SYMMETRY AND ISOMETRY, EXPLORATIONS AND

TRANSFORMATIONS OF EUCLIDEAN PLANE

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SYMMETRY AND ISOMETRY

What is Symmetry?

You probably have an intuitive idea of what symmetry means and can recognize it in various guises. For example, you can hopefully see that the letters N, O and M are symmetric, while the letter R is not. But probably, you can’t define in a mathematically precise way what it means to be symmetric — this is something we’re going to address.

Symmetry occurs very often in nature, a particular example being in the human body, as demonstrated by Leonardo da Vinci’s drawing of the Vitruvian Man on the right.

The symmetry in the picture arises since it looks essentially the same when we flip itover. A particularly important observation about the drawing is that the distance from the Vitruvian Man’s left index finger to his left elbow is the same as the distance from his right index finger to his right elbow. Similarly, the distance from the Vitruvian Man’s left knee to his left eye is the same as the distance

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from his right knee to his right eye. We could go on and on writing down statements like this, but the point I’m trying to get at is that our intuitive notion of symmetry—at least in geometry—is somehow tied up with the notion of distance.

What is an Isometry?

The flip in the previous discussion was a particular function which took points in the picture to other points in the picture. In particular, it did this in such a way that two points which were a certain distance apart would get mapped to two points which were the same distance apart. This motivates us to consider functions f which map points in the plane to points in the plane such that the distance from f (P) to f (Q) is the same as the distance from P to Q for any choice of points P and Q. Any function which satisfies this property is called an isometry. This comes from the ancient Greek words “isos”, meaning equal, and “metron”, meaning measure.

EXPLORATIONS AND TRANSFORMATIONS OFEUCLIDEAN PLANE

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Geometry is the study of those properties of a set which are preserved under a group of transformations on that set.

A transformation will be some function on points in the plane. That is, it will be some process whereby points are transformed to other points. This process could be the simple movement of points or could be a more complex alteration of the points. Transformations are basic to both a practical and theoretical understanding of geometry. Object permanence, the idea that we can move an object to a different position, but the object itself remains the same, is one of the first ideas that we learn as infants.

Felix Klein, one of the great geometers of the late nineteenth century, gave an address at Erlanger, Germany, in 1872, in which he proposed that geometry should be defined as the study of transformations and of the objects that transformations leave unchanged, or invariant.

This view has come to be known as the Erlanger Program. If we apply the Erlanger Program to Euclidean geometry, what kinds of transformations characterize this geometry? That is, what are the transformations that leave basic Euclidean figures, such as lines, segments, triangles, and circles, invariant? Since segments are the basic building blocks of many geometric figures, Euclidean transformations must, at least, preserve the “size" of segments; that is, they must preserve length.

In thislesson, we will investigate certain motions, called isometries. These are motions, or transformations, that preserve distance. There are four basic types of isometries of the plane. The purpose of looking at isometries is to have a better understanding of symmetry. In other assignments we will create pictures that demonstrate certain types of symmetry; by understanding isometries we will have an easier time recognizing the symmetry of figures.

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Euclidean Isometries

Definition 1. A function f on the plane is a transformation of the plane if f is a one-to-one function that is also onto the plane.

An isometry is then a length-preserving transformation. One important property of any transformation is that it is invertible.

Definition 2. Let f; g be functions on a set S. We say that g is the inverse of f if f(g(s)) = s and g(f(s)) = s for all s in S. That is, the composition of g and f (f and g) is the identity function on S. We denote the inverse by f-1.

It is left as an exercise to show that a function that is one-to-one and onto must have a unique inverse. Thus, all transformations have unique inverses.

A nice way to classify transformations (isometries) is by the nature of their fixed points.

Definition 3. Let f be a transformation. P is a fixed point of f if f(P) = P.

How many fixed points can an isometry have?

Theorem 1 . If points A;B are fixed by an isometry f, then the line through A;B is also fixed by f.

Proof: We know that f will map the line AB to the line f (A)f(B). Since A,B are fixed points, then line AB gets mapped back to itself.

Suppose that P is between A and B. Then, since f preserves betweenness, we know that f(P) will be between A and B. Also

AP = f(A)f(P) = Af(P)

This implies that P = f(P).

A similar argument can be used in the case where P lies elsewhere on line AB.

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ReflectionsA reflection can be thought of as getting a mirror image. A reflection

has a line of reflection. This line can be viewed as the line on which the mirror will sit in order to see the reflection of an image. This line is then unmoved by the reflection. Another way to think about a reflection is that you pick up an object, flip it over, and then place it down, keeping the reflection line unmoved in the end.

Definition 4. An isometry with two different fixed points, and that is not the identity, is called a reflection.

What can we say about a reflection? By Theorem 5.2 if A,B are the fixed points of a reflection, then the reflection also fixes the line through A,B. This line will turn out to be the equivalent of a “mirror" through which the isometry reflects points.

Theorem 2. Let r be a reflection fixing A and B. If P is not collinear with A,B, then the line through A and B will be a perpendicular bisector of the segment connecting P and r(P).

