I am flying on Monday to Novosibirsk in Siberia, and then a week later to Krk, an island in Croatia. The occasions are two interesting geometry conferences, where I will meet old friends, make new ones, and talk about my recent work. Travel is one of the joys of my life, and even a long plane flight is only half a burden, knowing that it ends in—a new place! It’s a chance to step out of the groove, in a small way, for a while.

The life of a research mathematician, if the truth be told (and I want to do that), is outwardly not very exciting. The routine is piled on thick; calculus and linear algebra lectures don’t change much from year to year, writing papers is rather boring and tedious, and the stable, conservative environment of most universities means that one’s working life is like a sheltered cocoon. Jobs are hard to come by, so career mobility for most of us is limited, and we get pretty familiar with our stomping grounds after a few decades.

The inward life, however, is a totally different story. Mathematics present us with such a rich and fascinating framework of ideas, concepts and challenges that I can think of no work that would give me even half the satisfaction I get from being a research mathematician. I would not trade my job for any other, not in law, politics, business or medicine. The search for patterns, the hunt for a key equation, the challenge in proving what our intuition tells us loud and clear* must* be true…until we find out a week later that our intuition was wrong and the story goes in a quite different direction! What a happy life to be able to think about interesting things; I am very grateful, and one of the purposes of this little blog is to share some of the rich ideas I get to think about, with you all.

At both conferences I will be talking about the *Rational Trigonometry of a Tetrahedron*, albeit in somewhat different forms. You are all familiar with ordinary trigonometry (the word means the study of triangle measurement); with those angles and lengths, cosines and sines, and a whole raft of complicated formulas. It’s useful, but not altogether pretty.

Hopefully you have heard that there is now a new and better way of tackling the whole subject, called *Rational Trigonometry*, discovered about 10 years ago by yours truly. This story of a completely new way of thinking about a very classical subject is rather interesting; as you might expect such a bold departure from tradition doesn’t occur very often in elementary mathematics. Is it really better? Does it make computation simpler? Does it lead to a lot of beautiful new mathematics? Yes, yes and yes. And the crucial question: is it what we ought to be teaching our young people in high schools?? Definitely yes!

I will be telling you more about this discovery of mine, and the reaction that it has gotten from colleagues etc. in due time. An interesting consequence is that new doors have been opened to me by this understanding; vistas and trails that lie before me now, undreamt of a decade ago. The possibility of a *new approach to the entire subject* is starting to emerge as the fog slowly settles—a more careful, honest, logical and beautiful mathematics, more closely aligned to computer science.

The consideration of “revolution” in the context of mathematics is to many practitioners unlikely and even heretical. The safe confines of academia enclose an even more secure installation of pure research mathematics, where orthodoxy, accepted practice and authority largely rule.

Perhaps my whole life has been building up to the realization that, even in mathematics, there are true paths, and false paths, and paths in-between, and that ultimately only I can decide which is which—for myself. If you allow me, I propose to take you on some little mathematical journeys, and show you new possibilities for thinking. Then you too can decide what is true, what is false, and what is in-between—for yourselves.