Thursday September 15

Symmetry Ideas
 Reflective symmetry: BI LATERAL SYMMETRY
T C O
0 I A
 Folding line: "axis of symmetry"
 The "flip."
 The "mirror."

 R(P) = P' : A Transformation. Before: P ....
After : P'
 If P is
on the line (axis), then R(P)=P. "P remains fixed by the reflection."
 If P is not on the axis, then the line PP' is
perpendicular to the axis and if Q is the point of intersection of PP' with
the axis then m(PQ) = m(p'Q).
 Definition: We say F has a reflective symmetry wrt a line l if there is a reflection
R about the line l where
R(P)=P' is still an element of F for every P in F....i.e.. R (F) = F.
l is called the axis of symmetry.
 Examples of reflective symmetry:
Squares... People

 Rotational Symmetry.
 Center of rotation. "rotational pole" (usually O)
and angle/direction of rotation.
 The "spin."

 R(P) = P' : A transformation
 If O is the center then R(O) = O.
 If the angle is 360 then R(P) = P for all P....
called the identity transformation.
 If the angle is between 0 and 360 then only
the center remains fixed.
 For any point P the angle POP' is the same.
 Examples of rotational symmetry.
 Now... what about finding all the reflective
and rotational symmetries of a single figure?
 Symmetries of playing card.... classify the cards by
having same symmetries.
Notice symmetry of clubs, diamonds,
hearts, spades

 Preview ... Possible tiles and tilings.
 One Regular Convex Polygon.
 Two or more Regular Convex Polygons.
 Other Polygon Tilings.
 Non polygonal tilings.
REVIEW: Recall our previous discussions for
a polygon with n sides.
When n = 3 this is a triangle, n=4, a quadrilateral, or
when n= 5, a pentagon.
The sum of the measures of the
interior angles of a triangle is 180 degrees.
Review Question:What about a quadrilateral? and a pentagon?
or an n sided polygon ( an "n gon")?
From the figure we saw that for a quadrilateral (n =4), which can be dissected
into two triangles,
the sum is 2*180= 360 degrees.
And for a pentagon (n=5)
which can be dissected into 3 triangles, the sum is 3*180=540 degrees.
In general: the sum
of the interior angles in a n sided polygon
is
(n2) *180 degrees.
A regular polygon
is a polygon where the sides are all of equal length
and the angles are all congruent (or of equal measure).
Review Question: what is the measure of an individual
angle in a regular polygon with n sides?
For a triangle, the individual angle is 180/3
= 60 degrees.
For a square, the individual angle is 360/4=90
degrees.
For a regular pentagon.... 3*180/5 = 540/5
=108 degrees.
Now for a HEXAGON (6 sides) the sum of the angles
is 720 degrees.
So ... for a REGULAR HEXAGON,
the individual angle is 4*180/6 =720/6 =120 degrees.
In general: The individual
angle for a regular polygon with n sides is (n2)*180/n degrees.
This can be expressed in other ways using algebra:
(n2)*180/n = [180 n  360] / n = 180  360/n.
name of polygon 

degrees of the interior
measure of each angle 
360 degrees divided
by # in Column 2 
equilateral triangle 
3

60 
360 / 3 = 120 
square 
4

90 
360/4= 90

regular pentagon 
5

3*180/5= 108

360/5= 72

regular hexagon 
6

4*180/6=120

360/6= 60

regular heptagon 
7

5*180/7

360/7

regular octagon 
8

6*180/8=135

360/8 = 45

regular dodecagon

12

10*180/12=1800/12=150

360/12=30

 Dissection of the plane Tilings
of the plane.
 One polygonal Tile: Quadrilateral Activity.

An online tool for making tilings of the plane
 Regular and semiregular Tilings of the plane.
 A tiling is a regular tiling if(i)it has a single tile shape that
is a regular polygon and (ii) the vertices and edges of the tiles coincide
(no overlapping edges)
 A tiling is a semiregular tiling if (ii) each tile shape is a regular
polygon, (ii) the vertices and edges of the tiles coincide (no overlapping
edges) and (iii) every vertex has the same polygon types arranged around
it.
 Wingeometry download!
and demonstrate tesselations.
 Naming tilings
(Math Forum)
 The numbers represent the number of sides in the poygons.
 The order indcates the order in which the poygons are arranged about
a vertex.
 Local considerations about a vertex. The sum of the
angles must equal 360 degrees.
 4.1 Ex. 3. How many tiles around single vertex:
3,4,5,6,7,8?...
 How can there be 6? Only one way: 333333
(6*60= 360)
 Why not 7? If 7 or more tiles aroung a single
vertex the sum of the angles must exceed 360 degrees.
 4.1 Ex. 4. Can there be 4 different tiles
around a single vertex? Not if they are all different since 60+90+108+120
= 378 >360.
 4.1 Ex. 5. If there are 4 tiles around a
vertex then ....At least two of the tiles must have the same number of sides.
 Arithmetic for vertices.
Example: Three regular polygons about a vertex with
n , k, and p sides.(180  360/n) + (180  360/k) + (180
 360/p) = 360
3*180 360( 1/n+1/k+1/p)= 2*180
1*180 = 360( 1/n+1/k+1/p)
SO....
180/360 = 1/n + 1/k + 1/p
or
1/n + 1/k +
1/p=1/2
So, for example, n=3, k=4
and p= 5 is not possible since
1/3+1/4+1/5
>1/2.
Number of polygons
around a vertex 
Equation for angle sum = 360 
Equivalent Arithmetic equation 
Solutions to the arithmetic equations. 

3: n , k, p 
180  360/n+180  360/k+180  360/p = 360 
1/n+1/k+1/p =1/2 
6 
6 
6 
5 
5 
10 
4 
5 
20 
4 
6 
12 
4 
8 
8 
3 
7 
42 
3 
8 
24 
3 
9 
18 
3 
10 
15 
3 
12 
12 


4: n, k, p, z 
180  360/n+180  360/k+180  360/p 180  360/z
= 360 
1/n+1/k+1/p +1/z =2/2 =1 
4 
4 
4 
4 
3 
3 
4 
12 
3 
3 
6 
6 
3 
4 
4 
6 


5: n, k, p, z, w 
180  360/n+180  360/k+180  360/p+180  360/z+180
 360/w = 360 
1/n+1/k+1/p +1/z+1/w =3/2 


 Local consideration about a polygon:
 Examples: If there is an equilateral triangle involved
with 2 other polygons, then the other two ploygons must have the same number
of sides. Because: if there were two different polygons aroung the triangle,
then on vertex would not have all three polygons sharing that vertex. (such
as 31015)
 Similar considerations can eliminate tilings using a single
pentagon and two other distinct tiles. (such as 4520).
 Semiregular Tilings: global results!
Look at the results using wingeometry.
 Student lesson
(Math Forum) a place for further explorations online.