Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects visually, and how this interpretation imbues a meaning to the mathematical structures. At this point I am simply trying to gather the opinions and musings of fellow mathematicians and logicians on this topic to see which ideas are the most interesting and worth writing about. I am particularly interested in the extreme cases of things which are very difficult to interpret visually, as well as the things which are best interpreted visually(fractals and impossible figures). Let me know what you guys think about this topic, Im sure this community has a wealth of interesting ideas.
Thanks! looking forward to hearing your thoughts
EDIT:
I realize my question is unclear, but Im not looking for an answer per se but more of some general thoughts on the subject. For example, when you say that there are far more mathematical objects which cannot be easily visualized, what aspects or properties make visualization difficult or impossible? Im interested in discussing the nature of visualization and its role in mathematics. Is visualization simply a tool to help us intuitively understand the more abstract concepts and relations? Or is visualization a process deeply connected to mathematics itself? If so is this due to our human intuition imbuing its own structural artifacts into algebra and logic?
I realize I am not asking a specific question, so answering in a concrete way is difficult, but as I said I am more interested in discussion and related examples of interesting phenomena.
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2$\begingroup$ No. It's very hard, if not impossible, to visualize non-measurable sets of $\Bbb R^n$. If one prefers a model where all sets are measurable, then there is a partition of the real line into strictly more parts than points which is impossible to visualize. Generally, most infinite objects, I feel, are impossible to visualize. $\endgroup$– Asaf Karagila ♦Mar 21, 2014 at 21:28
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$\begingroup$ I think very few mathematical objects can be represented visually. We imagine them, but those are just approximations, ideas, not accurate representations. And what does "visually" mean anyway? $\endgroup$– tomaszMar 21, 2014 at 21:28
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1$\begingroup$ I agree with tomasz it really depends on what you mean by visually. One example of something that you can see visually, but isn't what the object really looks like is the Cantor middle-third set. It is just a collection of unconnected points, but is always displayed as intervals in construction. For visualization should it be seeing its actual form or a useful form to think of it as? $\endgroup$– ruler501Mar 21, 2014 at 21:36
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1$\begingroup$ What I mean by visually is displaying objects and their relations with points, lines, planes, solids or any other visual object such that the abstract algebraic relations will hold on the visual representation. $\endgroup$– user1730099Mar 21, 2014 at 21:38
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$\begingroup$ In terms of infinite sets like the cantor set or menger sponge I dont mean visually interms of seeing the entire system or object, as the intervals of construction are a "graphical" or visual representation of a conceptual entity. I mean visual in a very general sense as in displaying an infinite line in a projective plane such that we can contain infinity within a bounded space, or even as simple as graphing an equation. Im mainly interested in the process of converting abstract logical and algebraic relations and entities into pictures or models which display those characteristics visually $\endgroup$– user1730099Mar 21, 2014 at 21:43
3 Answers
While it's easy to think about geometric shapes, circles and triangles, and other simple-enough lines through the plane, or even through a 3D-space, these sort of objects make but a tiny fraction of the mathematical universe.
In fact, while we think of objects which are not nice as those mentioned above, as "pathological", it is in fact the nice objects which are the scarce and the pathological objects which are common.
Even staying within the boundaries of the plane we can easily come up with intangible objects, or objects which are otherwise impossible to visualize. Here is maybe the simplest example of two very well understood objects, the rational numbers and the irrational numbers, on the real line.
Trying to visualize them, each is "like a line" but missing a lot of points, and in both cases, the points missing are tucked between the points that we have, and so it's impossible to see them. So if you think about them separately, you are probably going to have the same image in your head. Alas, these two are very different. One is countable, and the other is not. Meaning, there is no way to put them in a one-to-one correspondence (a list where each rational number appears once on the left column, and each irrational number appears once on the right column, and no rows have missing entries).
Going to a slightly more complicated object, but still somewhat tangible, consider a the Conway Base 13 function. This function has a graph which covers "almost" the entire plane (almost in the sense that the missing points are tucked deeply between those which are there).
So how can we imagine this correctly?
Going even further. Can we even visualize spaces whose dimension is larger than $3$? Maybeeeee $4$. But can we really see clearly into a $42$-dimensional space? What about an infinitely dimensional space? What about the distinction between two infinitely dimensional spaces whose dimension is different?
Going even further, we reach into the intangible objects. These are objects we can prove exist, mathematically, but we can't fully describe to begin with. Sets which we cannot assign "volume" to in a meaningful way, for example. Those have many different flavors, but none that we can fully visualize.
