Vol 1. No. 12
Editor: Dr. Larry Cusick.
"Not Knot" is a computer-generated video which illustrates some important ideas from knot theory and hyperbolic geometry. The object is to understand the space surrounding a mathematical knot by putting some kind of geometric structure on it. The video has some good illustrations and examples to help understand this process.
The audience will get a guided tour into computer-animated hyperbolic space. It proceeds from the world of knots to their complementary spaces -- what's not a knot. Profound theorems of recent mathematics show that most known complements carry the structure of hyperbolic geometry, a geometry in which the sum of three angles of a triangle always is less than 180 degrees. <\ul>
Problem 1.11: A square of side length 2, lying always in the first quadrant of the xy-plane moves so that two consecutive vertices are always on the x- and y-axes respectively. Find the locus of the midpoint of the square.
Solution to Problem 1.11: Let A and B be two consecutive vertices lying on the x- and y-axes, respectively. Let C be the center of the square and D and E the feet of perpendiculars from C to the x- and y-axes, respectively. Let O be the origin.
Then angle ACB is a right angle, and angle DCE is a right angle and triangle ACD is congruent to triangle BCE. Hence the x and y coordinates of C are equal, and OECD is a square. As the given square moves under the specified constraint, the center C moves back and forth along the segment from (1, 1) to (sqrt(2), sqrt(2)).
Correct solutions were received from Anar Ahmedov and Roden Macalma.
Solutions may be delivered to the math department office (for Dr. Cusick) or by e-mail at larry_cusick@csufresno.edu. no later than Thursday Thursday April 3, 3pm. There is a $75 dollar first prize and a $50 second prize to be awarded at the end of the semester to the student(s) who submit the most correct solutions.