11/18/2013

Symmetry

As are many other core concepts, symmetry is rather Chapter 11: hard to define, and we will not even attempt a proper definition until Section 11.6. We will start our The Mathematics discussion with just an informal stab at the of Symmetry mathematical (or geometric if you prefer) interpretation of symmetry.

11.1 Rigid Motions

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Example 11.1 Symmetries of a Triangle Example 11.1 Symmetries of a Triangle

The figure shows three triangles: (a) a scalene triangle Even without a formal understanding of what symmetry (all three sides are different), (b) an isosceles triangle, is, most people would answer that the equilateral and (c) an equilateral triangle. In terms of symmetry, triangle in (c) is the most symmetric and the scalene how do these triangles differ? Which one is the most triangle in (a) is the least symmetric. This is in fact symmetric? Least symmetric? correct, but why?

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Example 11.1 Symmetries of a Triangle Example 11.1 Symmetries of a Triangle

Think of an imaginary observer–say a tiny (but very In the case of the isosceles triangle (b), the view from observant) ant–standing at the vertices of each of the vertices B and C is the same, but the view from vertex A triangles, looking toward the opposite side. In the case is different. In the case of the equilateral triangle (c), of the scalene triangle (a), the view from each vertex is the view is the same from each of the three vertices. different.

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Symmetry Again Symmetry - Rigid Motion

Let’s say, for starters, that symmetry is a property of an The act of taking an object and moving it from some object that looks the same to an observer standing at starting position to some ending position without different vantage points. This is still pretty vague but a altering its shape or size is called a rigid motion (and start nonetheless. Now instead of talking about an sometimes an isometry ). If, in the process of moving observer moving around to different vantage points the object, we stretch it, tear it, or generally alter its think of the object itself moving–forget the observer. shape or size, the motion is not a rigid motion. Since in Thus, we might think of symmetry as having to do with a rigid motion the size and shape of an object are not ways to move an object so that when all the moving altered, distances between points are preserved: is done, the object looks exactly as it did before.

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Symmetry - Rigid Motion Symmetry - Rigid Motion

The distance between any two points X and Y in the In defining rigid motions we are completely result starting position is the same as the distance between oriented. We are only concerned with the net effect the same two points in the ending position . In (a), the of the motion–where the object started and where motion does not change the shape of the object; the object ended. What happens during the “trip” is only its position in space has changed. In (b), both irrelevant. position and shape have changed.

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Symmetry - Rigid Motion Symmetry - Rigid Motion

This implies that a rigid motion is completely defined Because rigid motions are defined strictly in terms of by the starting and ending positions of the object their net effect, there is a surprisingly small number of being moved, and two rigid motions that move an scenarios. In the case of two-dimensional objects in a object from the same starting position to the same plane, there are only four possibilities: A rigid motion is ending position are equivalent rigid motions–never equivalent to (1) a reflection , (2) a rotation , (3) a mind the details of how they go about it. translation , or (4) a glide reflection . We will call these four types of rigid motions the basic rigid motions of the plane .

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Symmetry - Rigid Motion Symmetry - Rigid Motion

A rigid motion of the plane–let’s call it M–moves each We will also stick to the convention that the image point in the plane from its starting position P to an point has the same label as the original point but with ending position P´, also in the plane. (From here on a prime symbol added. we will use script letters such as M and N to denote rigid motions, which should eliminate any possible It may happen that a point P is moved back to itself confusion between the point M and the rigid motion under M, in which case we call P a fixed point of the M.) We will call the point P´ the image of the point P rigid motion M. under the rigid motion M and describe this informally by saying that M moves P to P´.

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Reflection

A reflection in the plane is a rigid motion that moves Chapter 11: an object into a new position that is a mirror image of the starting position. In two dimensions, the “mirror ” is The Mathematics a line called the axis of reflection. of Symmetry From a purely geometric point of view a reflection can be defined by showing how it moves a generic 11.2 Reflections point P in the plane.

