# Where are the geometric figures in those "advanced geometry" textbooks?

It is me again to bother you. Since my last post I started to look at some seemingly "serious" mathematics for background study. Accidentally I went into a university bookshop and came across some "differential geometry", "algebraic geometry", "geometric analysis", etc. I am a writer, and my math skill remains at pre-calculus level, but I am (doubly) confident that the subject I learned long ago called "analytic geometry" contains quite a handful of geometric figures and coordinate systems. I was genuinely surprised that those "advanced geometry" textbooks I randomly picked-up contain very few figures, and radically different from what I expected if they have any. Although I work with literature and languages, the descriptions in math textbooks sound to me just like Alienese. I was hoping that the geometric figures might be enlightening somehow, and I was so so wrong.

Well, this is my question: where are the geometric figures in those "advanced geometry" textbooks?

PS The same applies to what I heard about topology. I learned it from wikipedia that mathematicians are like changing coffee cups to donuts in topology, so I guess I might see loads of "cups and donuts", or similar stuff, in topology textbook. I was rather disappointed when I open a book called "Introduction to General Topology", all that I read is about some set things.

PS2 When I was roaming around the web, I learned that a prestigious mathematician called Shinichi Mochizuki proves "abc conjecture". The abc conjecture I read from wikipedia sounds quite "algebra" to me, but on Shinichi Mochizuki's homepage he called himself "inter-universal geometer". I take that "inter-universal" is like something very powerful. But how come a geometer solves an algebra problem?

• There should be figures in Differential geometry books --- there certainly are in the textbook by O'Neill. But Algebraic Geometry is a different kettle of fish entirely. It's geometry often over spaces so abstract that pictures are hard to come by. It's a bit like asking why the pictures you remember from Run, Spot, Run aren't in the Collected Works of Shakespeare. Sep 1, 2013 at 7:21
• @GerryMyerson What is “Run, Spot, Run”? Sep 1, 2013 at 7:54
• > math textbooks sound to me just like Chinese -- Well, Chinese people are often annoyed to find their language used as an example of unintelligible nonsense. How about the old phrase "Greek to me"? This still might offend some people, but a smaller number, at least :-) Sep 1, 2013 at 8:06
• I was joking. Note the :-). These days, almost anything one says will offend someone or other, so ... Sep 1, 2013 at 8:25
• Dear @bubba: how right you are that today anything will offend someone somewhere! I wouldn't mind at all if people said to me that something is so difficult that it sounds French to them: on the contrary, I would be flattered to heve been so intelligent as to understand French when I was one year old :-) Sep 1, 2013 at 8:58

Analytic geometry, invented by Descartes and Fermat, was the beginning of algebraic geometry and of differential geometry.
Their contemporary Desargues's work, the mathematization of the discovery by Renaissance painters of perspective, was also an important source for the branch of algebraic geometry called projective geometry but had less influence until it was rediscovered by Poncelet (in Russian captivity as a prisoner of war, after Napoleon 's defeat).

Analytic geometry is still indispensable if you want to understand these branches of mathematics.
And drawings are capital in the study and understanding of modern algebraic geometry: I have rarely heard a talk at a conference on algebraic geometry which was not accompanied by them, with some mathematicians exercising real artistic talent and making heavy use of colored chalk, or more recently of beautiful computer animations.

For various reasons this is not always reflected in books: the most egregious example being Grothendieck and Dieudonné's Eléments de Géométrie Algébrique, which laid out the foundations of modern Algebraic Geometry six decades ago and does not contain a single drawing.
Note however that the excellent textbooks by Fulton, Hartshorne, Qing Liu, Perrin, Reid, Shafarevich, ... contain lots of magnificent illustrations.

Finally, I find it regrettable that many young learners of algebraic geometry are not exposed to computations in analytic geometry: many questions on this site are from obviously excellent and intelligent students who unfortunately have been exposed to much abstract concepts but lack practice in down-to-earth calculations, so that they have difficulty in, say, proving that a quadric in three space is birational to a plane.

