Intuitive crutches for higher dimensional thinking

  • I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-

    An engineer, a physicist, and a mathematician are discussing how to visualise four dimensions:

    Engineer: I never really get it

    Physicist: Oh it's really easy, just imagine three dimensional space over a time- that adds your fourth dimension.

    Mathematician: No, it's way easier than that; just imagine $\mathbb{R}^n$ then set n equal to 4.

    Now, if you've ever come across anything manifestly four dimensional (as opposed to 3+1 dimensional) like the linking of 2 spheres, it becomes fairly clear that what the physicist is saying doesn't cut the mustard- or, at least, needs some more elaboration as it stands.

    The mathematician's answer is abstruse by the design of the joke but, modulo a few charts and bounding 3-folds, it certainly seems to be the dominant perspective- at least in published papers. The situation brings to mind the old Von Neumann quote about " never understand things. You just get used to them", and perhaps that really is the best you can do in this situation.

    But one of the principal reasons for my interest in geometry is the additional intuition one gets from being in a space a little like one's own and it would be a shame to lose that so sharply, in the way that the engineer does, in going beyond 3 dimensions.

    What I am looking for, from this uncountably wise and better experienced than I community of mathematicians, is a crutch- anything that makes it easier to see, for example, the linking of spheres- be that simple tricks, useful articles or esoteric (but, hopefully, ultimately useful) motivational diagrams: anything to help me be better than the engineer.

    Community wiki rules apply- one idea per post etc.

    I've always thought the set of mathematicians are finite, and hence countable... :)

    ... from which I can only draw the conclusion that at least one mathematician has an uncountable amount of wisdom and experience. Kinda makes me jealous, really...

    Your joke reminds me of Burt Totaro's algebraic topology class, it must have been in 2001. He drew the standard picture of a 2-sphere on the board and wrote '$S^2$' next to it, but quickly started to discuss n-spheres. Someone put up their hand and asked 'What's an n-sphere?' Burt responded 'Oh it's easy, you just do this', erased the '2' on the board, replaced it by an 'n', and carried on talking.

    Not a great joke?!? That's basically the punchline for the funniest math joke I ever heard (usually they are such groaners). A mathematician and engineer go to a physics talk where the speaker discusses 23-dimensional models for spacetime. Afterwards the mathematician says "that talk was great!" and the engineer is shaking his head and is very confused: "The guy was talking about 23-dimensional spaces. How do you picture that?" "Oh," says the mathematician, "it's very easy. Just picture it in n dimensions and set n = 23." Laughed my a** off when I first heard that.

    The joke is on the mathematician, because as everyone knows, the space(-time) is 26-dimensional. Of course, if you are a hammer...

    There's a great joke that I'll repeat (abridged) from the joke topic: "A mathematician who studied a very abstract topic was annoyed that his peers in more applied fields always made fun of him. One day he saw a sign "talk today on the theory of gears", and said to himself "I'll go to that! what could be more practical than a talk on gears?" He arrives at the talk, eager to learn applicable knowledge. The speaker goes up to the podium and addresses the crowd: "Welcome to this talk on the theory of gear. Today I will be speaking about gears with an irrational number of teeth."

    A tangential requirement viz "Visualising functions with a number of independent variables" is asked here on MO.

    @KConrad: your joke reminds me of a musical version I witnessed. Most musicians have at some point learned to play triplets against duplets — one of your hands plays 2 evenly spaced notes per beat, the other hand plays 3. 3-against-4 is also not uncommon, and in ≥C20th music, higher divisions also occur. At this particular dinner, one pianist marvelled that another could play a perfectly even 13-against-14 rhythm; a composer present was surprised at the surprise. “But it’s easy, isn’t it? You just set up one hand playing 13, and the other playing 14, and then you put them together!”

    "To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." -- Geoff Hinton, man who goes directly to third Bayes.

