Notes Math 103 9_11

September 11
Discuss Project proposal... Guidelines will be on web by Friday! Form partnerships?

Review from last class:
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	degrees of the interior measure of each angle	360 degrees divided by # in Column 2
equilateral triangle	60	360 / 6 = 60
square	90	360/4= 90
regular pentagon	3*180/5= 108	360/5= 72
regular hexagon	4*180/6=120	360/6= 60
regular heptagon	5*180/7	360/7
regular octagon	6*180/8=135	360/8 = 45

Dissection of the plane--- Tilings of the plane.

One polygonal Tile: Quadrilateral Activity. (related to Assignment due next class!)
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 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: 3-3-3-3-3-3 (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

3 3 3 3 6

3 3 3 4 4

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 3-10-15)
Similar considerations can eliminate tilings using a single pentagon and two other distinct tiles. (such as 4-5-20).

Semiregular Tilings: global results!
Look at the results using wingeometry.
Student lesson (Math Forum) a place for further explorations on-line.