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 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-2 7e: 1.1 - 2 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? CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-3 7e: 1.1 - 3 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-4 7e: 1.1 - 4 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-5 7e: 1.1 - 5 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-6 7e: 1.1 - 6 1 11/18/2013 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-7 7e: 1.1 - 7 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-8 7e: 1.1 - 8 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-9 7e: 1.1 - 9 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-10 7e: 1.1 - 10 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 . CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-11 7e: 1.1 - 11 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-12 7e: 1.1 - 12 2 11/18/2013 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´. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-13 7e: 1.1 - 13 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-14 7e: 1.1 - 14 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-15 7e: 1.1 - 15 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-16 7e: 1.1 - 16 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-17 7e: 1.1 - 17 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-18 7e: 1.1 - 18 3 11/18/2013 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 . CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-19 7e: 1.1 - 19 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-20 7e: 1.1 - 20 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. CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-21 7e: 1.1 - 21 CopyrightCopyright © © 20142010 Pearson Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics,11.1-22 7e: 1.1 - 22 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.