Techniques for visualising $n$ dimension spaces

Question

Can you guys point me to the types of things I should read about (I'm a maths lay person really) if I want to learn about the current thinking how $n$ dimensional spaces can be visualised and thus navigated?

Sorry this question seems a bit vague, but I'm trying think of how I navigate through data in many dimensions and I feel that mathematicians would have solved these problems many many years ago.

Edit: Thank you for the informed debate: Let me clarify what I'm after. I'm looking at model for navigating data using a computer screen (so 2 dimensions). Add a third dimension (think zoom in zoom out). Ok so what about $n$ dimensions, what techniques could help me navigate them while confining my self o the 3d world of the computer model. Reading the link below to mathsoverflow was interesting and I now have some interesting ideas coming out.

Answer 1

When somebody says "high-dimensional space is hard to visualize", they are thinking of visualizing with the eyes. But mathematicians visualize with the brain!

I highly recommend the AMS article The World of Blind Mathematicians. Who could be better at visualizing things they can't see?

Answer 2

I'm looking at model for navigating data using a computer screen (so 2 dimensions). Add a third dimension (think zoom in zoom out). Ok so what about n dimensions, what techniques could help me navigate them while confining my self o the 3d world of the computer model.

There is a very large body of published work (both theoretical and applied) on multidimensional data visualization (also called multivariate or $n$-dimensional). Try a web search with these terms! Some example titles:

• Visualization of Multi-variate Scientific Data (Bürger & Hauser)
• A Taxonomy of Glyph Placement Strategies for Multidimensional Data Visualization (Ward)
• Handbook of Data Visualization (Chen, Härdle & Unwin)

and so on. Ward's paper (p. 10) offers some advice when viewing N-dimensional data with a 2-dimensional display, re selecting from the N(N − 1)/2 possible orthogonal views in the N-dimensional space (not including rotational variations).

Answer 3

Check out http://en.wikipedia.org/wiki/Parallel_coordinates if your data is discrete, it maybe very useful. If it is not, there are extensions but they are not always insightful.

Answer 4

You can use:

• Scatter plots for visualizing 1D, 2D, 3D clusters and 2D, 3D series.
• Bubble charts and Heat maps for visualizing 3D series
• Parallel coordinates for visualizing, coloring and filtering n-dimensional data.

PS: This question might be better suited to ux stackexchange

Answer 5

Your best thinking in $\mathbb{R}^2$ and 3 (never $\mathbb{R}$) for counterexamples, but that is it really. I mean everything's 'the same', especially if you consider (I know this is lin. algebra but the point is emphasized more strongly with open balls) open balls in $\mathbb{R}^n$. Literally everything you need to 'know' from an open ball will be 'seen' from a picture of a disc or sphere [i.e $\mathbb{R}^2$ = union of open balls centre zero and radius n; similar results for $\mathbb{R}^n$]. You can also 'move' open balls as $B(r,a) = a + rB(1,0)$. I don't want to go into too much detail and these examples are terrible but basically $\mathbb{R}^2$ and $\mathbb{R}^3$ will usually be fine for counterexamples, there is an obvious decomposition (though not 'best') of $\mathbb R^n$ into direct sums of the subspaces $\mathbb{R}^2$ and $\mathbb{R}^3$ (and $\mathbb{R}$), etc.

It is obviously best not to work in $\mathbb{R}^n$ often though, $(\mathbb{Z}_p)^n$; vectors with entries modulo $p$ for prime $p$ is an important example. There are two other obvious important vector spaces, but both of these are easy to visualize for counterexamples.

Answer 6

I would like to say that you can't visualize higher dimension in Mathematics. Even for n=4 it is impossible. Several barriers that stop you from doing it. If you take the 2-manifolds you have really three objects i.e. the sphere, the torus and finally projective space. The problem is even for closed surfaces you would think that it's easy to visualize, however the projective space can't be realized properly in $R^3$ i.e. can't be embedded into it(this mean it crosses itself like the Klein Bottle does). You would need to go into the fourth dimension.

This is a popular image of the Klein bottle. But, in reality this shouldn't intersect itself. However, it's impossible to visualize it because you would need to think in 4 dimensions. Mathematics has got around this problems by developing topology and algebraic methods.

The naive way to visualize the Klein Bottle is to find the points in which it intersects itself in $R^3$ and then give them another colour and say they are in the 4th dimension.

3-manifolds are even more impossible to visualize. 3-spheres would look crazy

There is no true way to visualize it. Everything is really just algebra mainly group theory mixed with topology.

