Tuesday September 20

REVIEW: Recall our previous discussions for a polygon with n sides.

In general: the sum of the interior angles in a n sided polygon is
(n-2) *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).

In general: The individual angle for a regular polygon with n sides is (n-2)*180/n degrees.
This can be expressed in other ways using algebra: (n-2)*180/n = [180 n - 360] / n = 180 - 360/n.


name of polygon	n	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 on-line 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 semi-regular 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 indicates the order in which the poygons are arranged about a vertex.

Recall Symmetries of an equilateral triangle:

Combining transformations to give new symmetries:

Example: Let V denote reflection across the line that is the vertical altitude of the equilateral triangle.

And consider a second symmetry, called R=R120, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees.

We first perform V to the figure and then perform R to the figure.
We will denote this V*R... meaning V followed by R.

The resulting transformation also leaves the equilateral covering the same position in which it started, and....

V*R = R2.

What about other products? This gives a "product" for symmetries.
If S and R are any symmetries of a figure then S*R is also a symmetry of the figure.

A "multiplication" table for Symmetries:

*	Id	R120	R240	V	G=R1	H=R2
Id	I
R120		R240		R1
R240		I		R2
V		R2	R1	I		R120
G=R1					I
H=R2						I

Do Activity on product of symmetries.
This shows that R240*V = R1
This "multiplicative" structure is called the Group of symmetries of the equilateral triangle.

Given any figure we can talk about the group of its symmetries.
Does a figure always have at least one symmetry? .....

Yes... The Identity symmetry.
Such a symmetry is called the trival symmetry.