# How were 'old-school' mathematics graphics created?

I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today:

Does anybody know how this graphic was created? Were equations used for the surfaces and then a plotting program used? How was the line-hatching achieved? Here is a another gorgeous picture from David Hilbert's "Geometry and the Imagination":

Again, how was this created? Was it done by hand, then scanned in?

More pressingly: how do I create these kinds of images? Certainly, most of us are familiar with Matlab, Geogebra, gnuplot, or other software for creating mathematical figures, as we are also familiar with vector-based programs like Inkscape and Adobe Illustrator. I've looked at 'old-school' programs like IPE (a little bit like XFig), but still, I don't find them as attractive as the examples above. There is then LaTeX solutions like TikZ. I guess they must surely be hand-drawn, but I would like to know about the process for how these were drawn (and the equipment used).

By way of note, there is an article here about trying to use 3d modeling programs and shaders to duplicate hand-drawn figures.

• I believe the figures were often made by professional artists, probably at the publishing company, based on sketches provided by the authors. Incidentally, I've noticed in books where computer generated graphs are used that (the more accurate) computer generated graphs are sometimes not as useful in conveying information as a less accurate schematic graph. For example, the graph of $y = \cos (\ln x)$ on p. 77 of Sergiy Klymchuk's Counterexamples in Calculus is very misleading as to the behavior of this function as $x \rightarrow 0^{+}.$ Jun 26 '14 at 15:41
• @DaveL.Renfro : I suspected they would be created by professional artists. It's a shame that modern graphics have gone on to replace these images, though it has likely improved the efficiency by tenfold. I'd love to see an article or video of this done---so I can better reproduce it myself. I wonder how one might go about this. Perhaps this should be posted in the Stack Exchange for graphics design?
– TSGM
Jun 26 '14 at 15:44
• Related on Academia.SE: How did authors prepare figures in their publications before the advent of computers? Jun 26 '14 at 18:05
• Related, possible duplicate: math.stackexchange.com/questions/627700/… Jun 26 '14 at 23:31
• @asmeurer: Let me check - they combined the series into a single book - the version I have. It looks like it is combo of computer graphics and also Diane Rogoli for drawings (I guess she is an artist and used hand drawn rough sketches). See: amazon.com/Dynamics-Geometry-Behavior-Ralph-Abraham/dp/… . Here is the intro with her name : aerialpress.com/DYN/DYN4Pt1Ch0X.pdf Jun 27 '14 at 1:52

Often the illustrations were drawn by hand, by the mathematicians themselves. The book A Topological Picturebook by George K. Francis (Springer, 1987) describes how one learns to do this:

This book is about how to draw mathematical pictures. … Theirs [the geometers of the 19th century] was a wonderfully straightforward way of looking at rather complex things, notably Riemann surfaces and geometrical constructions over the complex numbers. They drew pictures, built models and wrote manuals on how to do this. … I resolved to try to do the same for the mathematics of my contemporaries.

The first example is how to draw a hyperbolic paraboloid on the blackboard:

No software is used, but there are techniques one can learn.

• Lots of great answers, including some ideas of how these graphics can be created using Blender and NPR. However, the question was more about the 'old-school' process and this seems to be the most relevant answer. I'll accept it then.
– TSGM
Jun 27 '14 at 16:54
• @TSGM Consider accepting this answer, which made the same suggestion as me. My andwer was more detailed, but the other one was first.
– MJD
Jun 27 '14 at 17:30

Up until the 1990's, high-schools used to teach "technical drawing". Students would spend hours just drawing by hand isometric projections of various bits of machinery. To draw an ellipse, you would draw guidelines, a parallelogram, then several points of the circle, at various angle positions, 30, 45, 60 degrees, then sketch the curve and add lines.

• Up until the 1990s? They're still teaching it here! :)
– Bob
Jun 27 '14 at 12:40
• At my university in the 90's I learned how to draw those and much more interesting stuff like How to take the square root of a number just with a plain ruler (without marks) and compass among other mathematical calculations. Pitty if its not being taught now. Jun 27 '14 at 14:39

Non-photorealistic rendering techniques can be used for mathematical surfaces but it is still a hard job to make a nice mathematical illustration.

The images below are from A few good samples: shape & tone depiction for Hermite RBF implicits. See also Shape and tone depiction for implicit surfaces and Illustrating smooth surfaces for other examples.

• I'd be very pleased if you could write a small guide of how those images were generated. What 3d program was used? Was renderer? Were the surfaces generated using formalae then imported in some way, or were they created using the program's own objects? A small guide of how this was achieved would be very nice!
– TSGM
Jun 26 '14 at 16:32
• @TSGM, I've added some references to papers. These are only meant as examples. The field of NPR is a large one. As for software, there is Blender NPR, but I haven't tried it.
– lhf
Jun 26 '14 at 18:33

This book is intended to teach the reader how to draw such diagrams: http://www.amazon.com/Topological-Picturebook-George-K-Francis/dp/0387345426

• Beautiful book. Thank you!
– TSGM
Jun 26 '14 at 16:30

I think it's already explained how it was done in the past: manually.

Nowadays, you can use the free software Blender and use the Freestyle renderer introduced in version 2.67. One of the prominent examples is the sawshark.

On Youtube, you also find the animated version of the shark.

It can also be used for technical drawings like the airplane or the locomotive (can't embed because of unclear license).

So, as long as you can model your fomula in 3D, you can also render it in a sketchy way with Blender.