Some announcements. Game reflections deadline extended to Monday (4/4)
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1 Symmetry
2 Some announcements Game reflections deadline extended to Monday (4/4)
3 Some announcements Game reflections deadline extended to Monday (4/4) Next math talk on Wednesday (4/6) at 4pm. Speaker is Caroline Turnage of Duke. (Future talks on 4/13, 4/19, and 4/20.
4 Some announcements Game reflections deadline extended to Monday (4/4) Next math talk on Wednesday (4/6) at 4pm. Speaker is Caroline Turnage of Duke. (Future talks on 4/13, 4/19, and 4/20.
5 Next week I have a visitor from the University of Washington (Paul Smith). Here s some things you need to know. No regular office hours (appt. only). I will not assign any homework to be due during the week.
6 Next week I have a visitor from the University of Washington (Paul Smith). Here s some things you need to know. No regular office hours (appt. only). I will not assign any homework to be due during the week. Monday will be a regular class day. On Wednesday, Kent Vashaw (TA) will run a classwork session.
7 Next week I have a visitor from the University of Washington (Paul Smith). Here s some things you need to know. No regular office hours (appt. only). I will not assign any homework to be due during the week. Monday will be a regular class day. On Wednesday, Kent Vashaw (TA) will run a classwork session. No class on Friday.
8 Next week I have a visitor from the University of Washington (Paul Smith). Here s some things you need to know. No regular office hours (appt. only). I will not assign any homework to be due during the week. Monday will be a regular class day. On Wednesday, Kent Vashaw (TA) will run a classwork session. No class on Friday.
9 Finding symmetry Last time we discussed basic rigid motions: reflections, rotations, translations, and glide reflections.
10 Finding symmetry Last time we discussed basic rigid motions: reflections, rotations, translations, and glide reflections. Today we will start with identifying axes of reflections, rotocenters, angles of rotation, and vectors of translation.
11 Reflections Recall that a reflection is a rigid motion that moves an object into a new position that is a mirror image of the starting position. That is, it flips the object over a line, called the axis of reflection.
12 Reflections Recall that a reflection is a rigid motion that moves an object into a new position that is a mirror image of the starting position. That is, it flips the object over a line, called the axis of reflection.
13 Properties of reflections
14 Properties of reflections A reflection is completely determined by its axis l.
15 Properties of reflections A reflection is completely determined by its axis l. A reflection is completely determined by a single point-image pair P and P (unless P = P ).
16 Properties of reflections A reflection is completely determined by its axis l. A reflection is completely determined by a single point-image pair P and P (unless P = P ). A reflection has infinitely many fixed points (all points on l).
17 Properties of reflections A reflection is completely determined by its axis l. A reflection is completely determined by a single point-image pair P and P (unless P = P ). A reflection has infinitely many fixed points (all points on l). A reflection is an improper rigid motion.
18 Properties of reflections A reflection is completely determined by its axis l. A reflection is completely determined by a single point-image pair P and P (unless P = P ). A reflection has infinitely many fixed points (all points on l). A reflection is an improper rigid motion. When the reflection is applied twice, we get the identity motion.
19 Drawing a reflection If we are given the axis of symmetry l and a point P, we can locate P in the following way:
20 Drawing a reflection If we are given the axis of symmetry l and a point P, we can locate P in the following way: Draw a line from P perpendicular to l. Call the point of intersection R.
21 Drawing a reflection If we are given the axis of symmetry l and a point P, we can locate P in the following way: Draw a line from P perpendicular to l. Call the point of intersection R. The point P lies on l and is the same distance from R as P is from R (just in the opposite direction).
22 Drawing a reflection Here we are given a point P and an axis of reflection L.
23 Drawing a reflection We draw a line from P perpendicular to L.
24 Drawing a reflection We measure the distance from P to the intersection. The same distance in the opposite direction gives P.
25 Finding the axis of reflection Given a point P and its reflection P, we can identify the axis of reflection in the following way:
26 Finding the axis of reflection Given a point P and its reflection P, we can identify the axis of reflection in the following way: Draw a line l from P to P.
27 Finding the axis of reflection Given a point P and its reflection P, we can identify the axis of reflection in the following way: Draw a line l from P to P. Draw the perpendicular bisector of l (perpendicular to and splits in half).
28 Finding the axis of reflection Given a point P and its reflection P, we can identify the axis of reflection in the following way: Draw a line l from P to P. Draw the perpendicular bisector of l (perpendicular to and splits in half). This is the axis of symmetry.
