The Math Major

CSU Fresno Mathematics Department

Vol 3. No. 6 (November 11, 1998)

Editor: Dr. Larry Cusick.

Math Department Colloquium Series

Colloquium:

• Date & Location: Friday, November 13, 3:10 in McLane Hall 162.
• Speaker: Dr. Jennifer Taback, from the UC Berkeley Mathematics Department
• Title: "Hyperbolic Tessellations and Escher's Art"
• Description: Everyone is familiar with the regular Euclidean tessellations of the plane by equilateral triangles, squares and hexagons. What kind of geometry would we have to impose on the plane to make regular octagons tessellate, say 8 to a vertex? I will talk (interactively) about this type of geometry, called hyperbolic geometry, and contrast some basic properties of hyperbolic geometry with those of Euclidean geometry. We have all seen examples of hyperbolic geometry in some of Escher's tessellations. Together we will examine these drawings, and look for their hyperbolic symmetries.
  

Colloquium:

• Date & Location: Thursday, November 19, 1:10 in Science 139
• Speaker: Dr. Lesley Ward, from the Harvey Mudd College Mathematics Department
• Title: "Brownian motion and the shape of a region's boundary."
• Description:A random walker starts walking from a point P inside a region S. Let h(r) be the probability that the first place the walker hits the boundary is within distance r of P. If we know h(r) for every r, can we determine the shape of the boundary of S? In this talk, we'll see that the answer is no.

  If we know the boundary, what can we say about h(r)? We'll give some conditions h(r) must satisfy, and look at examples where we can compute h(r) exactly. Our main technique uses conformal mappings and harmonic measure; neither of these is a prerequisite for the talk.

Spring Math 181 Change

A Mathematical Quote

--Eric Temple Bell (1883-1960)

Problem Corner

f(x) = x if 0 <= x < 1 and 3 f(x/2) - 1/2 otherwise.

Prove that f is a continuous increasing function and find the unique positive solution x to the equation f(x) = 9.

Solution to Problem 3.5: f(x) = x for 1<= x < 0, and so f is continuous and increasing over this interval. Continuity at x = 1 can be established by noting that for x=1, f(1) = 3f(1/2) - 1/2 = 1 which agrees with the left hand limit. Similarly we can prove f is continuous and increasing over the interval 1<= x<= 2. Now proceed by induction, proving f is increasing and continuous for 2^k<=x <= 2^k+1 by appealing to the inductive step 2^k-1 <= x/2<= 2^k} with f(x) = 3 f(x/2) - 1/2.

To solve the equation f(x) = 9, we first note f(1) = 1, f(2) = 5/2, f(4) = 7 and f(8) = 39/2 and so the solution, x, must be between 4 and 8. This means we must iterate the definition of f three times to get f(x) = (27/8) x -13/2 = 9. The solution is

Correct solutions were received from Anar Ahmedov, Bryan Chaffe and John Jamisom.

Problem 3.6: (Due Thursday November 19, 3 p.m.) A pawn is placed in the central square of a 11x11 chessboard. Two players move the pawn in succession to any other square, but each move (beginning with the second) must be longer than the previous one. The player who cannot make such a move loses. Who wins an errorless game?

Solutions may be delivered to the math department office (for Dr. Cusick) or by e-mail.

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