmetries of the square. As we have already found eight distinct symmetries, there are exactly eight.
Exercises 1.4 1.4.1. Work out the products of the matrices E, R, R2,R3, A, B, C , D, and verify that these products reproduce the multiplication table for the symmetries of the square, as expected. (Instead of computing all 64 products, compute “sufficiently many” products, and show that your computations suffice to determine all other products.)
1.4.2. Find matrices implementing the six symmetries of the equilateral triangle. (Compare Exercise 1.3.1.) In order to standardize our notation and our coordinates, let’s agree to put the vertices of the triangle at .1; 0; 0/,
p
p
. 1=2; 3=2; 0/, and . 1=2; 3=2; 0/. (You may have to review some linear algebra in order to compute the matrices of the symmetries; review how to get the matrix of a linear transformation, given the way the transformation acts on a basis.) Verify that the products of the matrices reproduce the multiplication table for the symmetries of the equilateral triangle.
1.4.3. Let T .x/ D Ax C b be an invertible affine transformation of R3.
Show that T 1 is also affine.
The next four exercises outline an approach to showing that a sym metry of a rectangle or square sends vertices to vertices. The approach is based on the notion of convexity.
1.4.4. A line segment Œa1; a2 in R3 is the set Œa1; a2 D fsa1 C .1 s/a2 W 0 s 1g: Show that if T .x/ D Ax C b is an affine transformation of R3, then T .Œa1; a2/ D ŒT .a1/; T .a2/.
1.4.5. A subset R R3 is convex if for all a1; a2 2 R, the segment Œa1; a2 is a subset of R. Show that if R is convex and T .x/ D Ax C b is an affine transformation of R3, then T .R/ is convex.
1.4.6. A vertex v of a convex set R can be characterized by the following property: If a1 and a2 are elements of R and v 2 Œa1; a2, then a1 D a2.
That is, v is not contained in any nontrivial line segment in R. Show that if v is a vertex of a convex set R, and T .x/ D Ax C b is an invertible affine transformation of R3, then T .v/ is a vertex of the convex set T .R/.
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