    @PeterLeFanuLumsdaine Your comment reminded me of this:

    I like the idea of "more neighbours" in higher-D euclidean space. Geoff Hinton joked about being in a grocery store buying pizza. Tomato sauce and cheese were near the pizza-dough, but sardines were not. "Unfortunately it's not at 16-dimensional grocery store", because then everything related to pizza-dough could be next to pizza-dough. So I think of ℤ³ as a graph with each vertex having 6 edges. In ℤⁿ each node has 2n edges.

    @DoubleJay, I prefer the variant where the lecturer begins "Since the theory of gears with a real number of teeth is well known …".

    @DoubleJay I've heard it as "Welcome to this talk on the theory of gear. As the theory of gears with real number of teeth is well-known..."

  • I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of high-dimensional (or infinite-dimensional) vector spaces such as $\mathbb R^n$, there are plenty of ways to conceptualise these spaces that do not require visualising more than three dimensions directly.

    For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets of this space correspond to various classes of images.

    One can similarly interpret sound waves, a box of gases, an ecosystem, a voting population, a stream of digital data, trials of random variables, the results of a statistical survey, a probabilistic strategy in a two-player game, and many other concrete objects as states in a high-dimensional vector space, and various basic concepts such as convexity, distance, linearity, change of variables, orthogonality, or inner product can have very natural meanings in some of these models (though not in all).

    It can take a bit of both theory and practice to merge one's intuition for these things with one's spatial intuition for vectors and vector spaces, but it can be done eventually (much as after one has enough exposure to measure theory, one can start merging one's intuition regarding cardinality, mass, length, volume, probability, cost, charge, and any number of other "real-life" measures).

    For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable.

    More generally, many facts about low-dimensional projections or slices of high-dimensional objects can be viewed from a probabilistic, statistical, or signal processing perspective.

    This is an amazing answer. Thank you.

    What is meant by "real-life" measures?

    +1. Could you please elaborate in details on the second last paragraph regarding interpreting using the law of large numbers? Thank you!

    Sorry, perhaps the more accurate statement is that the law of large numbers implies that most of the mass of a gaussian measure (which can be thought of as a smoothed out version of a ball) lurks near a single sphere. (And similarly for most of the mass of a large cube.) It is still true that most of the mass of a large ball lurks near the boundary sphere, but to the probabilistic explanation of this is closer in spirit to large deviations than to the law of large numbers.

  • Here are some of the crutches I've relied on. (Admittedly, my crutches are probably much more useful for theoretical computer science, combinatorics, and probability than they are for geometry, topology, or physics. On a related note, I personally have a much easier time thinking about $R^n$ than about, say, $R^4$ or $R^5$!)

    1. If you're trying to visualize some 4D phenomenon P, first think of a related 3D phenomenon P', and then imagine yourself as a 2D being who's trying to visualize P'. The advantage is that, unlike with the 4D vs. 3D case, you yourself can easily switch between the 3D and 2D perspectives, and can therefore get a sense of exactly what information is being lost when you drop a dimension. (You could call this the "Flatland trick," after the most famous literary work to rely on it.)

    2. As someone else mentioned, discretize! Instead of thinking about $R^n$, think about the Boolean hypercube $\lbrace 0,1 \rbrace ^n$, which is finite and usually easier to get intuition about. (When working on problems, I often find myself drawing $\lbrace 0,1 \rbrace ^4$ on a sheet of paper by drawing two copies of $\lbrace 0,1 \rbrace ^3$ and then connecting the corresponding vertices.)

    3. Instead of thinking about a subset $S \subseteq R^n$, think about its characteristic function $f : R^n \rightarrow \lbrace 0,1 \rbrace$. I don't know why that trivial perspective switch makes such a big difference, but it does ... maybe because it shifts your attention to the process of computing $f$, and makes you forget about the hopeless task of visualizing S!

    4. One of the central facts about $R^n$ is that, while it has "room" for only $n$ orthogonal vectors, it has room for $\exp(n)$ almost-orthogonal vectors. Internalize that one fact, and so many other properties of $R^n$ (for example, that the $n$-sphere resembles a "ball with spikes sticking out," as someone mentioned before) will suddenly seem non-mysterious. In turn, one way to internalize the fact that $R^n$ has so many almost-orthogonal vectors is to internalize Shannon's theorem that there exist good error-correcting codes.