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Reflection Reflection

The image of any point P is found by drawing a line Points on the axis itself are fixed points of the through P perpendicular to the axis l and finding the reflection. point on the opposite side of l at the same distance as P from l.

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Example 11.2 Reflections of a Triangle Example 11.2 Reflections of a Triangle

The following figures show three cases of reflection of a triangle ABC . In all cases the original triangle ABC is In this figure, the axis of reflection l cuts through the ´ ´ ´ shaded in blue and the reflected triangle A B C is triangle ABC–here the points where l intersects the shaded in red. triangle are fixed points of the triangle. In this figure the axis of Reflection I does not intersect the triangle ABC .

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Example 11.2 Reflections of a Triangle Properties of Reflections

In this figure, the reflected triangle A´B´C´ falls on top The following are simple but useful properties of a of the original triangle ABC . The vertex B is a fixed point reflection. of the triangle, but the vertices A and C swap positions under the reflection. Property 1 If we know the axis of reflection, we can find the image of any point P under the reflection (just drop a perpendicular to the axis through P and find the point on the other side of the axis that is at an equal distance). Essentially a reflection is completely determined by its axis l.

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Properties of Reflections Properties of Reflections

Property 2 Property 3 If we know a point P and its image P´ under the The fixed points of a reflection are all the points on reflection (and assuming P´ is different from P), we the axis of reflection l. can find the axis l of the reflection (it is the perpendicular bisector of the segment PP ´). Once we Property 4 have the axis l of the reflection, we can find the Reflections are improper rigid motions, meaning that image of any other point (property 1). they change the left-right and clockwise- counterclockwise orientations of objects. This property is the reason a left hand reflected in a mirror looks like a right hand and the hands of a clock reflected in a mirror move counterclockwise.

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Properties of Reflections Identity Motion

Property 5 A rigid motion that is equivalent to not moving the If P´ is the image of P under a reflection, then ( P´)´ = object at all is called the identity motion. At first blush P (the image of the image is the original point). Thus, it may seem somewhat silly to call the identity motion when we apply the same reflection twice, every point a motion (after all, nothing moves), but there are very ends up in its original position and the rigid motion is good mathematical reasons to do so, and we will equivalent to not having moved the object at all. soon see how helpful this convention is for studying and classifying symmetries.

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PROPERTIES OF REFLECTIONS

■ A reflection is completely determined by its axis l. Chapter 11: ■ A reflection is completely determined by a The Mathematics single point-image pair P and P´ (as long as P´ ≠ P). of Symmetry ■ A reflection has infinitely many fixed points (all points on l). ■ A reflection is an improper rigid motion. 11.3 Rotations ■ When the same reflection is applied twice, we get the identity motion.

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Rotation Rotation

Informally, a rotation in the plane is a rigid motion that The figure illustrates geometrically how a clockwise pivots or swings an object around a fixed point O. A rotation with rotocenter O and angle of rotation rotation is defined by two pieces of information: (1) moves a point P to the point P´. the rotocenter (the point O that acts as the center of the rotation) and (2) the angle of rotation (actually the measure of an angle indicating the amount of rotation). In addition, it is necessary to specify the direction (clockwise or counterclockwise) associated with the angle of rotation.

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Example 11.3 Rotations of a Triangle Example 11.3 Rotations of a Triangle

The following illustrates three cases of rotation of a The rotocenter O is at the center of the triangle ABC . triangle ABC . In all cases the original triangle ABC is The 180º rotation turns the triangle “upside down. ” For shaded in blue and the obvious reasons, a 180º rotation is often called a half- reflected triangle A´B´C´ turn. (With half turns the result is is shaded in red. The rotocenter O lies outside the same whether we rotate the triangle ABC. The 90º clockwise or counter- clockwise rotation moved clockwise, so it is unnecessary “ the triangle from the 12 to specify a direction.) o’clock position” to the “3 o’clock position. ”

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Example 11.3 Rotations of a Triangle Properties of Rotations

The 360º rotation moves every point back to its original The following are some important properties of a position–from the rigid motion point of view it ’s as if the rotation. triangle had not moved. Property 1 A 360º rotation is equivalent to a 0º rotation, and a 0º rotation is just the identity motion. (The expression “going around full circle” is the well-known colloquial version of this property.)

