I was surprised to learn from John Stillwell's comment in answer to the question, "Can the unsolvability of quintics be seen in the geometry of the icosahedron?", that

There is not a single picture in the whole ...

of Felix Klein's 1884 book, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. And now that I have it in my hands, I can verify the lack of figures (but its remarkable clarity nonetheless).

By the time of David Hilbert's and Stephan Cohn-Vossen's 1932 Geometry and the Imagination (citing an earlier MO question), the use of figures had reached a high art. This high art is continued today in Tristan Needham's Visual Complex Analysis, with its stunning figures, e.g., (from this MSE answer):       
Needham p.135
To return to ancient times, it is clear that figures were valued to illustrate Euclid by 100 AD, and likely earlier:
Euclid papyrus
Finally, my question:

Has the use of figures to illustrate geometry waxed and waned over history in a fashion that could almost be graphed with respect to time, or am I plucking out idiosyncratic examples that do not point to any recognizable trends?

Perhaps this needs to be mapped country by country, different in France (during the Bourbaki period) than in Germany, etc.? Or perhaps any such attempt to capture gross trends is hopelessly historically naive?

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    Joseph: Even in France during "Bourbaki period" you can find Marcel Berger publishing 2-volume "Geometry" full of figures. – Misha Jun 29 '13 at 1:54
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    Misha's comment shows that the attitude to pictures depends more on the worldview of an individual mathematician, then on the period or country. Another example is E. Landau, who never included a picture in his books, and who lived in the same period and in the same country as Klein. – Alexandre Eremenko Jun 30 '13 at 6:37
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    Copied my comment on your previous question: There is a single figure, on the last page, in the German language edition of Klein's book. The figure is a lovely, labelled, rendering of the stereographic projection of the tiling of the sphere by (2,3,5) triangles. – Sam Nead Aug 20 '13 at 15:34
  • Sorry - I think that the figure only appears in the original German edition (as a fold out panel) and in a 1993 reprint (also in German). – Sam Nead Aug 20 '13 at 15:43
up vote 15 down vote accepted

Some other books of Klein (Fricke-Klein on authomorphic functions, his Lectures on hypergeometric functions) have a lot of excellent pictures. From reading many old mathematical books I cannot conclude that there was any trend or fashion that changed with time or from country to country. The explanation why some books have pictures and others do not is probably simpler: it always was (and still is!) very difficult to make good pictures. At the time of Klein, a skilled engraver had to be hired. Reproduction of the pictures probably substantially increased the cost of printing.

In our time, many mathematicians (including myself) do not know appropriate tools, and don't want to spend time required to produce good quality pictures. Reproduction of old pictures in new editions of the same books is probably also enormously difficult and expensive.

  • Excellent point that the effort and expense to create quality figures likely accounts for the variations better than historical trends! – Joseph O'Rourke Jun 29 '13 at 12:59
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    I agree with ALexandre about the situation today. If I have some text in mind, then I can easily produce, with TeX, a very presentable version of it. If I have some picture in mind, it will probably just stay in my mind, or at best in some rough, hand-drawn sketch. – Andreas Blass Jun 29 '13 at 16:12
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    I have been very happy with TikZ/PGF. See texample.net/tikz/examples It's a very very good tool to produce high-quality pictures. I think that it's been worth my time, not only for my writing/research, but also for my teaching and presentation (e.g. producing good handouts, beamer slides, etc.). – Marty Jun 30 '13 at 0:30

Maybe this isn't an answer, but below is a photograph of the tablet BM15285 (British Museum catalog #15285). It's a series of geometry problems, from c.1800 BCE (+/- 200 years?). There are plenty more Babylonian and Egyptian mathematical texts with illustrations.

I don't think the waxing and waning of figures in geometry is such an interesting question by itself, and I don't know what we'd learn from such a graph. What's in the figures tells us something about the geometry at the time/in the culture from where they came. The figures also reflect the writing and publishing industry at the time -- whether carving tablets, hand-drawing on papyrus, mass-printing and typesetting, etc..

enter image description here

You may enjoy Reviel Netz' book: The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge: Cambridge University Press, 1999.

It examines the use of letters diagrams in Greek deductive practices. Of course, my suggestion does not answer your question directly, but I am hoping that it sheds some light on the whys and hows of using figures over times.

The Sangaku problems of Japan's Edo period are worth mentioning here. Also see Sangaku: Reflections on the Phenomenon for some cultural history about it.

There isn't a single diagram in Weil's Foundations of Algebraic Geometry (1962, 363 pages) either.

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