    5. To get a feel for some high-dimensional object, ask questions about the behavior of a process that takes place on that object. For example: if I drop a ball here, which local minimum will it settle into? How long does this random walk on $\lbrace 0,1 \rbrace ^n$ take to mix?

    How does (4) follow from Shannon's theorem, which has to do with discrete codes rather than $R^n$? I think (4) is much more naturally thought of as a consequence of the Johnson-Lindenstrauss lemma, which is itself intimately connected to Dvoretzky's theorem. The latter, btw, is probably a natural candidate for "the one weird fact you need to know about high-dimensional Banach spaces".

    What is an almost-orthogonal vectors?

    Neat set of practically useful techniques, thanks.

    @FawzyHegab See this question, or just search "almost orthogonal vector exponential"

  • This is a slightly different point, but Vitali Milman, who works in high-dimensional convexity, likes to draw high-dimensional convex bodies in a non-convex way. This is to convey the point that if you take the convex hull of a few points on the unit sphere of R^n, then for large n very little of the measure of the convex body is anywhere near the corners, so in a certain sense the body is a bit like a small sphere with long thin "spikes".

    enter image description here

  • In general one uses a combination of projection, movie, and analogy from lower dimensions. I will try and exemplify each. Also, and importantly, think about linear algebra as an intrinsically geometric description. I will start from the last point first.

    The solutions to a single linear equation in (n+1) unknowns can be thought of as defining a hyperspace of n-dimensions in (n+1)-space. It also is the intersection between the graph of a function $y=\sum a_i x_i$ and the constant function $y=b$. Row reduction is trivial to implement, but gives a basis for the solution space of the associated homogeneous. Continue adding equations (generically) and continue to cut down the dimensions.

    Similarly, the binomial theorem expresses the volume of an n-dimensional cube in terms of a bunch of slices. I drew a picture here: and here:

    By a similar construction you can convince yourself that $\int_0^1 x^n dx = 1/(n+1)$ by considering the left hand side to be a pyramid in the (n+1)-cube. A total of (n+1) of these fill the (n+1)-cube. Abhijit Champanerkar and I posted a paper on the arXiv on that.

    In terms of projections, movies and analogies, consider a classical knot such as the braid closure of the braid word $s_1^3$. The movie of this consists of two pairs of points being born and a pair of these points dance about. The most immediate picture that I can think of is in a recent paper of Joan Licata in JKTR. The movie seems boring and hard to parse, but if you keep track of all details you can reconstruct the projection and the diagram (these two ideas are different: the diagram contains crossing information).

    Many knot theorist, when drawing a slice knot and disk, draw a circle with a pair of points connected by a bent chord in the interior. This is using the dimension analogy: you draw the analogue one dimension down rather than up.

    For knotted and linked surfaces, you can draw a movie which contains knots and links dancing, mating, and separating: bacterial voyeurism. When you draw this AND you keep careful track of critical information, you can reconstruct an accurate picture of the projection. We tell how to do this in the AMS book and the Springer book. See also my draft of the sphere eversion on page linked above but beware: the file is huge!!!

    If you want to study knots and links of 3-manifolds in 5-space you can make a movie of surfaces dancing. Singularity theory will help keep track of things.

    Finally remember when you draw a surface onto the plane you loose information. When you draw a solid in the plane, the information is contained in segments that are parallel to the kernel of linear approximations. These patterns persist. The standard projections of a hypercube contain a 2-d kernel.

    To summarize: Think about linear algebra geometrically. Take cross sections and use these to reconstruct projections, and consider carefully the information lost in the projections.

  • One way I always liked to think of $S^n$ is in terms of suspensions. While not particularly geometrically enlightening (though it can be topologically enlightening), is still an interesting way to think of them.

    Definition. The suspension $SX$ of a topological space $X$ is $(X\times[0,1])/\sim$ where $\sim$ collapses $X\times${0} to a point and $X\times${1} to a point. "Geometrically", this means we want to take $X$, and two "suspension" points, and then draw "lines" from the two points to all the points in $X$.

