Pure Geometry Study Guide: Books & Roadmap for Students

Introduction to Pure Geometry

Geometry is one of the oldest parts of mathematics. It has been studied and advanced by the greatest minds humankind has to offer. It has been described as a subject of great beauty. How do we approach such an amazing work of art as a student?

In this section, I will attempt to give you a basic road map towards pure geometry. I assume that you are familiar with high-school geometry and trigonometry. If not, see my guide on how to self-study high school stuff.

Geometry is a huge subject, and there are a lot of classical works on geometry. Not all of these classical works have survived the test of time.

The works of Apollonius and Archimedes are very beautiful but are much easier to study with modern language and modern notions, such as analytic geometry and calculus. Nevertheless, the history of geometry is important if one wishes to understand geometry today.

In modern times, pure geometry is very much neglected in the curriculum. Students go directly to the many related fields of algebraic geometry, differential geometry, topology, finite geometry, and so on. I find this regrettable: it is difficult to appreciate these subjects without a decent grounding in pure geometry or its history.

I have seen many students trying to learn algebraic geometry without having any idea of the classical notion of a projective space. This is a shame. As you will see in this guide, I think pure geometry is very important to know as a mathematician, even if you are eventually interested in a more modern type of geometry.

Where do we start learning pure geometry? Let’s start with a timeless book that almost any mathematician in history has read: Euclid’s Elements.

Euclid’s Elements — Hartshorne and Moise

Euclid’s books are truly wonderful. They mark the beginning of mathematics itself and geometry in particular. Much of the book can still be read very easily today, even though the books are millennia old.

Sadly, Euclid is not without flaws. While Euclid’s work is timeless, there has been considerable progress in geometry that shines a new light on Euclid’s results.

This is why I think it is important to read Euclid together with a decent commentary. Hartshorne’s book is perfect here. The first chapter guides the reader through the first four books of the Elements and gives valuable commentary on the books. Later chapters present Euclid’s results in the light of modern geometry: they present the Hilbert axioms, give a modern treatment of area (of rectilinear figures), treat hyperbolic geometry quite extensively, and much more.

A possible drawback of Hartshorne’s book is that some sections require advanced abstract algebra. Most of the book is still very readable with only a modest knowledge of abstract algebra, but some readers may prefer a somewhat less advanced book. For these readers, I highly recommend Moise’s Elementary Geometry from an Advanced Standpoint.

If you choose to read Hartshorne, read Hartshorne and Euclid concurrently: Hartshorne tells you when to read which parts of Euclid. If you choose Moise, first go through Euclid’s first four books and then read Moise afterward for a modern account.

I particularly like Joyce’s online edition of Euclid’s Elements: Joyce — Euclid’s Elements (online). Aside from giving Euclid’s original theorems and proofs, the site provides useful commentary. If you wish to buy Euclid in book form I recommend Euclid’s Elements Redux (Callahan), which comes with exercises to test your understanding. The book can also be downloaded legally for free: Euclid (Starrhorse mirror).

Hartshorne’s book can be found here: Geometry: Euclid and Beyond (Hartshorne), and Moise can be found here: Elementary Geometry from an Advanced Standpoint (Moise).

You will learn the following:

• Classical Euclidean plane geometry
• Construction problems
• The relation between fields and geometries (the first step towards algebraic geometry)
• The relation between groups and symmetries (the first step towards Lie theory)
• The algebraic solutions to many classical construction problems (Galois theory)
• The classical theory of area
• Hyperbolic geometry
• Platonic solids and other polyhedra

I strongly recommend reading the first four books of Euclid. If you desire, you can read the rest, but I do not think the later books are as necessary. Some later books have not stood the test of time as well; especially the Greeks’ refusal to accept irrational numbers made many later statements in Euclid awkward and difficult to grasp.

While reading Euclid’s Elements, you might want to get a more historical perspective on Greek mathematics. For this, I highly recommend Kline’s Mathematical Thought from Ancient to Modern Times (volume 1).

Brannan, Esplen & Gray — Geometry

After becoming acquainted with Euclidean geometry, it is time to meet other geometries. A very neat classification of modern geometries has been achieved by Klein’s Erlanger program, which connects groups and geometries. Brannan, Esplen, and Gray treat affine geometry, Euclidean geometry, projective geometry, elliptic (and spherical) geometry, hyperbolic geometry, and inversive geometry.

The book also gives a modern (coordinate-based) treatment of conic sections in both affine and projective frameworks. I think a book like this is a must-read for people going into differential or algebraic geometry.

A downside of the book is that most exercises are rather easy. However, the main text is excellent, and it is hard to find this collection of material in other books. The prerequisites include a good knowledge of linear algebra (up to diagonalization of matrices) and some basic group theory.

The book can be bought on Amazon here: Geometry (David Brannan).

After reading this book, you will know of:

• Conic sections and quadric surfaces
• Affine geometry
• Conic sections in affine geometry
• Projective geometry
• Conic sections in projective geometry
• Inversive geometry
• Hyperbolic geometry
• Spherical geometry
• Elliptic geometry
• The unification of these geometries via Klein’s program

These two books contain what I think every mathematician should know about pure geometry. You can continue reading on pure geometry if you wish. The following books are optional but recommended.

Bennett — Affine and Projective Geometry

In Hartshorne and Moise, you have encountered a relationship between Euclidean geometries and certain ordered fields. Algebraic geometry extends these relationships far more generally. It is not necessary to use the full machinery of algebraic geometry to relate geometries to arbitrary fields: weaker geometries still permit many neat results. Bennett’s book develops an axiomatic approach to affine and projective geometry and investigates when these geometries arise from algebraic objects such as fields.

In this book, you will read:

• An axiomatic approach to affine geometry
• The role of Desargues’ theorem in relating affine geometries to division rings
• The role of Pappus’ theorem in establishing commutativity
• An axiomatic approach to projective geometry
• An introduction to lattices and their relation to geometry
• The fundamental theorem of projective geometry, relating geometric collineations to algebraic matrices

Some modest abstract algebra is required for this book—mainly a basic knowledge of fields. A previous encounter with affine and projective geometry at the level of Brannan is recommended.

Projective and Cayley–Klein Geometries (Onishchik & Sulanke)

If you liked the unification of different types of geometries as in Brannan, then this book is a must-read. The book begins with a high-level approach to projective geometry using linear algebra in the first part. The second part covers Klein’s Erlanger program in modern language.

The book can be found here: Projective and Cayley–Klein Geometries (Onishchik & Sulanke).

This book is not for the faint-hearted. Easier introductions include Richter-Gebert’s Perspectives on Projective Geometry. I think it is a pity that Richter-Gebert’s book contains no exercises, which makes it harder to grasp the material. Another beautiful book is Kowol’s Projektive Geometrie und Cayley-Klein Geometrien der Ebene (Kowol), which has not been translated into English as of this writing.

Audin — Geometry (Advanced Perspective)

If you wish to see these pure geometries from a more mathematically advanced point of view, Audin’s book is excellent. It treats Euclidean, affine, and projective geometry using the language of topology, analysis, and group theory. It is a natural follow-up to Brannan for readers with more advanced mathematical background.

Audin’s book can be found here: Geometry (Michele Audin).