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Properties of Rotations Properties of Rotations

From property 1 we can conclude that all rotations Property 2 can be described using an angle of rotation between When an object is rotated, the left-right and 0º and 360º. For angles larger than 360º we divide the clockwise-counterclockwise orientations are angle by 360º and just use the remainder (a clockwise preserved (a rotated left hand remains a left hand, rotation by 759º is equivalent to a clockwise rotation and the hands of a rotated clock still move in the by 39º). clockwise direction). We will describe this fact by saying that a rotation is a proper rigid motion. Any In addition, we can describe a rotation using motion that preserves the left-right and clockwise- clockwise or counterclockwise orientations (a counterclockwise orientations of objects is called a clockwise rotation by 39º is equivalent to a proper rigid motion. counterclockwise rotation by 321º).

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Properties of Rotations Properties of Rotations

A common misconception is to confuse a 180º rotation with a reflection, but we can see that they Property 3 are very different from just observing that the In every rotation, the rotocenter is a fixed point, and reflection is an improper rigid motion, whereas the except for the case of the identity (where all points 180º rotation is a proper rigid motion. are fixed points) it is the only fixed point.

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Properties of Rotations Properties of Rotations

Property 4 Given a second pair of points Q and Q´ we can Unlike a reflection, a rotation cannot be determined identify the rotocenter O as the point where the by a single point-image pair perpendicular bisectors of PP ´ P and P´ it takes a second and QQ ´ meet. Once we point-image pair Q and have identified the Q´ to nail down the rotation. rotocenter O, the angle of The reason is that infinitely rotation α is given by the many rotations can move P measure of angle POP ´ (or to P´. Any point located on for that matter QOQ ´ –they the perpendicular bisector of are the same). the segment PP ´ can be a rotocenter for such a rotation.

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Properties of Rotations PROPERTIES OF ROTATIONS

■ A 360º rotation is equivalent to the identity motion. Note: In the special case where PP ´ and QQ ´ happen ■ A rotation is a proper rigid motion. to have the same ■ A rotation that is not the identity motion has perpendicular bisector, the only one fixed point, its rotocenter. rotocenter O is the intersection of PQ and PQ ´. ■ A rotation is completely determined by two point-image pairs P, P´ and Q,Q´.

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Translation

A translation consists of essentially dragging an object Chapter 11: in a specified direction and by a specified amount (the length of the translation). The two pieces of The Mathematics information (direction and length of the translation) are combined in the form of a vector of translation of Symmetry (usually denoted by v). The vector of translation is represented by an arrow–the arrow points in the direction of translation and the length of the arrow is 11.4 Translations the length of the translation.

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Translation Example 11.4 Translation of a Triangle

A very good illustration of a translation in a two- dimensional plane is the dragging of the cursor on a This figure illustrates the translation of a triangle ABC . computer screen. Regardless of what happens in Two “different” arrows are shown in the figure, but they between, the net result when you drag an icon on both have the same length and direction, so they your screen is a translation in a specific direction and describe the same vector of translation v. by a specific length. As long as the arrow points in the proper direction and has the right length, the placement of the arrow in the picture is immaterial.

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Properties of Translations Properties of Translations

The following are some important properties of a Property 2 translation. In a translation, every point gets moved some distance and in some direction, so a translation has Property 1 no fixed points. If we are given a point P and its image P´ under a translation, the arrow joining P to P´ gives the vector Property 3 of the translation. Once we know the vector of the When an object is translated, left-right and clockwise- translation, we know where the translation moves any counterclockwise orientations are preserved: A other point. Thus, a single point-image pair P and P´ is translated left hand is still a left hand, and the hands all we need to completely determine the translation. of a translated clock still move in the clock-wise direction. In other words, translations are proper rigid motions.

