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.
• Show FAPP video on Patterns
  
• as review of tilings so far and

• Introduces Penrose Tiles  K&M section 4.3 as source for possible protfolio/project ideas.

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  Duality Activity
  

• Reflective symmetry: BI LATERAL SYmMETRY
• Folding line: "axis of symmetry"
  
• The "flip."
  
• The "mirror."
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• R(P) = P'  : A Transformation. Before: P .... After : P'
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• If P is on the line (axis), then F(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) is still an element of F for every P in F....i.e.. R (F) = F.
• Examples of reflective symmetry:

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• Rotational Symmetry.
• Center of rotation. "rotational pole" (usually O) and angle/direction of rotation.
  
• The "spin."
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• R(P) = P' : A transformation
• If O is the center then F(O) = O.
• If the angle is 360 then F(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.
  
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•  Now what about finding all the reflective and rotational symmetries of a single figure?
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• What about combining transformations to give new symmetries: