Date: 11/05/97 at 15:06:21 From: Beth Auge Subject: Geometry - rotational symmetry I am looking for a precise definition of rotational symmetry of a figure in a 2-dimensional plane. I have checked in three different geometry books here in our district and they have different explanations and are somewhat contradictory. 1) According to one of the books, if a figure has one line of symmetry, it has rotational symmetry somewhere between 0 and 360 degrees. But where does that leave the isosceles trapezoid? The isosceles trapezoid does have one line of symmetry, but does not have rotational symmetry until it has rotated 360 degrees, which does not fit the definition. 2) Another book says a figure has rotational symmetry if it "looks the same." What kind of math definition is that? This is an 8th grade textbook that I am using in an advanced 7th grade pre-algebra class. 3) The geometry teacher here says he doesn't even use those two terms in connection with each other because symmetry and rotation are not interrelated. These kids are sharp and ask some high level questions. Can you help us out with this rotational symmetry idea? Sincerely, Beth Auge 7th grade math Jackson Hole Middle School Jackson, Wyoming
Date: 11/05/97 at 19:11:54 From: Doctor Tom Subject: Re: Geometry - rotational symmetry Hi Beth, Actually, definition 2 is probably the most useful, if it's stated correctly. In general, an object is "symmetric" if it "looks the same" after it has undergone some transformation or operation. Here's a more precise definition: an object (a set of points in the plane) has symmetry if you can find some transformation such that the set of points before the transformation is the same set as the set of points after the transformation. So we might say "looks exactly the same." Said another way, an object X has symmetry if there exists a non-trivial transformation T such that X = T(X). And you're right - line symmetry doesn't imply rotational symmetry. Let's look at some simple symmetries first. For example, you can have reflection or mirror symmetry if the object looks the same after being reflected in a mirror. A human is (roughly) mirror-symmetric, since our left and right sides are pretty much the same. In other words, if I took a photo of a person and showed it to you, you would have a hard time telling me where I had taken a photo of a person, or of a person's image in a "perfect" mirror, right? An isosceles triangle is mirror symmetric, but a triangle with sides of three different lengths is not. A circle is mirror symmetric, as is an ellipse. An interesting exercise is to look at the upper-case letters of the alphabet to see which ones are mirror symmetric. I get: A B C D E H I M O T U V W X Y If you reflect the other letters, it looks as if they have been drawn by someone with dyslexia. (Note that sometimes the mirrors are on the right of the letters and sometimes on the tops (or bottoms) to get the same reflected image.) But "reflecting in a mirror" is just one operation you can do to objects. How about rotating them by 180 degrees? Which ones look the same after that? I get: H I O S X Z These are symmetric under a 180-degree rotation. Some things are symmetric under a 90-degree rotation: squares, for example. In fact, squares have a bunch of symmetries - they are also symmetric under a 180-degree rotation, or under a reflection. A perfect pentagon is symmetric under a 72-degree (1/5 turn) rotation (or 144-degree rotation, or a 216-degree rotation, etc.). A polygon with 127 equal sides and angles has symmetry under a rotation of 1/127 turn (or 2/127 turn or 3/127 turn, ...). Some letters, (like F, G, P) don't have any standard geometric symmetries. Usually, in mathematics, "rotational symmetry" means that ANY rotation leaves the object looking (exactly) the same. On a plane, there aren't many things that are rotationally symmetric - circles are, or sets of circles with a common center. In three dimensions, however, there are thousands of interesting objects with rotational symmetry. Imagine taking any wire shape (bent however you like), holding the two ends, and spinning it so fast that it's a blur. The shape of that blur will have rotational symmetry. Common examples are spheres, cylinders, cones, tori (doughnut shapes), etc. Although it's not obvious from a 7th grader's point of view, symmetry is so important that huge areas of mathematics are basically devoted to the study of symmetry. For example, the equation: x^2 + y^2 + z^2 + xyz is symmetric in the variables, because if you jumble them around (put in x where you see y and y where you see x, for example), the equation is exactly the same. Or maybe a better way to look at it is this: if x, y, and z are three numbers, 3, 5, and 11, but you can't remember which is which, for symmetric equations it doesn't matter. However you decide to match them up, you'll get the same numerical result. The "object" is the equation; the operation is to jumble the variables. After the jumbling, the equation is exactly the same (well, the terms may be in a different order, but the numerical value is the same). And not just mathematics - physics too. Imagine the collision of two perfect balls - they come in, hit each other, and bounce off in different directions. Imagine taking a film of such a collision, and playing it backwards. Would the physics still be correct? Yes! Elastic collisions are symmetric with respect to "time reversal" - it's a weird concept, but it does fit the general definition - the object (the physical laws of elastic collisions) are symmetric if you look at them with time going backwards. This is just scratching the surface, but most other symmetries in physics are even more bizarre. -Doctors Tom and Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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