Math 4513 (Senior Mathematics Seminar)

Spring 2006


John Albert
Office: PHSC 1004, Ext. 5-3782.
Office hours: Mon, Wed and Thurs, 2:30-3:30 PM (or by appointment)

  • The take-home final is due by 5 pm on Thursday, May 11. You can return it to me any time before then by bringing it to my office and sliding it under the door if I'm not there. If you have trouble getting it under the door, you can also leave it with the receptionist in the math office.

    NOTE: There are a couple of misprints in the statement of problem 2. In both part a and part b, the expression (y^4 + 16) is wrong; it should be corrected to (y^4 + 4).

    Here are scans of the write-ups submitted with the presentations:

  • Adair/Tu/Yang: Liouville's transcendental number
  • Arrowood/Miller: Mathematical games
  • Cast/Whitman: Computing the digits of pi
  • Cheema/Jason Williams: Transcendence of pi
  • Clayton/McGoodwin: Irrationality of pi and e
  • Corp/Vong: Pappus' theorem
  • Farnan/Johnathan Williams: Descartes' method of tangents
  • Ford/Sposato: Ramsey theory
  • Garman/Sowell: Penrose tiles, Fibonacci numbers, and the golden ratio
  • Ha/Pipkin: Tilings of the plane
  • Hardman/Martinez: RSA algorithm
  • Hayes/LeGate: Rubik's cube

  • The midterm exam was on Friday, March 31. Here is a review sheet for the midterm.
  • Here is an explanation of how the presentations will be graded, and a current schedule of dates for the presentations.
  • I gave out two lists in class of suggested topics for course presentations. Part 1 contains mostly problems from geometry, and Part 2 contains mostly problems from number theory.
  • Syllabus for this course.
  • So far the only reference I have found online to the theorem of Hilbert which we discussed in class is the 1899 book by Hilbert himself, Foundations of Geometry. In it you will find, for example: (i) a definition of the terms "interior" and "exterior" of a polygon, and a theorem stating that every polygon in the plane has an interior and an exterior; (ii) axioms defining the notion of "congruence" of line segments, angles, and triangles, and proofs of theorems such as the side-angle-side condition for congruence of two triangles; (iii) a method for measuring the area of a polygon (by cutting it into triangles and adding the areas) which gives the same result no matter how the polygon is cut into triangles. Looking through the book you will immediately see that plane geometry is a surprisingly subtle subject. For example, Hilbert defines not only the "area" of a polygon but also the "content" of a polygon, and at first it is not at all apparent how the two notions are different. You will also see that Hilbert was in the habit of leaving most of the proofs to the reader!


  • Assignment 3 (due Wednesday, Mar. 22)
  • Assignment 2
  • Assignment 1