Regular and Semiregular 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
(i) 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.
 Question: What are the possible regular and semiregular tilings of the plane?
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

 Wingeometry download!
and demonstrate tesselations.
 Naming tilings
(Math Forum)
 The numbers represent the number of sides in the polygons.
 The order indicates the order in which the polygons 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 around 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 around 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.