Rotational Symmetry

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/