At the heart of metrical algebraic geometry there are hierarchies of beautiful algebraic formulas, starting in one dimension and then working their way up both in dimension and complexity. These can be viewed as generalizations of formulas of Archimedes, Ptolemy, Brahmagupta, Bretschneider, von Staudt and others, as well as classical results from Euclidean geometry.
One of the interesting aspects of our approach to these formulas, based on Rational Trigonometry, is the pursuit of generality: using general fields and arbitrary quadratic forms. Interesting connections emerge with the theory of special functions, differential geometry and operator analysis.
Most of these formulas would be challenging to find without the use of a computer: this is one reason why they have laid hidden for so long. We’ll see that there are many unanswered questions that invite exploration.
The talk will be accessible to undergraduates.