# Styles of visualization in geometry

Some people talk about visual thinkers and non-visual thinkers, but I am interested in a contrast within styles of visual thinking. There are people who readily visualize complicated flow charts and other diagrammatic graphics for information, but do not readily picture, say, a Möbius strip or an octahedron (let alone 4 dimensional objects).

I get this from the history of mathematics where some great examples (Richard Dedekind, Emmy Noether, Alexander Grothendieck) at first seem to be non-visual thinkers. They would seem to be algebra people rather than geometry people in terms of this question: Evidence for or against the claim that some students are "algebra people" and others are "geometry people" Yet each made great, explicit, deliberate contributions to geometry.

So I wonder if they are better described as visual in a different way than spatial visualization. Certainly Grothendieck made vast use of diagrams which are not really spatial diagrams (albeit they can be drawn on a blackboard). They are diagrams of conceptual relations.

Mathematics educators may know a lot more about this than I do, so I will not try to be more precise, unless people ask for that in comments.

• What a great question. I've often wondered about this. In fact, when I was younger I exhibited a striking difference between the neatness of my handwriting and my writing of mathematical symbols. The former was (and still is) nearly illegible and the latter was very neat. It felt, to me, that I was drawing the symbols rather than writing them. It seemed to me in some sense that I was processing these symbols differently than I processed written language. I've never found anything about this except for the remarks in Hadamard's Psychology of Invention in the Mathematical Field. – Jon Bannon Mar 17 '17 at 17:13
• Remark: Different learning styles (for example visual, auditory, doing by hands, etc.) are, to my non-expert knowledge, false in the sense that verification of them has not been successful. – Tommi Brander Mar 17 '17 at 18:31
• @TommiBrander Do you meant that some claims about learning styles are false? I do not think a style can be false. – Colin McLarty Mar 17 '17 at 19:21
• @ColinMcLarty False is probably a wrong word, but what I have read points to the direction that learning styles do not exist. – Tommi Brander Mar 18 '17 at 5:42
• The concept of reification might be related to this kind of "aesthetic chunking", or "impression chunking". It seems that commutative diagrams are somehow of the latter sort...as some sort of motion like diagram chasing is implicit in the diagram. Another example might be the interpretation of a free group as the fundamental group of a plane with two holes. It is possible to "draw a picture" of the latter, but the expression of the concept needs some temporal unfolding to be fully expressed. The music has to be played... – Jon Bannon Mar 18 '17 at 16:49

It seems the Stanford Encyclopedia of Philosophy has a useful article that at least tangentially addresses your interesting question:

"Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. [...] Reflection on a diagram such as Figure 9 does the work."

Giaquinto, Marcus. "The Epistemology of Visual Thinking in Mathematics." The Stanford Encyclopedia of Philosophy, Winter (2015). HTML link.

His last sentence:

Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it,1 we are not forced to traverse the steps blindly, link by link, feeling our way by touch.

Incidentally, the author of the encyclopedia entry, Marcus Giaquinto, wrote a book entitled, Visual Thinking in Mathematics (Oxford University Press, 2007).

1Weyl, H., 1995 [1932], “Topology and abstract algebra as two roads of mathematical comprehension”, American Mathematical Monthly, 435–460 and 646–651.

You might find the essay by Marjorie Senechal, "Visualization and visual thinking." Geometry's Future (1991): 15-21, published by the Consortium for Mathematics (COMAP), of interest. In particular she addresses the distinction between "visual thinking" and "visualization." (You can also find articles that cite this work by searching on "Google Scholar"; currently $24$ articles cite it.)