Nick Barlotta
MAE 311
Notes for 11/19/2009
1) We first discussed the 6 symmetries of a triangle which can be shown below. Let this be denoted
by
3 Rotations:
3 Flips:
We then discussed S₃, the symmetric group on 3 letters. In any set
elements in a finite set X, and X = {A,B,C,D,….},
, where n is the number of
has n! elements. This equation uses the fact that
there are n choices for the first position, n-1 for the second position, n-2 for the third position, and so
on.
3! = 6, so
must have 6 elements. These 6 elements are:
(ABC), (ACB), (BCA), (BAC), (CAB), (CBA).
Any composition of these six elements will give you another element in
and every element having an inverse,
. With (ABC) as the identity
is a permutation group.
Upon further discussion, we noticed that
and,
, the 6 symmetries of a triangle, are isomorphic
because there exists a naming from every element in
, to an element in
2) We then proceeded in trying to apply the same concept to
and
.
. We first started listing the
symmetries of a square. We found that there are 8 symmetries, which can be shown be below.
4 Flips:
4 Rotations:
We then tried the same concept to check if | | = |
Since 4! = 24, |
|
|?
|.
3) Since there were 6 symmetries for a polygon of size 3, and 8 symmetries for a polygon of size 4,
we came up with the following conjecture.
Conjecture: |
|, the size of a group of symmetries of a regular n-gon, is equal to 2n.
We came up with various methods in attempting to prove this conjecture.
Nick suggested that there are n rotations and n reflections. Combining these two categories of
symmetries, this leads us to 2n symmetries. This may have worked for a polygon of size 3 or 4, but how
do we know that compositions of these symmetries will not lead us to another symmetry that we have
not found?
By this statement, we can conclude that |
|≥ 2n.
Jimmy then suggested that if we send a vertex to any given vertex, then the 2 adjacent vertices only
have 2 possible choices. This must be true because a vertex that was adjacent before a flip or rotation
must still be adjacent after the symmetry.
Given the following polygon,
If vertex 1 is sent to vertex 5, then vertex 2 must either be sent to 6 or 4, leaving only 1 choice for vertex
8 to be sent to. Sending vertex 1 to all 8 of the possible vertices, and then adding the two choices at
each vertex leads us to 16 possible symmetries, which is what we expected.
4) We then looked at a third way of determining the number of symmetries in an n-gon. We broke
the symmetries down into 1 rotation, and a fixed reflection on any shape, where we reflected
over 1 axis.
Given n sides, we can let r =
, where r is a rotation. To find all the rotations, we can compose the
rotations with each other, and find that there are n rotations.
Possible rotations = {r,
,
, ... ,
,
} = n.
In order to find the rest of the symmetries, we can compose these possible rotations with a fixed
reflection about an axis. Let this reflection be denoted by R.
Composing the possible rotations with this reflection, we find that there are n reflections.
Possible reflections = {Rr, R , R , … , R
,R
} = n.
Example: R
Note:
1. If you compose any reflection with itself, this leads us to the identity. If we reflect about an axis
and then over that axis again, we are left with the original shape.
Example: (R * R) = e.
2. If you compose any rotation with itself, this leads us to another rotation that is already in our
given set of symmetries.
Example: (
*
)=
, where a+b is evaluated in
.
5) We then decided to create the table for the symmetries of a square to see check if we had
omitted any symmetries in our first analysis.
°
0°
90°
180°
270°
\
/
|
--
0°=id
0°
90°
180°
270°
\
/
|
--
90°
90°
180°
270°
0°
-|
\
/
180°
180°
270°
0°
90°
/
\
-|
270°
270°
0°
90°
180°
|
-/
\
\
\
-/
|
0°
180°
270°
90°
/
/
|
\
-180°
0°
90°
270°
|
|
\
-/
270°
90°
0°
180°
--/
|
\
90°
270°
180°
0°
Upon review of all the possible compositions, we found that we had not omitted any symmetry, and the
number of symmetries for a square, was indeed, 2n.
6) Towards the end of class, we began to discuss possible definitions of the term Congruence.
Angelo volunteered and gave us the following definition:
Congruence: The measures of all the corresponding sides and corresponding angles of
geometric objects are congruent.
We had a short discussion on the notion of congruence and planned to finish on Tuesday.
Congruence is defined differently by many different sources.
Definition on dictionary.com:
Congruence: a relation between two numbers indicating that the numbers give the
same remainder when divided by some given number.
Definition on Wikipedia.com
Congruence: Two figures are congruent if they have the same shape and size, but are in
different positions. (for instance, one may be rotated, flipped, or placed in another
position).
Definition on Mathopenref.com
Congruence: Two objects are congruent if they have the same dimensions and shape.
Very loosely, you can think of it as meaning 'equal', but it has a very precise meaning
that you should understand completely, especially for complex shapes such as polygons.
Sources:
Wikipedia.com
Dictionary.com
Mathopenref.com
MAT 312 Textbook