Tuesday, February 1
Discuss Project proposal... Guidelines
are on web! Form partnerships?
Dissection of the plane Tilings
of the plane:
Tessellations
 Possible tiles and tilings.
 One Regular Convex Polygon.
 Two or more Regular Convex Polygons.
 Other Polygon Tilings.
 Non polygonal tilings.
More on measurements
of angles and areas of polygons.
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.

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.
 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.
 Can there be 4 different tiles around
a single vertex? Not if they are all different since 60+90+108+120 = 378
>360.
 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.