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I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw figures like : enter image description here (Black is the set, red are the finite sub-covers)

My question is:

Am I handicapping myself by continuously drawing figures (limited to $\mathbb{R}^2$)?

I am tuned to imagine things which align neatly. For instance, if someone says imagine a triangle, I imagine an equilateral one and this usually prevents me from understanding subtle points.

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It depends on the mathematician, of course. – Mariano Suárez-Alvarez Jul 11 '12 at 19:18
As Mariano says, it depends on the mathematician. This works for some people and doesn't work for others. – Qiaochu Yuan Jul 11 '12 at 19:20
@QiaochuYuan I changed my question. – Inquest Jul 11 '12 at 19:24
@MarianoSuárez-Alvarez I changed my question. – Inquest Jul 11 '12 at 19:24
Figuring out complicated objects is all about handicapping yourself in the right way. For some reason, "everything is a blob in $\mathbb{R}^2$, except some things, which are blurrier blobs in $\mathbb{R}^2$" is a fantastic "handicap" for point-set topology. Obviously it's not going to cut it if you're computing the intersection of two specific 5-manifolds embedded in $\mathbb{R}^{10}$, but it's really good for point-set topology and analysis, even for infinite-dimensional spaces like function spaces. – user29743 Jul 11 '12 at 19:25
up vote 11 down vote accepted

I can't do mathematics at all without drawing and I have practically never given a course or a talk without making coloured doodles on the blackboard.
There are a few exceptions, as when I had to teach the Sylow theorems, but then I had the feeling that I didn't understand what I was talking about, although the proofs were (I hope!) correct in the logical sense.
The same goes for hard analysis, say linear partial differential equations: I tried to read Hörmander but gave up, because I couldn't get a gut feeling for the inequalities there (even though geometry is definitely present in that book).

I am in a field (algebraic geometry) where it is easy to make drawings and I started to understand scheme theory only when I saw Mumford's drawings of $Spec (\mathbb Z)$, $Spec (\mathbb Z[T])$ and his cartoonesque rendition of the spectrum $Spec(\mathcal O_{X,O})$ of the local ring at the origin of the plane $X=Spec(\mathbb A^2_k)$ over the field $k$, where the closed points of curves have disappeared and only their generic point is left behind , exactly like the grin of the Cheshire cat in Alice in Wonderland, the masterpiece of that wonderful mathematician who also loved drawings.
[If you have never seen a live scheme in its natural habitat, look at pages 72-75 of Mumford 's Red Book where the pictures I evoke above are to be found. Or here pages 111-112]

In conclusion, if you feel drawings help you, by all means go ahead: I find your version of a covering of a compact space truly ingenious an illuminating.
And to finish on a lighter note, there is this story of an engineer asking a mathematician how he could visualize 4-space: "Very easy, I imagine $\mathbb R^n$ and I specialize to $n=4$"

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I'm an engineer :| I love your answer. – Inquest Jul 11 '12 at 21:06
Thanks for your sense of humour, @Inquest. And since we are sharing confidences, I am a drop-out of an engineering school... – Georges Elencwajg Jul 11 '12 at 21:30
@GeorgesElencwajg In case it amuses you, my visualization of the Sylow theorems: Almost all proofs of the Sylow theorems involve the lemma -- let $X$ be a finite set with $|X| \neq 0 \mod p$ and let $G$ be a $p$-group acting on $X$. Then $G$ has a fixed point. $X$ is a blob which sometimes looks like a sphere and sometimes like a solid ball. $G$ looks like $S^1 \times S^1$. Orbits of size $p^2$ look like $2$-torii, orbits of size $p$ like circles, and fixed points are points. For Euler characteristic reasons, you can't write the two sphere or three ball as a union of circles and torii. – David Speyer Jul 11 '12 at 23:22
Thanks @David, this is incredibly imaginative: I would never have thought about this by myself. You convincingly demonstrate that making drawings is an art that can be taught. – Georges Elencwajg Jul 12 '12 at 9:11

I think that you should only worry at what you are doing at the moment. If it helps you to vizualize something to understand it you should do it. Even when you get to higher dimensions you can still vizualize stuff in your mind. I guess you should do whatever helps you understand. Its about making it easier, not harder.

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In response to the edit you made to the question I would like to say that you are correct in saying that imagining the "holotype" of an object is counterproductive because it will most likely lead you to over generalizations. For example someone who is looking at the centers of a triangle using a equilateral triangle might come to the conclusion that they are always the same point when in fact if any of the centers of a triangle is the same point, the triangle is equilateral – Carry on Smiling Jul 11 '12 at 20:38
Precisely my problem with imagination (the way I do it). A friend asked me to imagine a triangle and figure out a way to determine the difference between the orthocentre and centroid. I imagined an equilateral triangle and poseted that they would always co-incide though a second later I realized what went wrong. Sometimes, it's not as obvious though. – Inquest Jul 11 '12 at 21:09

The important point is that visualizing isn't a problem. If visualizations help you work faster and get more comfortable with a subject, more power to you. The issue with your example of the equilateral triangle isn't that it's visual, it's that you've chosen the simplest example possible and seem to have such tendency for simplicity/beauty. If you can remember to visualize extraneous cases and "push the boundaries" that way, then go for it. Personally I find visualizing good for gaining intuition, and then I switch to algebra for actual calculations and finding extreme cases. Visualization can be a fantastic tool, but it should just be one tool of many. It's one that many use liberally though, so don't take your preference for it as a shortcoming.

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The way we understand things has a lot to do with how we comprehend them. Some people are more visually inclined than others, and there is no problem with that. One learns, with time, how to "project" objects onto the mind and imagine them as shapes.

Do note that understanding how you understand things can be a key factor in finding the parts of mathematics which you may enjoy more and possible succeed in more often. For example, I actually have a problem with "low dimensional" visualization and drawings often make things harder for me.

On the other hand I come from set theory where things are weird enough that the very little drawings have little to do with the objects (as they carry very little geometry). There are "usual" drawings in set theory but the intuition one draws from them is different, in my experience, than the intuition one draws from drawing open, closed or compact sets (the best example I saw was in the Ph.D. dissertation of Ioanna Dimitriou where she sketched how generic and symmetric extensions look like, and how a permutation of a forcing poset look like).

I think that the most important tip I can suggest is that unless you deal with specific finite objects (e.g. graphs with a tame number of vertices), drawing something is only going to take you so far with intuition. At a certain point one must return to the definitions and work with them. The key point is to be creative and figure out how to formalize what you drew on a piece of paper into pure mathematical context.

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