Math 444 Assignment 9 (Due Monday, December 2)

• Read handout from Chapter 5 of Farmer, Groups and Symmetry. 
• Also read (again) in Brown for big ideas: groups, basic unit (Farmer's term; sometimes called fundamental domain), 17 wallpaper groups and how to recognize them (last page of Farmer).

Problem 9-1

On page 57 of the Farmer handout are 6 wallpaper patterns. For each of these patterns:

• Tell the name of the symmetry group of the pattern (use page 58).
• Indicate clearly for each pattern the mirror lines of line reflection (solid lines), invariant lines of glide reflection (dashed lines), centers or rotation (use the icons of triangle for 120, hexagon for 60, square for 90 and rhombus for 180) and translations (draw some arrows to show one parallelogram that is a basic region for the translations).

Problem 9-2

Return to Cases 0, 1, 2, 3 in Lab 8.  Do this exercise for each case separately and clearly.

Note: Go to this link for the patterns that are the Answers for for Lab 8.

• Use square graph paper to draw (freehand is OK) the square with the shaded triangle. 
• Now draw in additional squares to form a basic unit for the translations that are symmetries of the whole pattern.  If your figure includes more than this, outline and label clearly a basic unit.
• On the same figure, outline and label clearly a basic unit for the whole symmetry group.
• Use the last page of Farmer to tell the name of the symmetry group of the whole pattern.

Problem 9-3

The symmetries of a cube will be discussed in class Wednesday.  These symmetries include many plane reflections.  The cube is cut up by these planes of symmetry into congruent tetrahedra (not regular) which are pyramids with vertex the center of the cube and base = isosceles right triangle on a face.

• Construct a net for this tetrahedron.  Labeling angles and lengths.
• Make a second copy of the net, cut it out and make a model of such a tetrahedron.  Bring it to class Monday.

Problem 9-4

Based on the discussion in class and this outline, write a proof of Napoleon's theorem.  Main points to explain and justify:

• Product F of three 120-degree rotations is a translation (possibly identity).
• One point is fixed by F -- for some P, F(P) = P.
• F is the identity.
• If F is the identity, the centers of the three 120-degree rotations form an equilateral triangle.