One of the difficulties of modern algebraic geometry, and the cause of its reputation as a hard subject, is the necessity to learn to use quite abstract tools (cohomology of sheaves, commutative algebra,...) but also simultaneously to be able to understand beautiful classical results like Pascal's mystic hexagram or Cayley-Bacharach's bound on the dimension of the complete linear system associated to a divisor on a curve.

• > I have rarely heard a talk at a conference on algebraic geometry which was not accompanied by them -- Yes, for some reason, conference talks have more pictures than books do. Why is that, I wonder? Is it related to the authoring tools? Or some misguided ideas about the level of formalism that's required in a book, but not in a talk? Sep 1, 2013 at 8:40
• Dear @bubba: I think the explanation in your fine answer is spot on: it is difficult to make illustrations in a book! Might it also be, as you conjecture, that authors subconsciously censor themselves because drawings look less serious? I have always been horrified by the fact that students fresh from high school (lycée in France) never spontaneously make drawings while solving problems. Some students told me that their teachers had discouraged them because a drawing is not a rigorous proof. True, but beside the point... Sep 1, 2013 at 8:51
• > drawings look less serious -- Yes. Maybe. More like "Run, Spot, Run", and less like the works of Shakespeare :-) Sep 1, 2013 at 9:07
• It is also very, very difficult to make good pictures. While giving a talk one can afford to present amateurish drawings, but there is a certain expectation that drawings in books be drawn by professional illustrators, and they are expensive, and it is a not trivial problem to get them to draw what the mathematician wants. Sep 1, 2013 at 9:10

This bothers me, too. Pictures don't provide rigorous proofs of anything (usually) but they can certainly be very helpful in explaining the ideas behind a proof.

I suspect that the shortage of pictures stems from some combination of the following:

(1) Some authors are not very good at drawing pictures, and don't have anyone to help them. Drawing is a specialized skill, and doesn't necessarily correlate well with mathematical ability. Professional illustrators are expensive.

(2) The tools used to produce many advanced books are not very good for drawing pictures (in my opinion, anyway). A lot of people use packages like TikZ and Asymptote, which essentially require you to write a program to describe your picture. It's hard work.

(3) Many mathematicians love abstraction and generality. Pictures of two and three dimensional things are not so hard to draw, but drawing $n$-dimensional or infinite-dimensional objects is much more difficult.

The deficiency isn't universal -- there are some examples of books where pictures play a major role. I highly recommend this book by Tristan Needham, for example.

A differential geometry book with lots of good pictures is this one by Gray. It's title is "Modern Differential Geometry" but I suspect some people would consider it old-fashioned, because it's very non-abstract. Anyway, old-fashioned or not, I like it.

Another remarkably visual book is this one by Koenderink. Arguably, it's not an "advanced mathematics" book, but the pictures are wonderful. They are almost all hand-drawn (by the author, I think). The book even includes a chapter that tries to teach us how to draw good pictures.

• Grateful for your suggestion! That book indeed sounds visual! Do you have more similar examples? Sep 1, 2013 at 7:44
• Added another one Sep 1, 2013 at 7:54

Marcel Berger's 'A panoramic View of Riemannian Geometry' is very visual (and excellent !). It is a rather comprehensive book but with less proofs than more abstract texts.

Vladimir Arnold's books about differential equations and his 'Mathematical Methods of Classical Mechanics' are very nicely exposed. His book about 'Catastrophe Theory' contains more graphics.

But to start perhaps that a more introductory and general text will be more appropriate like Ian Stewart's very visual 'Concepts of Modern Mathematics'.

Since we are making lists of advanced recent books with very nice illustrations (in colors) see :
- Mumford's 'Indra's pearls'
- Audrey Terras' 'A stroll through the garden of graph zeta function' : a draft of the book is available from the link there.

Fractals books offer perhaps the nicest graphics :
- Peitgen, Jürgens, Saupe's 'Chaos and Fractals'
- Prusinkiewicz and Lindenmayer's 'The Algorithmic Beauty of Plants'
- Ebert, Musgrave, Peachey, Perlin and Worley's 'Texturing & Modeling' (whole planets with satellites !)
but this is perhaps getting us too far from the initial subject...

Different presentations may come from the favorite references of the authors themselves : Klein and Poincaré (and many others) as opposed to more abstract (axiomatic) schools...