Week 11 - Symmetry

rotation is defined by:

(1) 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).

the result of applying the glide reflection

(slide 55 - 57)

vector of translation

(usually denoted by v) The two pieces of information (direction and length of the translation) are combined in the form of a vector of translation. The vector of translation is represented by an arrow-the arrow points in the direction of translation and the length of the arrow is the length of the translation.

PROPERTIES OF ROTATIONS

-A 360º rotation is equivalent to the identity motion. - A rotation is a proper rigid motion. - A rotation that is not the identity motion has only one fixed point, its rotocenter. - A rotation is completely determined by two point-image pairs P, P´ and Q,Q´

PROPERTIES OF GUIDE REFLECTIONS

-A guide reflection has no fixed points. - A guide reflection is an improper rigid motion. - A guide reflection is completely determined by two point-image pairs P, P´ and Q,Q´. - When a guide reflection with vector v and axis of reflection l is followed with a translation with vector -v and the same axis of reflection l we get to the identity motion.

PROPERTIES OF TRANSLATIONS

-A translation is completely determined by a single point-image pair P and P´. - A translation has no fixed points. - A translation is a proper rigid motion. - When a translation with vector v is followed with a translation with vector -v we get to the identity motion.

PROPERTIES OF REFLECTIONS (summary)

1. A reflection is completely determined by its axis l. 2. A reflection is completely determined by a single point-image pair P and P´ (as long as P´ ≠ P). 3. A reflection has infinitely many fixed points (all points on l). 4. A reflection is an improper rigid motion. 5. When the same reflection is applied twice, we get the identity motion.

Property 6

A 360º rotation is equivalent to a 0º rotation, and a 0º rotation is just the identity motion.

Property 14 of glide reflections

A fixed point of a glide reflection would have to be a point that ends up exactly where it started after it is first translated and then reflected. This cannot happen because the translation moves every point and the reflection cannot undo the action of the translation. It follows that a glide reflection has no fixed points.

Property 15

A glide reflection is a combination of a proper rigid motion (the translation) and an improper rigid motion (the reflection). Since the translation preserves left-right and clockwise-counterclockwise orientations but the reflection reverses them, the net result under a glide reflection is that orientations are reversed. Thus, a glide reflection is an improper rigid motion.

determining a glide reflection

A glide reflection is completely determined by two point-image pairs P, P´ and Q, Q´. Given a point-image pair P and P´ under a glide reflection, we do not have enough information to determine the glide reflection, but we do know that the axis l must pass through the midpoint of the line segment PP´.

reflection

A reflection in the plane is a rigid motion that moves an object into a new position that is a mirror image of the starting position. (From a purely geometric point of view a reflection can be defined by showing how it moves a generic point P in the plane.)

M

A rigid motion of the plane-let's call it M-moves each point in the plane from its starting position P to an ending position P´, also in the plane.

identity motion

A rigid motion that is equivalent to not moving the object at all

symmetry

A symmetry of an object (or shape) is any rigid motion that moves the object back onto itself.

equilateral triangle

A triangle with three congruent sides

Z2 symmetry

An object having only two rotation symmetries (the identity and a 180º rotation symmetry) (slide 86)

proper rigid motion

Any motion that preserves the left-right and clockwise-counterclockwise orientations of objects

translation example

Dragging of the cursor on a computer screen. Regardless of what happens in between, the net result when you drag an icon on your screen is a translation in a specific direction and by a specific length.

identifying the rotocenter

Given a second pair of points Q and Q´ we can identify the rotocenter O as the point where the perpendicular bisectors of PP´and QQ´ meet. (slide is 41)

determining the axis of a reflection

Given a second point-image pair Q and Q´, we can determine the axis of the reflection: It is the line passing through the points M (midpoint of the line segment PP´) and N (midpoint of the line segment QQ´). (slide 61)

Property 5

If P´ is the image of P under a reflection, then (P´)´ = P (the image of the image is the original point). Thus, when we apply the same reflection twice, every point ends up in its original position and the rigid motion is equivalent to not having moved the object at all.

Property 10

If we are given a point P and its image P´ under a translation, the arrow joining P to P´ gives the vector of the translation. Once we know the vector of the translation, we know where the translation moves any other point. Thus, a single point-image pair P and P´ is all we need to completely determine the translation.

Property 2

If we know a point P and its image P´ under the reflection (and assuming P´ is different from P), we can find the axis l of the reflection (it is the perpendicular bisector of the segment PP´). Once we have the axis l of the reflection, we can find the image of any other point (property 1).

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.

Property 11 of translations

In a translation, every point gets moved some distance and in some direction, so a translation has no fixed points.

Property 8

In every rotation, the rotocenter is a fixed point, and except for the case of the identity (where all points are fixed points) it is the only fixed point.

finding the axis when midpoints of PP´ and QQ´ are the same point

In the event that the midpoints of PP´ and QQ´ are the same point M, we can still find the axis l by drawing a line perpendicular to the line PQ passing through the common midpoint M. (slide 63)

Properties of Rotations Note:

In the special case where PP´ and QQ´ happen to have the same perpendicular bisector, the rotocenter O is the intersection of PQ and PQ´. (slide 42)

axis of reflection

In two dimensions, the "mirror"

rotation

Informally, a rotation in the plane is a rigid motion that pivots or swings an object around a fixed point O.

fixed point

It may happen that a point P is moved back to itself under M(swirly), in which case we call P a fixed point of the rigid motion M(swirly).

