# Course:Harris, Fall 08: Diary Week 2

(short week)

Wed:

• Groups seemed unclear that U2 could be anything other than a Rectangle.
• Groups finished up Tessellations (First Look) Exploration, 15-20 minutes.
• Question: What's a regular hexagon? Or other polygon?
• All sides congruent.
• All angles congruent.
• Are those redundant, i.e., does one imply the other?
• Example looked at of hexagon with all sides congruent, but not convex.
• Groups looked at Kali in Symmetric Figures Exploration.
• Considered what a symmetry of a figure is:
• a rigid motion of the plane that moves the figure onto itself
• Kali "rosette group" buttons induced
• upper row:
• rotational symmetries
• lower row:
• rotational symmetries
• flip symmetries

Fri:

• Turned back Tessellations I Exploration and Polygon Exercises.
• Exercises (but not Explorations, as a rule) can be redone for additional credit.
• We spent most of the period going over #3 of Polygon Exercises (parallelograms are the same as quadrilaterals with opposite angles congruent).
• I noted that there are (or should be, from Quadrilaterals Exploration) two logical directions:
• (both pairs of opposite angles congruent) ==> (both pairs of opposite sides parallel)
• proved this last week, using:
• Parallel lines cut by a third line implies opposite interior angles are congruent.
• (We didn't say where this theorem comes from.)
• Angle-sum of a triangle is 180 degrees.
• (Proved using opp-int-ang theorem, plus construction of a parallel line through a point not on the line--Parallel Postulate.)
• (both pairs of opposite sides parallel) ==> (both pairs of opposite angles congruent)
• Use opp-int-ang theorem three times:
• for parallel horizontal sides cut by one of the diagonal sides;
• for parallel horizontal sides cut by the other diagonal side; and
• for parallel diagonal sides cut by one of the horizontal sides.
• Also use angles making up straight line are supplements (add to 180).
• Put it all together, and if one angle is A, the neighboring angle is 180 - A, the next one is A, and the next one is 180 - A.
• Groups started in on Symmetry of Stars/Polygons Exploration, but question was raised about Symmetry Groups:
• The Symmetry Group for a figure is all the symmetries that apply to it.
• A symmetry is a rigid motion of the plane. The ones applicable to compact figures are:
• rotation of the plane about a point (rotation center);
• reflection of the plane across a line (the reflection axis).
• If there are just n-fold rotation symmetries, then that group is Cyclic and called Cn.
• If there are n-fold rotation symmetries and also reflection symmetries, then that group is Dihedral ("two-faced") and called Dn.
• We'll finish up Symmetry of Stars/Polygons Exploration on Monday (5 minutes?).
• Rosette Exercises are due Monday.