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     6. Epilogue: Pólya and Ourselves - Mathematics, Tea and Cakes6

Professor George Pólya (1887 - 1985) emigrated to the United States in 1940 and joined the Mathematics Department at Stanford University in 1942. Although the rest of his professional life was spent at Stanford, he made many trips abroad to accept visiting appointments for short periods of time.  During Pólya's visit to the ETH (Zürich) in 1966 he shared an office with Peter Hilton (and PH was a guest at his 80th birthday party, held in Zürich, in 1967).

In 1969 Pólya was invited by Gerald Alexanderson (Mathematics Department Chairman at Santa Clara University - then and now) to give a colloquium talk at SCU. While there Pólya met Jean Pedersen and was fascinated by (a) the models in her office (some of which are described in [Rec]) and (b) by her lack of knowledge about their symmetry and their usefulness in exemplifying some of the mathematics of polyhedral geometry. After this initial meeting Pedersen visited George and Stella Pólya at their Palo Alto home once a week until his death in 1985. 7. Pedersen and her husband Kent (who shares Pólya's birthday - except for the year!) were guests at Pólya's  90th birthday party, held at Stanford, in 1977 and the Pólyas were guests at the Pedersen's home for Thanksgiving dinner for many successive years.

A typical visit, for Pedersen,  included a discussion with Pólya about mathematics. After an hour or so Stella would appear with tea and cakes, or cookies, and the three of them would turn their attention to current events and politics. It was during this time that Pedersen learned about proper rotation groups (knowledge that Pólya acquired from Felix Klein himself) and the Pólya Enumeration Theorem, about Euler's famous formula connecting vertices, edges and faces of a polyhedron, and about the formula Descartes discovered concerning the total angular deficiency of a polyhedron. Pedersen found herself studying very hard8 and looking forward to discussing the new-found aspects of her own models.  Pólya and Pedersen also discussed pedagogy and, in fact, Pedersen was Pólya's last co-author (see [PP]).


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Figure 9

In 1978 Pedersen was asked to try to get George Pólya and Peter Hilton together9 in Seattle at the joint annual meeting of the American Mathematical Society and the Mathematical Association of America, to discuss "How to and How Not to Teach Mathematics". The suggestion was that Hilton should discuss "How Not to Teach Mathematics" and this would be followed by Pólya giving "some Rules of Thumb for Good Teaching". Pólya agreed to participate on the condition that Pedersen would handle the travel details of getting him to and from Seattle. Hilton also gave only conditional approval for the plan. Hilton's idea was that it would be much more interesting, and effective, if he were to demonstrate a thoroughly bad mathematics lecture (instead of simply talking about it). Hilton also suggested that Pedersen should be the moderator for the program.

All conditions were met and the Seattle presentation duly took place. It was a tremendous success. Hilton's part was hilarious and some said it nearly ruined the rest of the meeting as participants saw many of Hilton's intentional errors unintentionally repeated by some of the other speakers. Pólya's contribution was, as you might expect, superb and had the unmistakable mark of a master teacher. A month or so later Pedersen was asked by the National Council of Teachers of Mathematics to arrange that the Hilton-Pólya performance be repeated at their San Diego meeting in the fall of 1978 so that it could be videotaped. After a few more meetings with tea and cakes, and some long distance calls, this was done.

At the San Diego meeting Pedersen invited Hilton to visit SCU in October to give a colloquium talk.  He did, and when he saw the models in Pedersen's office they again sparked long discussions, but this time the discussions centered on the differences between the ways geometers and topologists classify surfaces.

In 1982, while Peter Hilton was on sabbatical leave as a visiting professor at the ETH and Pedersen was visiting there for a quarter, they began looking seriously at the paper-folding. Hilton suggested to Pedersen that she should try to devise a really systematic way of constructing the polygons from the folded strips (since the 2n+1-gons seemed to have very special features that didn't generalize). The first result of Hilton's suggestion was the FAT-algorithm. This innocent-looking algorithm, in fact, opened the flood gates for both the development of the general folding procedures and the number theory that grew out of the paper-folding.

After 1978 whenever Hilton visited SCU he went with Pedersen to visit the Pólyas and together they continued the tradition of mathematics, tea and cakes. In 1981 Hilton and Pedersen, along with Alexanderson, cooperated with Pólya to bring out the Combined Edition of Mathematical Discovery (see [P2]).10 During many of the tea parties at Pólya's home, Pólya talked about his idea of homologues, and on one occasion told us that he had never written about them and that someday he would like us to write about them - in fact, he extracted a promise from us that we would do so. Thus we are very grateful to Denes Nagy for giving us such a splendid opportunity to fulfill our promise to our dear friend and teacher George Pólya, and to convey the flavor, and a few of the details, of our friendly relationship with him.


6 We present this more personal epilogue to our article at the express request of Denes Nagy.
7 After George Pólya's death, Pedersen continued to visit Stella Pólya at least once a week until her death in 1989, just before her 94th birthday.
8 Figure 9 is an example, in Pólya's own handwriting of a page he once gave Pedersen saying "see if you can figure out what it means". It is connected with what we've been writing about in this article, so we leave the reader to do Pólya's homework  assignment for the week! The only hint Pólya gave was to say that H = Hexahedron (cube).
9 This was how Hilton and Pedersen met and began a collaboration that has resulted in over 70 papers and four books - to date!
10 Alexanderson updated the references, Hilton wrote a foreword, and Pedersen provided an expanded (and less esoteric) index.


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