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### Rotational Symmetry

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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

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
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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Associated Topics:
High School Basic Algebra
High School Geometry
High School Physics/Chemistry
High School Symmetry/Tessellations

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