Drawing in Mathematics: From Inverse Vision to the Liberation of Form by Gemma Anderson, Dorothy Buck, Tom Coates and Alessio Corti (2015)

LEONARDO Journal, October 2015, Vol. 48, No. 5, Pages 439-448

Posted Online September 23, 2015.
The paper focuses, rather than geometry in art or computer visualisation, on drawing as a shared mode of communication in art, mathematics and the wider world. The argue that its shared role in mathematics and in art as a fundamental mode of understanding, and as language, “make possible a new mode of collaboration between artists and mathematicians, in which the different logics of the two disciplines coexist on equal terms.”
For the artist (Anderson), the collaboration provides access to beautiful and otherwise inaccessible geometries and the opportunity to experience and transform them, integrating them into her knowledge of form. For the mathematicians (Buck, Coates and Corti), the collaboration allows a new form of creativity, giving material form to purely conceptual objects, and brings their research to a wider audience in
a nondidactic way.
They describe mathematical creativity as a dialogue between two characters: the Thinker, existing in the world of linear logical thinking, and the Drawer, who operates in the world of the imagination and of inverse vision. They focus on the drawing of internally imagined objects (illustration, then?? Or is something more interesting going on?).
We illustrate this by showing some drawings made by Alessio Corti as a teenager. There are five regular (or Platonic) three-dimensional solids in Euclidean geometry (Fig.2). Having read somewhere that in four dimensions [5] there are six regular solids and that one of them is made of 120 regular dodecahedra, Corti tried to prove this fact using the Euler formula in four dimensions, but he could not do it. Eventually he decided that the only way for him to prove the existence of this 120-cell was to draw it (Fig. 3).
Interesting – to prove it, huh?
…the drawing that we speak of here is the drawing of visual objects. There has been much written about the visual and spatial representation of nonvisual scientific objects, for example on the visualization of statistical data or on the visual representation of processes and of relationships between concepts [6].
[6] See, for example, E. Tufte’s magisterial work on statistical visualization (The Visual Display of Quantitative Information, Graphics Press, 1983; Envisioning Information, Graphics Press, 1990; Visual Explanations: Images and Quantities, Evidence and Narrative, Graphics Press, 1997); M. Lynch, “Discipline and the Material Form of Images: An Analysis of Scientific Visibility,” Social Studies of Science, SAGE, London, Beverly Hills and New Delhi, Vol. 15 (1985) pp. 37–66; T. Hankins, “Blood, Dirt, and Nomograms: A Particular History of Graphs,” Isis, Vol. 90, No. 1 (1999) pp. 50–80.
Sounds like that ^ should be worth a read.
They give examples of drawing supplementing and giving rise to new thinking, organising new thoughts that can then be made rigorous by the Thinker, and also changing the person doing the drawing.
The following quote describes a collaboration between a molcular biologist and topologist:
These drawings are far less precise than the text and professional vocabulary in a typical mathematical research article, but they allowed Buck and her colleague to share and develop the essence of the ideas involved. In fact, the lack of precision here is an advantage: The ability to highlight mathematically interesting aspects while suppressing unimportant detail makes drawing a more useful tool than, for example, faithful computer-generated imagery […]
Because the professional vocabulary of both of these fields is highly technical, and because topologists and molecular biologists typically share no technical training, drawing serves as a vital bridge between the two disciplines.
They describe
  • the taxonomy of fano varieties that Anderson did at first
  • the sliceforms collaboratively generated
  • the creation of ceramic knots, and talk about how different representations can highlight different elements

The conclusion talks about these drawings as an end in themselves, and the mathematicians are positive about the opportunity to engage in outreach without the usual didactic relationship.

The artworks that we produce are true to the mathematical objects they represent, even as they carry none of the technical context where those objects originate. The Fano models and knot sculptures give body and weight to forms that had no previous physical existence, and it is precisely by giving these forms a body that is stripped of the original scientific meaning that we can bring contemporary mathematical research to a wider audience in a direct and unmediated way. Through drawing and modeling, the forms are liberated and can exist and function on different levels.
Interesting that they’ve ended up producing attractive mathematical artefacts… There’s something of Giaquinto’s understanding/formalising dialectic in here, though they don’t reference him at all. There is a good overview of the usual spirals golden section etc art-maths stuff previously published in this and other journals in the notes. A glossary is a useful tool in this sort of paper.

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