This follows the outline for 90-degree centers done in class. The concept of conjugation was introduced. If A_s is one rotation and B_t is another, then there is another rotation C_t centered at C = A_s(B). This rotation C_t is the product A_sB_t (A_s)^(-1), or A_sB_tA_(-s). For example, if we denote by U and V the rotations by 90 degrees centered at A and B, respectively, in the figure below.
C_90 = UVU^(-1) = UVUUU. Also E_180 = U^(-1)VVU = UUUVVU.
In class, you received two handouts of tessellations of the plane by triangles. Two points A and B are marked on each sheet, but one says A_60 and B_60 and the other says A_120 and B_120. For the first sheet we take U = A_60 and V = B_60 and for the other take U = A_120 and V = B_120
The exercise is the same in each case (the answers will not be the same, but will be related).
Let G be the set of isometries obtained by all possible products and products or products, etc., formed from U and V and their inverses. This is a group of transformations (see Brown, this means that if H and K are in G, so is HK and the inverses of H and K). Since U and V are rotations, some of the isometries in G are also rotations and some are translations. Let T be the set of translations in G.
(a) The goal is to mark all the centers of rotation C for all rotations in G. You already have studied products such as UV and VU, etc. The conjugation idea shows how to use the UVU^(-1) idea to get rotations centered at points obtained by rotating other centers. So you can rotate all the centers you already have by any rotation in G. The pattern of centers of rotations in G should be a symmetric pattern related to one of the tessellations of the plane by regular polygons.
(b) Now indicate the translations in T in this way. Start by coloring the point A blue. Then also color blue all other centers that can be obtained from A by a translation in T. Then color the point B red. (Did you color it blue? Should you have?) Then also color red all other centers that can be obtained from B by a translation in T. If there is a center which is not yet colored, color it another color and color the same color the centers which are translates of this point by T. Continue until no centers are uncolored.
You were given a handout that poses part of this problem. Answer the following questions.
(a) Given a kite, can a circle always be circumscribed around the kite? If so, prove it. If not, tell for which kites such a circle can be constructed (and prove it).
(b) Given a kite, can a circle always be inscribed in the kite? If so, prove it. If not, tell for which kites such a circle can be constructed (and prove it).
You got a handout with a tiling by one big square superimposed on a tiling by two squares. Observe that the pattern can be produced either by cutting out the squares of the single-square tessellation with sides c or by cutting out the small squares and the big squares with sides a and b of the two-square tessellation. There are several consequences: (1) the area c^2 of the single-square = the sum of the areas of the other two squares= a^2 + b^2. This actually is a proof of the Pythagorean theorem. (See the Wed. Handout from Hidden Connections, Double Meanings for more.) (2) If you cut up the single-square of side c along the segments coming from sides of the two other squares, you can treat this as a jigsaw puzzle. You can cut up the square of side c and reassemble it as two squares of sides a and b. Or you can reverse your thinking, start with the two squares from a previous handout, cut the apart and assemble them as a single square.
(a) Carry out this process for the square. Using the handout or otherwise, cut up a square and show how to reassemble the pieces as two smaller squares.
(b) Now take the same approach with the tessellation by rotationally symmetric crosses (Greek crosses). Show how to cut apart a cross and reassemble it as a square. (Hint: The square has something to do with the centers of symmetry.)