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Properties of Translations PROPERTIES OF TRANSLATIONS Property 4 The effect of a translation with vector v can be ■ A translation is completely determined by a undone by a translation of the same length but in the single point-image pair P and P´. opposite direction. The vector for this opposite ■ A translation has no fixed points. translation can be conveniently described as –v. Thus, a translation with vector v followed with a translation ■ A translation is a proper rigid motion. with vector –v is equivalent to the identity motion. ■ When a translation with vector v is followed with a translation with vector –v we get to the identity motion.

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Glide Reflection

A glide reflection is a rigid motion obtained by Chapter 11: combining a translation (the glide) with a reflection. Moreover, the axis of reflection must be parallel to the The Mathematics direction of translation. Thus, a glide reflection is of Symmetry described by two things: the vector of the translation v and the axis of the reflection l, and these two must be parallel. 11.5 Glide Reflections

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Glide Reflection Example 11.4 Glide Reflection of a Triangle The footprints left behind by someone walking on soft sand are a classic example of The figures on the next two slides illustrate the result of a glide reflection: right and applying the glide reflection with vector v and axis l to left footprints are images of the triangle ABC . We can do this in two different ways, each other under the but the final result will be the same. combined effects of a reflection and a translation.

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Example 11.4 Glide Reflection of a Example 11.4 Glide Reflection of a Triangle Triangle

The translation is applied first, moving triangle ABC to If we apply the reflection first, the intermediate position the triangle ABC gets moved A*B*C* . The reflection is then to the intermediate position applied to A*B*C* giving the A*B*C* and then translated to the final position A´B´C´. final position A´B´C´.

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Properties of Glide Reflections Properties of Glide Reflections

Property 1 Given a second point-image pair Q and Q´, we can A glide reflection is completely determined by two determine the axis of the reflection: It is the line point-image pairs P, P´ and Q, Q´. Given a point- passing through the points M (midpoint of the line image pair P and P´ under a glide reflection, we do segment PP ´) and N (midpoint of the line segment ´ not have enough information to determine the glide QQ ). reflection, but we do know that the axis l must pass through the midpoint of the line segment PP ´.

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Properties of Glide Reflections Properties of Glide Reflections

Once we find the axis of reflection l, we can find the In the event that the image of one of the points–say P´– under the midpoints of PP ´ and QQ ´ reflection. This gives the intermediate point P*, and are the same point M, we the vector that moves P to P* is the vector of can still find the axis l by translation v. drawing a line perpendicular to the line PQ passing through the common midpoint M.

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Properties of Glide Reflections Properties of Glide Reflections

Property 3 Property 2 A glide reflection is a combination of a proper rigid A fixed point of a glide reflection would have to be a point that ends up exactly where it started after it is motion (the translation) and an improper rigid motion first translated and then reflected. This cannot (the reflection). Since the translation preserves left- happen because the translation moves every point right and clockwise-counterclockwise orientations but and the reflection cannot undo the action of the the reflection reverses them, the net result under a translation. It follows that a glide reflection has no glide reflection is that orientations are reversed. Thus, fixed points. a glide reflection is an improper rigid motion.

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Properties of Glide Reflections PROPERTIES OF GUIDE REFLECTIONS Property 4 To undo the effects of a glide reflection, we need a ■ A guide reflection has no fixed points. second glide reflection in the opposite direction. To be more precise, if we move an object under a glide ■ A guide reflection is an improper rigid motion. reflection with vector of translation v and axis of reflection l and then follow it with another glide ■ A guide reflection is completely determined reflection with vector of translation –v and axis of by two point-image pairs P, P´ and Q,Q´. reflection still l, we get the identity motion. It is as if the ■ When a guide reflection with vector v and axis object was not moved at all. of reflection l is followed with a translation with vector –v and the same axis of reflection l we get to the identity motion.

