Name: __________________
Straight and Symmetries 4
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Student 1: None of us has been able to draw a figure that has ½turn symmetry and only one reflection symmetry. Can you draw one
for us?
Mathematician:
Student 2:
Maybe it is not possible to draw such a figure.
But surely we can if we only tried long enough!
Mathematician: Let’s try to answer this question by exploring more
about symmetries.
Student 3:
OK. Just what is a symmetry?
Mathematician:
weeks back?
Student 4:
Good question.
Do you remember what we said a few
We first made some definitions:
DEFINITIONS. An isometry (i-SOM-e-tree) is a transformation that
preserves distances (and angle measures). A symmetry of a figure
is an isometry of a region of space that takes the figure (or the
portion of it in the region) onto itself. (dictionary terms)
Student 1: We used transparencies in order to make sure that
distances and angle measures were not changed.
Student 2:
Let’s summarize what we learned about symmetries of
straight lines and circles:
A straight line has:
–
translation symmetry; also called constant curvature
–
½-turn symmetry about every point on the line
–
mirror symmetry in the line; also called reflectionin-the-line symmetry, or bilateral symmetry
–
mirror symmetry perpendicular to the line at any
point on the line; also called reflection-perpendicularto-the-line symmetry at any point on the line
A circle has:
–
rotation symmetry about its center; also called
constant curvature.
–
mirror symmetry along any diameter
Mathematician: Good! So, the isometries that we used were
translations, rotations, and reflections. For finite figures (like
circles or the figures at the beginning of class) we need only
rotations or reflections. Translation symmetry only works for
figures like straight lines that go on and on indefinitely. Later,
we will also explore another symmetry, called a glide, for figures
that go on and on.
Name: __________________
Straight and Symmetries 4
Student 3:
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Can you show us an example of glide symmetry?
Mathematician: Sure!
and on indefinitely:
Look at a decoration that we imagine goes on
This has glide symmetry – it can move forward or backward but not
by any distance. It must go from bump to bump.
Student 4:
Cool!
Mathematician:
Now let’s look at symmetries of a rectangle.
Student 1: A rectangle seems to have ½-turn symmetry about some
point in the center.
Student 2:
How do we find that point?
Mathematician: I suggest we first draw in a diagonal of the
rectangle ABCD and mark its midpoint P:
D
A
P
B
C
Student 3: This diagonal is a transversal of the parallel sides,
AD and BC, and thus the marked angles, CBD and ADB, are
congruent. We also showed this in class.
Student 4:
We also showed that, AD and BC, have a common
Name: __________________
Straight and Symmetries 4
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perpendicular thru P.
Mathematician: And now, if we do a ½-turn about the center point
P then B→D, (read “B goes to D”), CBD→ADB, C→A. To help see
this, I suggest that everyone copy the above diagram on a blank
sheet of paper and perform the ½-turn of the paper.
Student 1: Because of the ½-turn we know that P is the midpoint
of the common perpendicular.
Student 2: But also the sides, CD and BA, are parallel and so the
same applies to them
Student 3:
So, CD and BA, also have a common perpendicular.
Mathematician:
Good, so finally we have the picture:
D
A
P
C
B
Student 1: I notice that the rectangle has ½-turn symmetry and 2
reflection symmetries. How is this going to help us decide if there
are any figures with ½-turn symmetry and only 1 reflection
symmetry?
Mathematician:
to show.
First let’s formulate carefully what it is we want
Student 2: We want to show that there cannot be any figure with
½-turn symmetry and only 1 reflection symmetry.
Name: __________________
Straight and Symmetries 4
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Mathematician: Yes, that is what we want to show but that is
negative – trying to prove that something does not exist. We first
need to turn it into a positive statement.
Student 3:
Like what?
Mathematician: How about: “If a figure has ½-turn symmetry and a
reflection symmetry, then it must also have at least one more
reflection symmetry.”? Will this do what we want?
Student 4: Yes, this would show that no figure with ½-turn
symmetry has only 1 reflection symmetry.
Student 1: But how can we possibly show this for ALL figures.
Maybe there are figures that no one has thought of yet.
Mathematician:
this Theorem:
Good point.
Let’s try.
We are trying to prove
Theorem. If a figure has ½-turn symmetry and a reflection
symmetry, then it must also have at least one more reflection
symmetry.
Student 2:
What does “theorem” mean?
Mathematician: The original meaning of “theorem” was “that which
has been seen”. We want to see why it is true.
Student 3:
How do we see it?
Mathematician: We devise a communication that convinces and
answers – “why?” Such a communication is called a proof. So now
we can define “theorem” to be a statement which has a proof.
Student 4: But this does not sound like the “2-column proofs” that
we have heard about.
Mathematician: That is true. A 2-column proof is only one way of
attempting to give a communication that convinces and answers –
“Why?” There are many other ways. Proofs can involve pictures and
motions and imaginations, in addition to words.