Z1 symmetry

Objects whose only symmetry is the identity (all objects have at least this) (slide 88)

finding the image of a point

Once we find the axis of reflection l, we can find the image of one of the points-say P´- under the reflection. This gives the intermediate point P*, and the vector that moves P to P* is the vector of translation v. (slide 62)

Points on the axis

Points on the axis itself are fixed points of the reflection.

Property 4

Reflections are improper rigid motions, meaning that they change the left-right and clockwise-counterclockwise orientations of objects.

Types of symmetries

Since there are only four basic kinds of rigid motions of two-dimensional objects in two-dimensional space, there are also only four possible types of symmetries: 1. symmetries, 2. rotation symmetries, 3. translation symmetries, 4. glide reflection symmetries

rigid motion

The act of taking an object and moving it from some starting position to some ending position without altering its shape or size (Since in a rigid motion the size and shape of an object are not altered, distances between points are preserved)

(rigid motion rule)

The distance between any two points X and Y in the starting position is the same as the distance between the same two points in the ending position

Property 13

The effect of a translation with vector v can be undone by a translation of the same length but in the opposite direction The vector for this opposite translation can be conveniently described as -v. Thus, a translation with vector v followed with a translation with vector -v is equivalent to the identity motion. (slide 50)

slide 31

The figure illustrates geometrically how a clockwise rotation with rotocenter O and angle of rotation moves a point P to the point P

Property 3

The fixed points of a reflection are all the points on the axis of reflection l.

Glide reflection example

The footprints left behind by someone walking on soft sand are a classic example of a glide reflection

The image of any point

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

Property 17

To undo the effects of a glide reflection, we need a second glide reflection in the opposite direction. To be more precise, if we move an object under a glide reflection with vector of translation v and axis of reflection l and then follow it with another glide reflection with vector of translation -v and axis of reflection still l, we get the identity motion. It is as if the object was not moved at all.

same set of symmetries

Two different-looking objects can have exactly the same set of symmetries

Property 9

Unlike a reflection, a rotation cannot be determined by a single point-image pair P and P´ it takes a second point-image pair Q and Q´ to nail down the rotation. (slide 40)

image

We will call the point P´ the image of the point P under the rigid motion M(swirly) and describe this informally by saying that M(swirly) moves P to P´.

Property 7

When an object is rotated, the left-right and clockwise-counterclockwise orientations are preserved (a rotated left hand remains a left hand, and the hands of a rotated clock still move in the clockwise direction)

Property 12

When an object is translated, left-right and clockwise-counterclockwise orientations are preserved: A translated left hand is still a left hand, and the hands of a translated clock still move in the clock-wise direction. In other words, translations are proper rigid motions.

symmetry example

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 symmetry (this does not 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 )

half-turn

a 180º rotation

glide reflection

a rigid motion obtained by combining a translation (the glide) with a reflection. Moreover, the axis of reflection must be parallel to the direction of translation. Thus, a glide reflection is described by two things: the vector of the translation v and the axis of the reflection l, and these two must be parallel.

determining the translation

a single point-image pair P and P´ is all we need to completely determine the translation.

scalene triangle

a triangle with no congruent sides

isosoceles triangle

a triangle with two congruent (equal) sides

translation

essentially dragging an object in a specified direction and by a specified amount (the length of the translation).

D4 symmetry

four reflections plus four rotations (slide 79)

P

in rigid motion, a starting position on the plane

P'

in rigid motion, an ending position on the plane

D1 symmetry

objects having a single reflection symmetry plus a single rotation symmetry (the identity) Notice that it doesn't matter if the axis of reflection is vertical, horizontal, or slanted (slide 87)

Z4 symmetry

objects having four rotations only (slide 82)

M and N (swirly)

script letters such as M and N to denote rigid motions, which should eliminate any possible confusion between the point M and the rigid motion M (swirly).

Reflections of a Triangle

slide 20 - 22

rotation degrees

slide 36

the translation of a triangle ABC

slide 47

8 Symmetries of the Square

slide 72 - 74

symmetry for starters

symmetry is a property of an object that looks the same to an observer standing at different vantage points. (ant and triangles)

basic rigid motions of the plane

there are only four possibilities: A rigid motion is equivalent to (1) a reflection, (2) a rotation, (3) a translation, (4) a glide reflection

improper rigid motions

they change the left-right and clockwise-counterclockwise orientations of objects

symmetry type

two objects or shapes are of the same symmetry type if they have exactly the same set of symmetries.

equivalent rigid motions

two rigid motions that move an object from the same starting position to the same ending position are equivalent rigid motions (We are only concerned with the net effect of the motion-where the object started and where the object ended. What happens during the "trip" is irrelevant.)

confusing a 180º rotation with a reflection

we can see that they are very different from just observing that the reflection is an improper rigid motion, whereas the 180º rotation is a proper rigid motion.