29 Drawing a reflection Here we are given a point P its image P.
30 Drawing a reflection Draw a line from P to P.
31 Drawing a reflection Draw the perpendicular bisector of that line.
32 Rotations Recall that a rotation is a rigid motion that pivots an object around a fixed point O.
33 Rotations Recall that a rotation is a rigid motion that pivots an object around a fixed point O. A rotation is described by three pieces of data:
34 Rotations Recall that a rotation is a rigid motion that pivots an object around a fixed point O. A rotation is described by three pieces of data: The rotocenter O.
35 Rotations Recall that a rotation is a rigid motion that pivots an object around a fixed point O. A rotation is described by three pieces of data: The rotocenter O. The angle of rotation.
36 Rotations Recall that a rotation is a rigid motion that pivots an object around a fixed point O. A rotation is described by three pieces of data: The rotocenter O. The angle of rotation. The direction of the rotation (clockwise or counterclockwise).
37 Rotations Here is an equilateral triangle rotated 90 about the point O.
38 Rotations Here is an equilateral triangle rotated 90 about the point O.
39 Properties of rotations
40 Properties of rotations A rotation is completely determined by two point-image pairs P, P and Q, Q.
41 Properties of rotations A rotation is completely determined by two point-image pairs P, P and Q, Q. A rotation that is not the identity motion has only one fixed point (the rotocenter).
42 Properties of rotations A rotation is completely determined by two point-image pairs P, P and Q, Q. A rotation that is not the identity motion has only one fixed point (the rotocenter). A rotation is a proper rigid motion.
43 Properties of rotations A rotation is completely determined by two point-image pairs P, P and Q, Q. A rotation that is not the identity motion has only one fixed point (the rotocenter). A rotation is a proper rigid motion. A 360 rotation is equivalent to the identity motion.
44 Drawing a rotation Given the rotocenter O and the angle of rotation α (and direction) we can find the image of a point P in the following way: Draw a line L from P to O.
45 Drawing a rotation Given the rotocenter O and the angle of rotation α (and direction) we can find the image of a point P in the following way: Draw a line L from P to O. Draw a line L through O such that the angle between L and L is α (helps to use a protractor). The point P is the point on L that is the same distance from O as P is from O
46 Drawing a rotation We are given a rotation with rotocenter O and angle of rotation 90 clockwise. We want to find the image of the point P.
47 Drawing a rotation Draw a line from P to O.
48 Drawing a rotation Draw a new line through O. such that the two lines make a 90 angle.
49 Drawing a rotation The point P is the point on the new line that is the same distance from O as P is from O.
50 Finding the rotocenter Given points P, Q and their images P, Q, we can identify the rotocenter in the following way:
51 Finding the rotocenter Given points P, Q and their images P, Q, we can identify the rotocenter in the following way: Draw the perpendicular bisector of the lines connecting P, P and Q, Q.
52 Finding the rotocenter Given points P, Q and their images P, Q, we can identify the rotocenter in the following way: Draw the perpendicular bisector of the lines connecting P, P and Q, Q. The point where the bisectors meet is the rotocenter.
53 Drawing a rotation Here we are given points P, Q and their images P, Q.
54 Drawing a rotation We draw lines between P, P and Q, Q.
55 Drawing a rotation Now we draw perpendicular bisectors for each of these lines.
56 Drawing a rotation The intersection points of the bisectors is the rotocenter.
57 Symmetry A symmetry of an object is a rigid motion that moves the object back onto itself.
58 Symmetry A symmetry of an object is a rigid motion that moves the object back onto itself. Every object has one symmetry:
59 Symmetry A symmetry of an object is a rigid motion that moves the object back onto itself. Every object has one symmetry: the identity motion.
60 Symmetry A symmetry of an object is a rigid motion that moves the object back onto itself. Every object has one symmetry: the identity motion. The idea is this: with no labels, an object should not appear any different.
61 Symmetries of regular polygon Triangle:
62 Symmetries of regular polygon Triangle: a triangle has six symmetries.
63 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square:
64 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square: a square has eight symmetries.
65 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square: a square has eight symmetries. Pentagon:
66 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square: a square has eight symmetries. Pentagon: a pentagon has ten symmetries.
67 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square: a square has eight symmetries. Pentagon: a pentagon has ten symmetries. These symmetries are known as dihedral symmetries and denoted D 3, D 4, and D 5, respectively.
68 Symmetries of regular polygon Triangle: a triangle has six symmetries. Square: a square has eight symmetries. Pentagon: a pentagon has ten symmetries. These symmetries are known as dihedral symmetries and denoted D 3, D 4, and D 5, respectively. Other objects can have these symmetry types. For example, a propeller has symmetry type D 4.
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