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Symmetry

With an understanding of the four basic rigid motions Chapter 11: and their properties, we can now look at the concept of symmetry in a much more precise way. Here, The Mathematics finally, is a good definition of symmetry, one that of Symmetry probably would not have made much sense at the start of this chapter:

11.6 Symmetries and Symmetry Types

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One Way to Think of Symmetry

SYMMETRY You observe the position of an object, and then, while you are not looking, the object is moved. If you can ’t tell that the object was moved, the rigid motion is a A symmetry of an object (or shape) is any rigid symmetry. It is important to note that this does not motion that moves the object back onto itself. necessarily force the rigid motion to be the identity motion. Individual points may be moved to different positions, even though the whole object is moved back into itself. And, of course, the identity motion is itself a symmetry, one possessed by every object and that from now on we will call simply the identity.

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Four Types of Symmetry Example 11.6 The Symmetries of a Square

Since there are only four basic kinds of rigid motions of What are the possible rigid motions that move the two-dimensional objects in two-dimensional space, square in Fig. 11-17(a) back onto itself? there are also only four possible types of symmetries: reflection symmetries , rotation symmetries , translation symmetries , and glide reflection symmetries .

Fig. 11-17(a)

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Example 11.6 The Symmetries of a Example 11.6 The Symmetries of a Square Square

First, there are reflection symmetries. For example, if we The square has rotation symmetries as well. Using the

use the line l1 as the axis of reflection, the square falls center of the square O as the rotocenter, we can back into itself with rotate the square by an angle of 90º. This moves the A points A and B to B, B to C, C to D and D to A. interchanging places and Likewise, rotations with C and D interchanging rotocenter O and angles of places. It is not hard to 180º, 270º, and 360º, think of three other respectively, are also reflection symmetries, with symmetries of the square. axes l2, l3, and l4 as shown. Notice that the 360º rotation is just the identity.

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8 Symmetries of the Square Example 11.7 The Symmetries of a Propeller All in all, we have easily found eight symmetries for the square. Four of them are reflections, and the Let ’s now consider the symmetries of the shape shown– other four are rotations. Could there be more? What if we combined one of the reflections with one of the a two-dimensional version of a boat propeller (or a rotations? A symmetry combined with another ceiling fan if you prefer) with four blades. symmetry, after all, has to be itself a symmetry. It turns out that the eight symmetries we listed are all there are–no matter how we combine them we always end up with one of the eight.

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Example 11.7 The Symmetries of a Example 11.7 The Symmetries of a Propeller Propeller

Once again, we have a shape with four reflection There are four rotation symmetries with rotocenter O and angles of 90º, 180º, 270º, and 360º, respectively. symmetries, the axes of reflection are l1, l2, l3, and l4. And, just as with the square, there are no other possible symmetries.

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Objects with the Same Symmetries Objects with the Same Symmetries

An important lesson lurks behind Examples 11.6 and 11.7: Two different-looking objects can have exactly The symmetry type for the square, the propeller, and the same set of symmetries. A good way to think each of the objects shown is called D4 (shorthand for about this is that the square and the propeller, while four reflections plus four rotations). certainly different objects, are members of the same “symmetry family ” and carry exactly the same symmetry genes. Formally, we will say that two objects or shapes are of the same symmetry type if they have exactly the same set of symmetries.

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Example 11.8 The Symmetry of Type Z4 Example 11.8 The Symmetry of Type Z4

Let ’s consider now the propeller shown. This object is Here we still have the four rotation symmetries (90º, 180º, only slightly different from the one in Example 11.7, but 270º, and 360º), but there are no reflection symmetries! from the symmetry point of view the difference is This makes sense because the individual blades of the significant. propeller have no reflection symmetry.