Student 1: How do we find a proof of our theorem so that it
convinces and answers – “why?”
Mathematician: Let’s imagine we have a figure with ½-turn
symmetry and a reflection symmetry.
Name: __________________
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Student 2: We do not have to imagine this, we already constructed
many examples.
Mathematician: That is true, but we want to do this with any
figure with ½-turn symmetry and a reflection symmetry. We need to
imagine doing this with any such figure even those that no one
thought of yet.
Student 3: That is weird!
what it is?
Mathematician:
How can we imagine it if no-one knows
Well, what DO we know about this figure?
Student 4: We only know that it has ½-turn symmetry and a
reflection symmetry.
Mathematician: OK, so let’s imagine only the point that is the
center of the ½-turn symmetry and the line that is the line of
reflection symmetry. In our imagination everything else about the
figure is fuzzy.
Student 1: OK, so now I am imagining a dashed line of reflection
and a dot that is the center of the ½-turn with everything else
fuzzy. I can draw it like this:
Mathematician: OK, now copy this onto a transparency and explore
what you can tell about the fuzzy figure. What happens if we do a
half turn of the transparency about the dot?
Student 2: Well, because the dot is the center of ½-turn symmetry
of the fuzzy figure, then after I do the ½-turn the fuzzy figure
on the transparency must be back on top of the figure on the paper.
Student 3:
So, if that is true then the situation looks like this:
Name: __________________
Straight and Symmetries 4
Line of reflection
symmetry on paper
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Line of reflection
symmetry on
transparency
Student 4: So, if all of this is true, then there must be a second
line of reflection symmetry.
Mathematician: Good. Look at the example of the decoration with
glide symmetry. This decoration is an example of this sort.
Student 1: I am not convinced!
the dot not being on the line.
What we have done here depends on
Mathematician: Very good! So, we have not finished the proof. We
have so far only shown that if the dot is not on the line then
there is another line of reflection symmetry. Now we have to look
at the other case: The center of ½-turn symmetry is on the line
of reflection symmetry. Here’s the picture for this case:
Student 2: In that case the above argument does not work because
the half turn will just take the line of reflection onto itself.
Mathematician: OK, now lets draw a rectangle on the paper so that
its center of ½-turn symmetry and line of refection symmetry are
the same as for our imagined fuzzy figure.
Name: __________________
Straight and Symmetries 4
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D
A
C
B
Student 3: So, the rectangle has the same ½-turn symmetry as the
imagined fuzzy figure and also has the same reflection symmetry
about the dashed line.
Mathematician: Now trace this onto the transparency and first
reflect (flip) about the dashed line and then rotate a ½-turn
about the dot. I suggest each of you fill in the missing letters.
Reflection over the dashed line
½-turn about the dot
A → D
A → C
B →
______
B → ______
C → ______
C → ______
D → ______
D → ______
Student 4: So this reflection in the dashed line followed by the
½-turn about the dot, takes:
A → B, B → A, C → D, D → C
Mathematician:
What do you notice about this?
Student 1: I notice that this is the same arrow diagram as the
other reflection symmetry of the rectangle -- the reflection about
the centered horizontal line.
Mathematician:
Good!
Student 2: We already know that this is a symmetry of the
rectangle, but how do we know that it is a symmetry of the fuzzy
figure?
Mathematician: Think back – what is a symmetry?
when in doubt we refer back to the definitions.
In mathematics,
Student 3: A symmetry of a figure is an isometry that takes the
figure onto itself.
Name: __________________
Straight and Symmetries 4
Mathematician:
isometry?
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So is the reflection followed by the ½-turn an
Student 4: Yes, because we did it with the transparency and the
transparency keeps all lengths and angles the same.
Student 1: I see it because both the reflection and the ½-turn
preserve lengths and angles and so one followed by the other must
preserve lengths and angles.
Mathematician: So does the reflection followed by the ½-turn take
the fuzzy figure onto itself?
Student 2: Sure! The reflection takes the fuzzy figure onto
itself because that was what we knew about the dashed line. And
then the ½-turn does the same thing.
Student 3: So, the reflection followed by the ½-turn is a
symmetry of the fuzzy figure. And we see that it is also a
reflection about a horizontal line thru the dot.
Student 4: We are done! We have shown that if any figure has ½turn symmetry and a reflection symmetry then it must have (at
least) one more reflection symmetry.
Mathematician: Very, very good. We now have a proof of the
theorem. I suggest you all go back over the proof and summarize it
for yourself and make sure it convinces you.
The proof of this theorem involved two different cases. One case
involved the line of symmetry not passing through the point of half turn
symmetry and the other case involved the line of symmetry passing
through the point of half-turn symmetry. In your own words describe
what the argument is for each case being true.
Case 1.
Name: __________________
Straight and Symmetries 4
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Case 2.
Was this proof convincing to you?
Why or Why not?