Math 103 Contemporary Math Tuesday, February 8, 2005
Review from last class FAPP video on Tilings of the plane.
Symmetry Ideas Reflective symmetry: BI LATERAL SYMMETRY T C O 0 I A • Folding line: "axis of symmetry" • The "flip.“ • The "mirror."
R(P) = P': A Transformation Before: P .... After : P' If P is on the line (axis), then R(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 lif there is a reflection R about the line l where R(P)=P' is still an element of Ffor every P in F.... • i.e.. R (F) = F. • l is called the axis of symmetry. • Examples of reflective symmetry:Squares... People
Rotational Symmetry • Center of rotation. "rotational pole" (usually O) and angle/direction of rotation. • The "spin.“
R(P) = P' : A transformation • If O is the center then R(O) = O. • If the angle is 360 then R(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.
Single Figure Symmetries • Now... what about finding all the reflective and rotational symmetries of a single figure? • Symmetries of playing card.... • Classify the cards having the same symmetries. Notice symmetry of clubs, diamonds, hearts, spades. • Organization of markers.
Why are there only six? • Before: AAfter : A or B or CSuppose I know where A goes:What about B? If A -> A Before: B After: B or C If A ->B Before:B After: A or C If A ->C Before: B After: A or BBy an analysis of a "tree" we count there are exactly and only 6 possibilities for where the vertices can be transformed.
Tree Analysis Identity B C A Reflection C B C Reflection A B A Rotation C B Rotation A C Reflection A B
What about combining transformations to give new symmetries Think of a symmetry as a transformation: Example: V will mean reflection across the line that is the vertical altitude of the equilateral triangle.Then let's consider a second symmetry, R=R120, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees. We now can think of first performing V to the figure and then performing R to the figure. We will denote this V*R... meaning V followed by R.[Note that order can make a difference here, and there is an alternative convention for this notation that would reverse the order and say that R*V means V followed by R.]Does the resulting transformation V*R also leave the equilateral covering the same position in which it started?
Symmetry “Products” • V*R = ? • If so it is also a symmetry.... which of the six is it? • 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.
Activity • Do Activity. • This shows that R240*V = ? • 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 trivial symmetry.So we can compare objects for symmetries.... how many?Does the multiplication table for the symmetries look the same in some sense?