A Geometry of Points and Polygons

I am giving a talk, and I plan on posting my slides here afterwards:

  • Date: Wednesday, 22 April 2026
  • Time: 20:00 Muscat Time (GMT 16:00)
  • Speaker: Prof. David Pierce, Mimar Sinan Ince Sanat Universitesi
  • Title: A Geometry of Points and Polygons
  • Moderator: Mohammad Shahryari – Sultan Qaboos University – Oman
  • Access Link: https://meet.google.com/kdv-rakt-uut

I have cut and pasted the details from the email announcement, which came with a poster. There is a curious error. My university here in Istanbul is actually Mimar Sinan Güzel Sanatlar Üniversitesi. In English, the name is Mimar Sinan Fine Arts University. If you translate “fine” back into Turkish, you will normally get not güzel “beautiful,” but ince “thin.”

Below is the abstract for the talk that I submitted last September (2025). The talk that I have prepared is pretty faithful to it. I can say more briefly that the emphasis is on doing Descartes’s work with ratios of lengths, rather than with lengths themselves; and on developing the geometry of ratios by means of areas. As much of this development as possible is made visual in the slides. Anyway, here is the submitted abstract:

In La Géométrie of 1637, René Descartes gives a geometrical foundation for algebra by interpreting a field in a Euclidean plane. At least, that is what we can understand him as doing, but his work has gaps. These are filled in different ways by David Hilbert (1862–1943), Alfred Tarski (1901–83), and Emil Artin (1898–1962). Tarski’s plane is explicitly just a set of points, with a ternary relation of betweenness and a quaternery relation of equidistance, satisfying certain axioms, written in first-order logic (that is, each axiom is finite, with quantification over individuals only). Artin does not use the quaternary relation, but the elements of his field are algebraic, in the sense of being operations on the set of points. We can make the field elements more purely geometric by understanding them as equivalence classes of ordered triples of collinear points. The equivalence relation is proportional division.

Artin’s axioms include versions of theorems named for Desargues (1591–1661) and Pappus (4th century). That Desargues’s Theorem follows from Pappus’s Theorem is shown by Hessenberg (1874–1925). Moreover, Pappus proves his theorem by means of Book I alone of the Elements of Euclid, without any notion of proportion. The theory of area is needed: in particular, that triangles on the same base are equal, if and only if the line joining their apices is parallel to the base.

In a two-sorted structure then, with points and areas, we can axiomatize a plane in which a field is interpreted. This approach is of interest, both for for providing a new example of the model-theoretic concept of a companionable theory, and because many results of Greek mathematics, notably those of Apollonius on conic sections, rely essentially on areas. The proofs do not readily translate into the formalism of Descartes, in which the constants and variables are only for lengths.

Descartes’s solution of a five-line locus problem does not require areas. Neither does it require his formalism, but it can be proved, perhaps more clearly, in the ancient manner.

And here is the bio submitted at the same time:

Read Euclid, Apollonius, and Descartes at St John’s College, Santa Fe, New Mexico, USA. Doctorate from the University of Maryland, College Park, USA. After postdoctoral positions in the USA and Canada, worked at Middle East Technical University, Ankara, Turkey; since 2011, at Mimar Sinan.