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Example 11.8 The Symmetry of Type Z4 Example 11.9 The Symmetry of Type Z2

As can be seen here, a vertical reflection is not a Here is one last propeller example. Every once in a symmetry, and neither are any of the other reflections. while a propeller looks like the one here, which is kind of

This object belongs to a new symmetry family called Z4 a cross between the previous two examples–only (shorthand for the symmetry type of objects having four opposite blades are the same. rotations only).

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Example 11.9 The Symmetry of Type Z2 Example 11.9 The Symmetry of Type Z2

This figure has no reflection symmetries, and a 90º The only symmetries of this shape are a 180º rotation (turn it upside down and it looks the same!) and the rotation won ’t work either. 360º rotation (the identity).

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Example 11.9 The Symmetry of Type Z2 Example 11.10 The Symmetry of Type D1

An object having only two rotation symmetries (the One of the most common symmetry types occurring in identity and a 180º rotation symmetry) is said to be of nature is that of objects having a single reflection symmetry type Z . Here are a few additional examples symmetry plus a single rotation symmetry (the identity). 2 ’ This symmetry type is called D1. Notice that it doesn t of shapes and objects with symmetry type Z2. matter if the axis of reflection is vertical, horizontal, or slanted.

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Example 11.11 The Symmetry of Type Z1 Example 11.12 Objects with Lots of Symmetry Many objects and shapes are informally considered to In everyday language, certain objects and shapes are have no symmetry at all, but this is a little misleading, said to be “highly symmetric ” when they have lots of since every object has at least the identity symmetry. rotation and reflection symmetries. Here two very Objects whose only symmetry is the identity are said to different looking snowflakes, but from the symmetry have symmetry type Z . 1 point of view they are the same:

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Example 11.12 Objects with Lots of Example 11.12 Objects with Lots of Symmetry Symmetry

All snowflakes have six reflection symmetries and six This figure shows a decorative ceramic plate. It has nine

rotation symmetries. Their symmetry type is D6. (Try to reflection symmetries and nine rotation symmetries, find the six axes of reflection symmetry and the six and, as you may have guessed, its symmetry type is

angles of rotation symmetry.) called D9.

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Example 11.12 Objects with Lots of Properties of Glide Reflections Symmetry In each of the objects in Example 11.12, the number Finally, we have an architectural blueprint of the dome of reflections matches the number of rotations. This of the Sports Palace in Rome, Italy. The design has 36 was also true in Examples 11.6, 11.7, and 11.10. reflection and 36 rotation symmetries (symmetry type Coincidence? Not at all. When a finite object or

D36 ). shape has both reflection and rotation symmetries, the number of rotation symmetries (which includes the identity) has to match the number of reflection symmetries! Any finite object or shape with exactly N reflection symmetries and N rotation symmetries is

said to have symmetry type DN.

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Example 11.13 The Symmetry Type D∞ Example 11.14 Shapes with Rotations, but No Reflections Are there two-dimensional objects with infinitely many symmetries? The answer is yes–circles. A circle has We now know that if a finite two-dimensional shape has infinitely many reflection symmetries (any line passing rotations and reflections, it must have exactly the same through the center of the circle can serve as an axis) as number of each. In this case, the shape belongs to the well as infinitely many rotation symmetries (use the D family of symmetries, specifically, it has symmetry type center of the circle as a rotocenter and any angle of DN. However, we also saw in Examples 11.8, 11.9, and 11.11 shapes that have rotations, but no reflections. In rotation will work). We call the symmetry type of the this case, we used the notation Z to describe the circle D (the ∞ is the mathematical symbol for N ∞ symmetry type, with the subscript N indicating the “ ” infinity ). actual number of rotations.

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Example 11.14 Shapes with Rotations, but Types of Symmetries No Reflections We are now in a position to classify the possible The figure on the left shows a flower with five petals that symmetries of any finite two- dimensional shape or

has symmetry type Z5. The figure on the right shows an object. (The word finite is in there for a reason, which airplane turbine with 24 blades that has symmetry type will become clear in the next section.) The possibilities

Z24 . Notice that the absence of reflections is the result of boil down to a surprisingly short list of symmetry types: some twist or bump on the petals or blades.

