Math 460 - Senior Seminar - Home Page

Journey through Genius

Fall 2008

"Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone."

Put on the web by Dave Rusin. 

Instructor

Brian Shelburne 329-E  BDK Science

Class Meetings 

Th 2:10 - 3:40 - Room 320

Textbook

Journey Through Genius by W. Dunham

Course Syllabus

Course Objectives: This is a capstone course for mathematics majors. Its purpose is to let you think about and reflect on what mathematics is and to tie together your years of studying mathematics. Dunham's book, Journey Through Genius, covers the story of mathematics from the 5th century B.C.E. up to the 20th century C.E. by looking at some famous problems and theorems and the mathematicians who worked on them. The book is many things. It is a selective history of mathematics. It is a look at some of the more famous and colorful characters who were mathematicians. It presents rigorous but readable proofs of some interesting and famous theorems in mathematics.

 

In addition the course will make use of a number of on-line resources; two in particular are The MacTutor History of Mathematics Archive at The University of St. Andrews in  Scotland (http://www-history.mcs.st-andrews.ac.uk/history/index.html) which contains a wealth of biographical and historical essays on mathematicians and mathematics and Kevin Brown’s Mathpages.com (http://www.mathpages.com) which is source of articles on a number of unusual and interesting mathematical topics.  

 

Dunham’s book is well written, fun to read. Coupled with your explorations of on-line resources this course will give you a deeper appreciation for the unique endeavor we call mathematics.

 

Week - Date

Assignment

1 – Th Aug 28

Exercise 01

2 -  Th Sept 4

Exercise 02

3 – Th Sept 11

Exercise 03

4 – Th Sept 17

Exercise 04

5 – Th Sept 24

 

6

 

7

 

8

 

9

 

10

 

11

 

12

 

 

 

AI do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinarily, whilst the great ocean of truth lay all undiscovered before me.@ - Isaac Newton

 

 

Links to Chapter Topics from Dunham's Journey Through Genius

Thales: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the “T” link to find the link to Thales

Pythagoras: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "P" link to find the link to Pythagoras

The Three Famous Problems of Classical Greek Mathematics: Go to http://www-history.mcs.st-andrews.ac.uk/Indexes/Greeks.html . The first three items under the column to the left are links to

  1. Squaring the circle
  2. Doubling the cube
  3. Trisecting an angle  

Some solutions to these problems are covered in Three Famous Problems.pdf

What are constructable numbers? See On Constructable Numbers.pdf

·         Chapter 2 - Euclid's Proof of the Pythagorean Theorem (ca. 300 BCE)

Euclid's Elements On-Line. "The" Textbook on Geometry written by Euclid around 300 B.C.E. is brought forward into the 21th Century C.E

Eudoxus: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "E" link to find the link to Eudoxus

Euclid: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "E" link to find the link to Euclid.

Eves, Howard & Newsom, Carroll, An Introduction to the Foundations and Fundamental Concepts of Mathematics Revised Ed., Holt Rinehard & Winston, 1971; Appendix A.6 The Eudoxian Resolution of the First Crisis in the Foundation of Mathematics; pp 332 - 334

An Article on Non-Eucludean Geometry: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry.html  

Prime numbers: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Prime_numbers.html

The Golden Ratio: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html to find a construction of the regular pentagon

Archimedes: Goto http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "A" link to find the link to Archimedes

Kevin Brown's Archimedes on Spheres and Cylinders: http://mathpages.com/home/kmath343.htm

A History of Pi: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_through_the_ages.html

Archimedes’ result that the surface area of a sphere, Ssphere = 4πr2, is also equal to the lateral surface area of a cylinder that encloses it (i.e. a cylinder with radius r and height 2r), has an interesting application in cartography. If we project all points on the sphere radially outward from a north-south axis (not from the center), we obtain the Lambert cylindrical equal area projection of the sphere onto a plane which leaves all areas the same. See http://en.wikipedia.org/wiki/Lambert_cylindrical_equal-area_projection. Thus, unlike the “standard” Mercator projection map, Greenland and Africa are correctly proportioned with respect to their areas (although Greenland is smeared across the northern half of the projection) – TANSTAAFL!    

At this point we consider the influence of Arab, Indian and Chinese cultures on mathematics

Arab Mathematics: Forgotten Brilliance: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html

An Overview of Chinese Mathematics: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_overview.html

An Overview of Indian Mathematics: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html

A History of Zero: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

Tartaglia vs Cardano: Goto: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Tartaglia_v_Cardan.html

Quadratic, Cubic and Quartic Equations: Goto: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations.html

Internet Resources for the History of Complex Numbers: Goto: http://math.fullerton.edu/mathews/c2003/HistoryComplexBib/Links/HistoryComplexBib_lnk_1.html

Gottfried Wilhelm von Leibniz: Goto http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Leibniz.html 

A History of the Calculus: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html

Beyond Mere Convergence - A paper by James Sellers of Penn State discussing the convergence of other series by Jakob Bernoulli: Goto http://www.math.psu.edu/sellersj/p25.pdf

Fermat Numbers: Goto Wolfram's MathWorld http://mathworld.wolfram.com/FermatNumber.html to learn (?) about Fermat Numbers and Fermat Primes

Who was Sophie Germain? Goto http://www.pbs.org/wgbh/nova/proof/germain.html

Grabiner, Judith V. "Who Gave You The Epsilon? Cauchy and the Origins of Rigorous Calculus".The American Mathematical Monthly Vol 90, No. 3 (Mar.1983), pp. 185-194. Available on-line thru JSTOR. Goto http://www.jstor.org/search/ and Search using the title of the article.

The Function Concept: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Functions.html

20th Century Mathematicians: Goto http://www-history.mcs.st-andrews.ac.uk/history/BiogIndex.html and follows the links to Kurt Godel and Alan Turing

Hilbert's 23 Problems  - Goto http://aleph0.clarku.edu/~djoyce/hilbert/toc.html

The Millennium Problems

Other Links from St Andrews Site: http://www-history.mcs.st-andrews.ac.uk/history/index.html

History Topics: Alphabetical Index: Goto http://www-history.mcs.st-andrews.ac.uk/Indexes/Hist_Topics_alph.html to obtain a list of mathematical topics

Links to Other Interesting & Useful Web Sites

Wolfram MathWorld: Goto http://mathworld.wolfram.com/ An excellent site for any technical questions about mathematics.

Mathpages.com: A site for lots of interesting articles on various mathematical subjects broken down by category. Useful for finding topics for "quick presentations"

Euclid's Elements On-Line. "The" Textbook on Geometry written by Euclid around 300 B.C.E. is brought forward into the 21th Century C.E

National Curve Bank: An archive for interesting and famous curves

Other Interesting Sites

The Ohio Section of the MAA - Find out what's happening "mathematically" in Ohio

The Math Archives - A great source for links to all sorts of interesting mathematical sites.

Return to Alternate Home Page

 

 

"This, therefore is mathematics: she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes the oblivion and ignorance which are ours by birth" - Proclus 5th Century C.E.