Thursday, July 23, 2009

self referential

There is a fascinating and ever-growing body of literature comprised of works by mathematicians writing about what it means to be a mathematician. Sometimes these works touch on the topic while exploring the questions "what is mathematics?" or "how should math be taught?" but a significant number of mathematicians write about what they do without reference to these other questions.

Mathematicians who write in this vein are likely the best examples of reflective practitioners, and are probably better teachers, researchers, and scholars as a result of thinking carefully about what they do and taking the time to explain it to others. The "others" that this sort of literature is intended for are worth considering - who are these mathematicians trying to explain themselves to? Rarely are these works intended for complete outsiders, rather they are often intended as a set of Letters to a Young Mathematician, or are addressed to other academics (scientists, and engineers, mainly) that the authors feel they should have more in common with. Often these writings are addressed to fellow mathematicians, perhaps just to reassure them that they are all in the same boat. Sometimes these writings are a sort of meta-communication - mathematicians explaining to other mathematicians about how they should really be explaining things to the man in the street.

I started thinking about this kind of writing after seeing the film I want to be a Mathematician (mentioned here) about Paul Halmos, and reading some essays by Freeman Dyson and Timothy Gowers (mentioned here). Another excellent essay more sharply focused on the topic is Through a Glass Darkly by Steven Krantz.

Those who study the sociology of science, of academics, and of scholarly mathematicians in particular (yes, there are sociologists who study mathematicians), must have to treat these texts in a special way. Obviously, these are valuable documents, but are mathematicians the best observers of mathematicians?

Tuesday, July 21, 2009

Euler on arXiv

Jordan Bell has been posting his translations of some of Euler's works on the arXiv.

Two recent additions are: An observation on the sums of divisors and
On the infinity of infinities of orders of the infinitely large and infinitely small.

Bell has also written the paper Euler and the pentagonal number theorem.

To find other Euler translations by Bell, you can search arXiv under Euler, or search at the Euler Archive, which links to the papers Bell and others have translated.

Wednesday, July 15, 2009

Annotated Math Blogroll 1

Before starting this blog of bits of recreational math I had no idea about the wealth of math-related blogs that are out there. It turns out that there are lots of blogs provided by broad range of practitioners and enthusiasts. Each has its own particular lens, but taken collectively they give a surprisingly rich image of what mathematics is.

I've put this first group of blogs into three sets: enthusiasts, expositors, and educators. Most blogs could be in all categories, the groupings are intended to reflect how I've interpreted the bulk of the postings.

Group 1: Enthusiasts
In the enthusiasts category there are professional mathematicians, educators, amateurs and writers who post on a variety of math topics in a way that is generally accessible.

Brian Hayes writes a wonderful column "Computing Science" for American Scientist (for anyone who misses what Scientific American used to be like, this is what you should be reading). Brian correctly sees computing as a mathematical activity which is both practical and recreational. I think that many of his readers see his column as the natural successor to Martin Gardner's Mathematical Games column.

The Math Factor Podcast
A playful and interesting podcast and blog - mathematics exposition and promotion at its best.

Travels in a Mathematical World
What can I do with my math degree? Where can math take me? Well, if you have ever wondered about these things, this blog and its accompanying podcast will help you find some answers.

Tanya Khovanova's Math Blog
For numbers in the range of 0-9999, Tanya Khovanova's Number Gossip gives Wolfram Alpha a run for the money. Her blog presents interesting insight into mathematical life.

Division By Zero
The author of Euler's Gem provides a nice blend of technical and accessible posts, often with applets to demonstrate the concepts that he's discussing.

Group 2: Expositors

In the expositor category I've put blogs by researchers and professional mathematicians who post about their current work, or subjects that interest them. Generally highly technical research-level mathematics.

What's New
Terrance Tao is a Fields medal winning mathematician who's output is staggering (just count how many articles he contributed to the Princeton Companion to Mathematics to get a general idea). In this blog he talks about his current research - varied and intense.

Gower's Weblog
Fields medalist Timothy Gowers (editor of the Princton Companion to Mathematics, mentioned above) maintains a blog that ranges over the technical, the general and the personal. On the leading edge of online mathematics collaboration, another interesting math resource that he is involved with is the Tricki.

Gyre and Gimble
Charles Wells provides a portmanateau of a blog that contains his thoughts on mathematics and language. Wells is the coauthor of Toposes, Triples, and Theories, a classic Category Theory text that is available free online, and author of the Handbook of Mathematical Discourse.

Combinatorics and More
A fascinating blog that really can be described as 'combinatorics and more'. The "more" is quite interesting - take for example the author's surreptitious involvement in the "String Theory Wars" as described in his recent book Gina Says.

Mathematics under the Microscope
The pictures of children who evnetually become leading mathematicians are a surprisingly poignant element of Alexandre Borovik's book of the same name.

Group 3: Educators

Math educators blog about school math topics (where school could be K-12 or college), and about math education.

Let's Play Math!
The original home of the Math Teachers at Play blog carnival. This blog is a key hub for finding many other math-related sites and blogs.

The "unofficial" blog of the Nazareth College math department. Their recent series of posts on ways to multiply should win some sort of award.

Mathematics Education Research Blog
What are all those math education journals saying, right now? Reidar Mosvold's synopses and pointers to papers are a great resource.

Tuesday, July 7, 2009

Rhombic Hexecontahedron

If you ask Wolframalpha about its logo, it will tell you that it is a rhombic hexecontahedron. I had fun making one of these recently (here is a pdf of the pattern I used - you will need to print 3 copies onto card stock to make the model shown above).

One neat thing about the rhombic hexecontahedron is that it is a stellation of the rhombic triacontahedron (described a bit in this post), and both polyhedra have the same type of faces - golden rombs (how unusual is it for a stellation to have the same polygonal faces as the polyhedron that it is obtained from?).

A neat project would be to create a set of golden rhombohedra that could be used to build both the hexecontahedron and the tricontahedron. According to the mathworld entry, 20 acute rhombohedrons can be assembled to make up the hexacontahedron, and according to Coxeter (Regular Polytopes, page 27), you can build the tricontahedron with 10 acute and 10 obtuse rhombohedrons.

Another construction is suggested by the 'ball of whacks' toy - which shows that you can build up the tricontahedron with 30 rhombic pyramids.