Reflection and Symmetry

The word symmetry is usually used to refer to objects that are in balance. Symmetric objects have the property that parts of the object look similar to other parts. The symmetric parts can be interchanged, thus creating a visual balance to the entire figure. How can we use transformations to mathematically describe symmetry?

Perhaps the simplest definition of mathematical symmetry is the one that most dictionaries give: an arrangement of parts equally on either side of a dividing line. While this type of symmetry is not the only one possible, it is perhaps the most basic in that such symmetry pervades the natural world. We will call this kind of symmetry bilateral symmetry.

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Definition 5. A figure F in the plane is said to have a line of symmetry or bilateral symmetry if there is a reflection r that maps the figure back to itself having the line as the line of reflection. For example, the line l is a line of symmetry for triangle ABC since if we reflect the triangle across this line; we get the exact same triangle back again.

Where is bilateral symmetry found in nature? Consider the insect body types in Fig. 1. All exhibit bilateral symmetry. In fact, most animals, insects, and plants have bilateral symmetry. Why is this case? Living creatures need bilateral symmetry for stability. Consider an animal that needs to be mobile, that needs to move forward and backward. To move with the least expenditure of energy, it is necessary that a body shape be balanced from side to side so that the creature does not waste energy keeping itself upright. Likewise, an immobile living creature, such as a tall pine tree, needs to be bilaterally symmetric in order to keep itself in a vertical equilibrium position.

Figure 1.Insect and Arachnid Symmetry

Definition 6.Lines that are mapped back to themselves by a transformation f are called invariant lines of f.

Note that we do not require that points on the line get mapped back to themselves, only that the line as a set of points gets mapped back to itself. Thus, a line may be invariant under f, but the points on the line need not be fixed by f.

Definition 7. A polygon is a regular polygon if it has all sides congruent and all interior angles congruent.

TranslationsA translation is a shift in some direction.

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Definition 8. An isometry that is made up of two reflections, where the lines of reflection are parallel, or identical, is called a translation.

What can we say about a translation?

Theorem 3. Let T be a translation that is not the identity. Then, for all points A ≠B, if A, B, T(A), and T(B) form a quadrilateral, then that quadrilateral is a parallelogram.

Translational Symmetry

Translational symmetry, being infinite in extent, cannot be exhibited by finite living creatures. However, we can find evidence of a limited form of translation symmetry in some animals and plants. For example, the millipede has leg sections that are essentially invariant under translation (Fig. 2).

Figure 2.Millipede

Also, many plants have trunks (stems) and branching systems that are translation invariant. Whereas translational symmetry is hard to find in nature, it is extremely common in human ornamentation. For example, wallpaper must have translational symmetry in the horizontal and vertical directions so that when you hang two sections of wallpaper next to each other the seam is not noticeable. Trim patterns called friezes, which often run horizontally along tops of walls, also have translational invariance (Fig. 3).

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Fig. 3 Frieze Patterns

RotationsA rotation comes equipped with two

pieces of information: a point about which the rotation is made, and an angle that says how far to rotate. This point, sometimes called the center of rotation, is fixed, while every other point moves.

Definition 9. An isometry that is made up of two reflections where the lines of reflection are not parallel will be called a rotation.

Theorem 4. An isometry R ≠id is a rotation iff R has exactly one fixed point.

Definition 10. The point O of intersection of the reflection lines of a rotation R is called the center of rotation. The angle ø defined by angle AO R(A) for A≠O is called the angle of rotation.

Rotational Symmetry

Definition 11. A figure is said to have rotational symmetry (or cyclic symmetry) of angle ø if the figure is preserved under a rotation about some center of rotation with angle ø.

Rotational symmetry is perhaps the most widespread symmetry in nature.

Rotational symmetry can be found in the very small, such as this radiolarian illustrated by Ernst Haeckel in his book Art Forms in Nature[18], to the very large, as exhibited by the rotationally symmetric shapes of stars and planets.

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Many flowers exhibit five-fold symmetry, the rotational symmetry of the regular pentagon. Let's prove that the regular pentagon has five-fold symmetry.

Theorem 5.The regular pentagon has rotational symmetry of 72 degrees.

Glide Reflections

It is possible to combine isometries to produce other isometries. However, some combinations do not produce anything new. For example, if you do a translation, then do another translation, the result is again a translation. However, there is one further type of isometry that comes from combining two isometries. If you perform a reflection followed by a translation, you get an isometry different from any of the previous types. This is called a glide reflection.

In this picture the triangle on the right is reflected across the line and then translated downward. The result can be considered as taking the first triangle, flipping it across the line and then placing it down where the final triangle resides.

References:

http://mathworld.wolfram.com/Isometry.html

https://en.wikipedia.org/wiki/Isometry

http://www.regentsprep.org/regents/math/geometry/gt5/properties.htm

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http://www.cut-the-knot.org/pythagoras/Transforms/Isometries.shtml

http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/isometryRn.pdf

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