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Types of Symmetries Types of Symmetries

DN ZN This is the symmetry type of shapes with both rotation This is the symmetry type of shapes with rotation and reflection symmetries.The subscript N (N = 1, 2, 3, symmetries only. The subscript N (N = 1, 2, 3, etc.) etc.) denotes the number of reflection symmetries, denotes the number of rotation symmetries. which is always equal to the number of rotation symmetries. (The rotations are an automatic D∞ consequence of the reflections–an object can ’t This is the symmetry type of a circle and of circular have reflection symmetries without having an equal objects such as rings and washers, the only possible number of rotation symmetries.) two-dimensional shapes or objects with an infinite number of rotations and reflections.

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Pattern

We will formally define a pattern as an infinite “shape” Chapter 11: consisting of an infinitely repeating basic design called the motif of the pattern. The reason we have The Mathematics “shape” in quotation marks is that a pattern is really of Symmetry an abstraction–in the real world there are no infinite objects as such, although the idea of an infinitely repeating motif is familiar to us from such everyday 11.7 Patterns objects as pottery, tile designs, and textiles.

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Pattern Symmetry in Patterns

Just like finite shapes, patterns can be classified by Here are a few examples: their symmetries. The classification of patterns according to their symmetry type is of fundamental importance in the study of molecular and crystal organization in chemistry, so it is not surprising that some of the first people to seriously investigate the symmetry types of patterns were crystallographers.

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Symmetry in Patterns Border Patterns

Archaeologists and anthropologists have also found Border patterns (also called linear patterns) are that analyzing the symmetry types used by a patterns in which a basic motif repeats itself particular culture in their textiles and pottery helps indefinitely in a single direction, as in an architectural them gain a better understanding of that culture. frieze, a ribbon, or the border design of a ceramic pot. We will briefly discuss the symmetry types of border and wallpaper patterns.

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Border Patterns Border Patterns - Translations

The most common direction in a border pattern A border pattern always has translation symmetries – (what we will call the direction of the pattern) is they come with the territory. There is a basic horizontal, but in general a border pattern can be in translation symmetry v (v moves each copy of the any direction (vertical, slanted 45º, etc.). motif one unit to the right), the opposite translation –v and any multiple of v or –v.

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Border Patterns - Reflections Border Patterns - Reflections

A border pattern can have (a) no reflection In this last case the border pattern automatically symmetry, (b) horizontal reflection symmetry only, (c) picks up a half-turn symmetry as well. In terms of vertical reflection symmetries only, or (d) both reflection symmetries, these figures illustrate the only horizontal and vertical reflection symmetries. four possibilities in a border pattern.

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Border Patterns - Rotations Border Patterns - Rotations

Like with any other object, the identity (i.e., a 360º rotation) is a rotation symmetry of every border Thus, in terms of rotation symmetry there are two kinds pattern, so every border pattern has at least one of border patterns: those whose only rotation rotation symmetry. The only other possible rotation symmetry is the identity (a) and those having half-turn symmetry of a border pattern is a half-turn symmetry in addition to the identity (b). (180º rotation). Clearly, no other angle of rotation can take a horizontal pattern and move it back onto itself.

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Border Patterns - Glide Reflections Border Patterns - Glide Reflections

A border pattern can have a glide reflection On the other hand, the border pattern shown does not symmetry, but there is only one way this can happen: have horizontal reflection symmetry (the footprints do The axis of reflection has to be a line along the center not fall back onto other footprints), but a glide by the of the pattern, and the reflection part of the glide vector w combined with a reflection along the axis l reflection is not by itself a symmetry of the pattern. result in an honest-to-goodness glide reflection This means that a border pattern having horizontal symmetry. An important property of the glide reflection reflection symmetry such as the one shown is not symmetry is that the vector w is always half the length considered to have glide reflection symmetry. of the basic translation symmetry v.