    So we can imagine the suspension of the circle easily. $S^1\times[0,1]$ is the cylinder, and $S^1 \times${0} is the circle on the "bottom" of the cylinder, and $S^1\times${1} is the circle at the "top". We identify these circles with points, which collapses the top and bottom of the cylinder to points. This clearly gives us $S^2$!

    Generally, it is not hard to show $SS^n = S^{n+1}$.

    So, now let's try to imagine $S^3$, which is three dimensional, so shouldn't be too hard to think about. We start with the 2-sphere, considered embedded in $\mathbb{R}^3$, and two "suspension" points. But we can already tell this is going to look weird if we choose two points, say, above the north and south poles. So, let's pick one point inside the sphere.

    The set of all lines from the point in the center of the sphere to the sphere is the solid sphere. Now, we want to deal with the second point. Say, we place this above the north pole, and connect each point on the sphere to it.

    enter image description here

    Since these lines are supposed to go to all points of the sphere, we should imagine this diagram shows the lines dense in the space around the sphere... but this is looking crowded, so let's move this extra point all the way off to infinity, and redraw this picture,

    enter image description here

    again imagining in this crudely drawn picture the lines cover the whole sphere.

    But now these lines cover, in addition to the surface of the sphere and the point in the center, all of $\mathbb{R}^3$! And, all we've got left is the point at infinity.

    So, we've just shown $S^3$ is $\mathbb{R}^3\cup$ {$\infty$}.

    Now, how can we imagine $S^4$? We do the same thing again! Draw all lines (this time, since we can't quite imagine the bigger space to embed this in, we can instead think of formal linear combinations) from two points to every point in $\mathbb{R}^3$, and to the extra point at infinity.

    I won't try to draw that one, but, thinking of it isn't too hard (although things get geometrically confusing if you try to do this process too many times!) But at the very least, you can convince yourself the spheres all have a relatively simple structure.

    Generically, one can construct other spaces by suspensions, cones (suspensions over one point), joins (drawing lines between two arbitrary topological spaces), wedges ($\vee$) (quotient of disjoint unions), and smash products $X\times Y/X\vee Y$, which, with are simple enough that in some cases, with enough effort, one can use them to visualize what lots of types of higher dimensional spaces look like! For more, grab your favorite algebraic topology book.

    If it helps make sense of this explanation, I'm a physicist ;).

    The images aren't available.

    @MartinBrandenburg: Worse, now they're showing ads instead. I'm going to edit them out; hopefully the original author can replace them with properly hosted versions.

    I used to retrieve the images: I hope they are right now.

  • This post is not a direct answer to your question, but rather a movie recommendation.

    "Dimensions" try to help you visualize the 4th dimension by projecting it onto the familiar two and three dimensions. It also contains a part depicting two-dimensional beings trying to conceive of a third dimension, so that the viewer can visualize the easier and analogous situation first.

    The graphics are pretty and it is totally free. Check it out!

    Dimensions, movie

  • Any kind of varying visual property of a surface (e.g. color, texture, opacity) can be used to describe one extra dimension. This really only helps in low dimensions, but it's quite effective!

    To give an example, Hatcher describes visualizing the embedding of the Klein bottle into four space by letting most of the bottle be blue, but having it "blush" as it passes through itself.

    What are the different types of embeddings of the Klein bottle into four space -- for instance, in what ways is a Klein bottle that is blue all over except for the blush different from a rainbow colored Klein bottle?

    You can color it any way you want and it'll basically be the same, so long as 1) the parts with spatial overlap have different colors and 2) the colors (i.e. position in 4th dimension) are continuous.

    DoubleJay, I don't think that can be right: there must be more than one embedding of a Klein bottle in $\mathbb{R}^4$. Maybe it's true that if there's only a single circle of self-intersections then it's a standard embedding?