If you really care to learn all this it's best to pick up any decent book on Topology and read it. Armstrong Basic Topology is the best place to start. Would like to add that the 4th dimension is the hardest dimension possible you can get. http://en.wikipedia.org/wiki/Exotic_R4

Answer 7

I'm not so experienced with problems in N dimensions but i can suggest one thing: do not confuse the geometry with an analitical process. Also do not confuse when you are talking about mathematical analysis visualizing it with the help of the geometry and viceversa.

The geometry is a really old branch of the math, sometimes someone describe it as a science apart, the thing is that this discipline was born when the human knowledge had to dial with nature very closely and directly, like was during the ancient egipt or in greece. The geometry was born from the observation of the reality, nothing more and nothing less, and the way to express geometry consist in the use of the mathematical language because it is universal and unambiguous.

The analitic process is born to approach and to try to solve other kinds of problems and has to dial with concepts that are not properly available in nature like the concepts of approximation, N dimensions, the laws that we think are the correct ones to describe the natures, the concept of infinite, the infinitely small and the infinitely big, and so on.

The analitic process is really focused on the human needs and only this, if you should describe the lenght of a piece of wood to make a bridge with your hands, you probably do not need to have a really precise measurement, you probably do not need it at all, but if you have to abstract that bridge into a project you probably have to deal with an approximation of that measure, so you need a value, simply because you have to put a quantity on a piece of paper, and thanks to this you can skip from geometry to an analitical process .

You simply can't imagine more than 3 dimension because N>3 does not belong to the world you are in, geometrically speaking.

Answer 8

I sometimes like to pervert the standard "perspective" 2-D rendering of the $x,y$ & $z$ axes in $\mathbb{R}^3$ by drawing $n$ axes through a common origin all with highly acute angles between them, which I imagine are all pairwise perpendicular (for each of the $\binom{n}{2}=\frac{n(n-1)}{2}$ pairs of axes) and sometimes even indicate this with the little right angle indicators (parallel to each alternate axis in a pair) -- as if I am looking from some yet other unspecified, distant dimension. But it requires of course some imagination, and complements rather than replaces other, non-visual ways of conceptualizing $\mathbb{R}^n$.

Answer 9

For 4 dimensions, a Terrasact might be useful:

I could be wrong (I'm not a Math major), but this seems to achieve some sense of visualization for 4 dimensions. It seems analogous to "parallel coordinates" which another answer mentions, projecting higher dimensional space into a 3d one (instead of a 2d one).

Good question by the way, I'm wondering this myself for D3.js visualization.

Answer 10

Arbitrary $n$ dimensions would be a tall order to represent generically because really it depends on what data you're wanting to visualise and what relationships you're trying to present/highlight. Assuming you're targeting a $2d$ space for visualisation (you're thinking of paper or a screen and not building lego or some other physical model), the simplest and most nearly generic approach i can think of is to add a dimension to a method previously found for $n-1$ data by using a brightness or lightness intensity scale of a distinct colour hue as a scalar to represent the new dimension. Arguably, you could use whatever you have not used already from a bunch of techniques. The scalars could be colour, intensity/whiteness, a letter/character for small discreet representations, cross-sectional cell position in a series of cells such as might be used to show an explosion of cross sections eg the brain. You could even use pixel (datum point) size as a measure, or perhaps shape. Sure, the more digital and less analogue choices have constraints, but as previously mentioned out depends on what relationships you're trying to find or highlight. Contour lines can help visualise terrain nicely. And their representation of height isn't even absolute let alone analogue. They iterate through a relative incrementing scale with only the occasional annotated reference height.

Obviously this 'generic' approach has limitations too, but if we think about it, arguably most of us already see already in at least $5$ dimensions: we see via an obstructed $2d$ window onto a physical third. But within that world oft dismissed as $3d$, we see individual distinct colours, offering another dimension. We see brightness differentiation on those hues. Another dimension. We see it over time, changing, offering another dimension. So yes it seems we're prejudiced by our perceptions of physicality and matter in our assessment that we live in $3d$, yet informational other dimensions are lurking all around us. We can use these analogues as inspiration for our own multidimensional representations.

• Eg. A $3$-dimensional bump map can be represented in $2d$ by having a different intensity for the height/depth. $3d$ is often represented in drawings orthogonal. 'Point ambiguity' arrives almost immediately yet it's the popular approach. Consider the other attributes mentioned and see how far it gets you. If you're searching for a relationship, time and instance can be fantastic. Mapping combinations of dimensions in $2d$ cells can often reveal a trend between any two at a glance. Add brighteness and rgb, fewer cells... Moore potential relationships to see!