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Border Patterns - Glide Reflections Symmetries of Border Patterns

1. The identity : All border patterns have it. This border pattern has a vertical reflection symmetry as well as a glide reflection symmetry. In these cases a 2. Translations : All border patterns have them. There half-turn symmetry (rotocenter O) comes free in the are a basic translation v, the opposite translation –v, bargain. and any multiples of these.

3. Horizontal reflection : Some patterns have it, some don’t. There is only one possible horizontal axis of reflection, and it must run through the middle of the pattern.

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Symmetries of Border Patterns Symmetries of Border Patterns

4. Vertical reflections : Some patterns have them, some 6. Glide reflections : Some patterns have them, some don’t. Vertical axes of reflection (i.e., axes don’t. Neither the reflection nor the glide can be perpendicular to the direction of the pattern) can symmetries on their own. The length of the glide w is run through the middle of a motif or between two half that of the basic translation The axis of the motifs. reflection runs through the middle of the pattern v.

5. Half-turns : Some patterns have them, some don’t. Rotocenters must be located at the center of a motif or between two motifs.

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Border Patterns - Symmetry Families Border Patterns - Symmetry Families

■ 11 . This symmetry type represents border patterns ■ mm . This symmetry type represents border patterns that have no symmetries other than the identity and with both a horizontal and a vertical reflection translation symmetry. symmetry. When both of these symmetries are present, there is also half-turn symmetry. ■ 1m . This symmetry type represents border patterns with just a horizontal reflection symmetry. ■ 12 . This symmetry type represents border patterns with only a half-turn symmetry. ■ m1 . This symmetry type represents border patterns with just a vertical reflection symmetry. ■ 1g . This symmetry type represents border patterns with only a glide reflection symmetry.

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Border Patterns - Symmetry Families Border Patterns - Symmetry Families

■ mg . This symmetry type represents border patterns Summary of the seven border pattern symmetry with a vertical reflection and a glide reflection families. symmetry. When both of these symmetries are present, there is also a half-turn symmetry.

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Wallpaper Patterns Wallpaper Patterns - Translations

Wallpaper patterns are patterns that fill the plane by Every wallpaper pattern has translation symmetry in at repeating a motif indefinitely along several (two or least two different (nonparallel) directions. more) nonparallel directions. Typical examples of such patterns can be found in wallpaper (of course), carpets, and textiles.

With wallpaper patterns things get a bit more complicated, so we will skip the details. The possible symmetries of a wallpaper pattern are as follows:

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Wallpaper Patterns - Reflections Wallpaper Patterns - Rotations

A wallpaper pattern can have (a) no reflections, (b) In terms of rotation symmetries, a wallpaper pattern reflections in only one direction, (c) reflections in two can have (a) the identity only, (b) two rotations nonparallel directions, (d) reflections in three (identity and 180º), (c) three rotations (identity, 120º, nonparallel directions, (e) reflections in four nonparallel and 240º), (d) four rotations (identity, 90º, 180º, and directions, and (f) reflections in six nonparallel 270º), and (e) six rotations (identity, 60º, 120º, 180º, 240º, directions. There are no other possibilities. Note that and 300º). There are no other possibilities. Once again, particularly conspicuous in its absence is the case of note that a wallpaper pattern cannot have exactly reflections in exactly five different directions. five different rotations.

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Wallpaper Patterns - Glide Reflections Surprising Fun Fact!

A wallpaper pattern can have (a) no glide reflections, In the early 1900s, it was shown mathematically that (b) glide reflections in only one direction, (c) glide there are only 17 possible symmetry types for wallpaper reflections in two nonparallel directions, (d) glide patterns. This is quite a surprising fact–it means that the reflections in three nonparallel directions, (e) glide hundreds and thousands of wallpaper patterns one reflections in four nonparallel directions, and (f) glide can find at a decorating store all fall into just 17 reflections in six nonparallel directions. There are no different symmetry families. other possibilities.

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