  • In "Regular polytopes", H.S.M. Coxeter writes:

    Only one or two people have ever attained the ability to visualize hyper-solids as simply and naturally as we ordinary mortals visualize solids, but a certain facility in this direction may be acquired by contemplating the analogy between one and two dimensions, then two and three and so (by a kind of extrapolation) three and four. This intuitive approach is very fruitful in suggesting what results should be expected. However, there is some danger of being led astray unless [...]

    You should read the whole section 7.1 to find out. As a bonus you will find there the following quote by Poincaré, whose origin I am afraid I was unable to verify:

    Un homme qui y consacrerait son existence arriverait peut-être à se peindre la quatrième dimension.

  • One extremely useful trick for visualising a certain class of simple 4- and 6-dimensional spaces is the toric moment map picture.

    (a) The basic example is a 2-sphere $\{x^2+y^2+z^2=1\}$, which you equip with a linear height function $(x,y,z)\mapsto z$. Now instead of drawing the sphere you draw its image (an interval). Under this map, the sphere is a family of circles being collapsed to points.

    (b) The next basic example is $S^2\times S^2$, which maps to a square: away from the edges, the preimage of a point is a 2-torus; over the edges away from the corners the preimages are circles; over the corners the preimages are points. Over each edge, there is a sphere whose projection to that edge is the one we saw in (a). The diagonal sphere $\{(x,x)\ :\ x\in S^2\}$ (respectively antidiagonal sphere $\{(x,-x)\ :\ x\in S^2\subset\mathbf{R}^3\}$) map to the diagonal/antidiagonal in the square and intersect each torus fibre in the diagonal/antidiagonal circle.

    (c) $\mathbf{CP}^2$ with homogeneous coordinates $[x:y:z]$ projects to a triangle $\{a+b\leq 1,\ a,b\geq 0\}$ via $[x:y:z]\mapsto(|x|^2/T,|y|^2/T)$, $T=|x|^2+|y|^2+|z|^2$. Over each edge there is a sphere: you usually think of the sphere over the hypotenuse as being ``at infinity'' ($z=0$). These spheres are complex lines. If you cut out the spheres living over edges, everything retracts down to the fibre over the barycentre (which is again a torus).

    In general what you're drawing is the image of a symplectic $2n$-manifold $X$ with a Hamiltonian action of the $n$-dimensional torus $T$ (in these cases, $n=1,2$) under the map $X\to X/T$. This is always a convex polytope whose vertices and be $\mathbf{Z}$-linearly identified with the vertex of the positive orthant in $\mathbf{R}^n$ (the Delzant property). A six-manifold projects to a 3-d polytope: $\mathbf{CP}^3$ becomes a standard simplex, for example. Various natural operations like blow-up can be easily visualised (chopping off corners of polytopes); certain natural singularities can be understood (by allowing non-Delzant vertices), for example the small resolution and flop of a 3-fold node has a nice toric picture (see the picture near the end of this blog post). High degree algebraic curves can be visualised using their amoebas.

    Even more generally (as others in this thread have said), high-dimensional spaces can be visualised by their projections to other, simpler spaces. The most interesting and important part of this information is the singularities of the projection maps. This is the moral of Morse theory, Cerf theory, Picard-Lefschetz theory and, in this instance, of toric geometry, where the singularities of the moment maps occur along the faces and edges of the image polytope and give you a rich collection of important submanifolds for free.

    More philosophically, I would say the key in developing a geometric intuition is in learning to draw simplified, lower-dimensional and possibly misleading pictures, provided you understand exactly how misleading the pictures are. For example, in the above example the diagonal and antidiagonal are disjoint in $S^2 \times S^2$ but their images intersect in the square.

  • In 5 or more dimensions handlebody decomposition and the associated handle moves. In dimension 4 Kirby calculus. In dimension 3 Heegaard splittings. (Dimensions less than 3 are left as an exercise for the interested reader.)

    Handlebody decomposition and the associated handle moves are covered elegantly in Kosinski's Differential Manifolds while Kirby calculus is covered in Stipsicz and Gompf's 4-Manifolds and Kirby Calculus. Both books touch upon Heegaard splittings.

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