And all that is within the confines of "things related to the real numbers". I haven't even began to talk about the wild wild universe that lies beyond that.
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$\begingroup$ When I say visualize I dont mean to ask if these entities can be cleanly represented as lines and points so that we can see the whole picture. For example I would consider the menger sponge as something impossible to "see" on its whole, as it appears to have volume, yet does not. However, we can look at several iterations of its construction which is enough for us to intuitively "see" how it would look. Also flavors and and colors can be represented as axes, and higher dimensional figures can be visually approximated by looking at shadows in lower dimensions. $\endgroup$ Mar 21, 2014 at 21:47
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$\begingroup$ You wrote what you "don't ask", how about clarifying what you do ask instead? As for colors and shadows, sure they can help for four, and maybe five dimensions. But what help is that if your space has more dimensions than atoms in the observable universe? $\endgroup$– Asaf Karagila ♦Mar 21, 2014 at 21:58
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$\begingroup$ (Your title, by the way, all mathematical relations and entities, and that includes so much more than you can imagine. And then some more. So you should really clarify your question.) $\endgroup$– Asaf Karagila ♦Mar 21, 2014 at 22:02
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$\begingroup$ Yes you are right, I edited my question and changed the title. I like the idea of something having more dimensions than atoms in the universe, this is the kind of thing I would like to examine as something which would require very creative methods of visualization. $\endgroup$ Mar 21, 2014 at 22:43
I think that "vision" in mathematics is more deep that in common sense. When a matematician "visualises" something it is not just a two-dimensional picture. It is a picture in a much more rich world.
For instance, there are famous mathematicians which don't see (I don't know the word in english). And some of them do... GEOMETRY! (and at least one 4-dimensional geometry)
To visualise is like "tranlsate in a world that you are confortable with".
Visualisation's power is proportional to how many examples and counterexamples you have "seen" in your life.
In this sense, visualising is absolutely fundamental for mathematicians.
Here my favourite "difficult thing to visualise":
Russel paradox: The set of sets that does not contain itlsef, does contain itself?
In general, can you immagine the universe? (the mathematical universe, the class of all the sets) is it bounded in your "vision"?
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$\begingroup$ I've heard of mathematicians who lost their eyesight (Even at a very early age) and constructed some beautiful geometry, supposedly being able to visualize things outside of the real world with less distraction, but do you have any examples of mathematicians who were born blind (and thus never had the chance to experience vision) and contributed in geometry? $\endgroup$ Mar 21, 2014 at 22:59
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$\begingroup$ This is exactly what Im talking about, I would love to hear more of your thoughts on the subject as your statement "To visualise is like 'tranlsate in a world that you are confortable with'." is precisely what I am attempting to examine in my paper. For those things which are difficult to visualize im interested in what aspects of those things make visualization difficult? Thanks for your answer $\endgroup$ Mar 21, 2014 at 23:00
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$\begingroup$ Well, I don't know if the examples I have in mind born blind (now I know the word!) or became blind. One of the most famous geometer in 4-dim, Emmanuel Giroux is blind. As a curiosity, the italian 3d-geometer Bruno Martelli is not blind, bur he discovered when he was 20 years hold that he had no 3d-vision. Then he gained his 3d-vision 20 years later at his 40... $\endgroup$ Mar 21, 2014 at 23:10
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$\begingroup$ Again for 3d. The famous "3d magic eye" pictures are interesting in my opinion: because at the beginning are very difficult to "see", but once you have see the image, you cannot stop to "see". $\endgroup$ Mar 21, 2014 at 23:15
In basic levels, it is quite possible.For example, you may prove
$$(a+b)^2=a^2+2ab+b^2$$ or find the solution of $$x^2+bx+c=0$$ by geometricly.
In general,it seems quite diffucult.The reason is completly technical since you can not imagine $4$ dimensional vector space but you can make computation on it.
For example, you can say $(1,1,0,-1)$ and $(1,0,0,1)$ are perpendicular in $R^4$.But actually,we can not visualize it, we just see that inner product of these vectors are $0$ so they are perpendicular.
And when we increase the abstruction,give a genereal meaning to vectors space, we see that all continious function defiened from $[0,1]$ to $[0,1]$ constitute a vector space. It has infinite dimension and $sin(2\pi x)$ and $cos(2\pi x)$ are two vectors perpendicular to each other.You see that being perpendicular gains more meaning than in geometry.
Even if we can not complitly visualize every object in math, geomety help us in many different areas of math as a strong tools.The best examples is usage of graph theory in algebraic topology as a strong tool. You may want to search about algebraic topolgy.
