1.4. SYMMETRIES AND MATRICES11DefinerkD.r1/kfor any positive integerk. Show thatrkDr3kDrm, wheremis the unique element off0; 1; 2; 3gsuch thatmCkis divisible by 18.104.22.168.Here is another way to list the symmetries of the square card thatmakes it easy to compute the products of symmetries quickly.(a)Verify that the four symmetriesa; b; c;anddthat exchange thetop and bottom faces of the card area; ra; r2a;andr3a, in someorder. Which is which? Thus a complete list of the symmetriesisfe; r; r2; r3; a; ra; r2a; r3ag:(b)Verify thatarDr1aDr3a:(c)Conclude thatarkDrkafor all integersk.(d)Show that these relations suffice to compute any product.1.4. Symmetries and MatricesWhile looking at some examples, we have also been gradually refining ournotion of a symmetry of a geometric figure. In fact, we are developinga mathematical model for a physical phenomenon — the symmetry of aphysical object such as a ball or a brick or a card. So far, we have decided
This is the end of the preview.
access the rest of the document.