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:

From Berry and Howls in Nonlinearity 3 (1990) 281-291

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”:

Hilbert's book

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.


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:

Hyperbolic paraboloids

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

Source : Link , Question Author : TSGM , Answer Author :

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