(2) Lo-res art with stacked Rubik’s Cubes
(3) And a Mobius music box
Thanks to Ravi Vakil, Jim McBride, and Fritz Obermeyer for the links.
I just came across the following interesting question of Nati Linial.
If a two-dimensional simplicial complex has vertices and faces, does it necessarily contain an embedded torus?
I want to advertise this question to a wider audience, so I’ll explain first why I think it is interesting.
First of all this question makes sense in the context of Turán theory, a branch of extremal combinatorics. The classical Turán theorem gives that if a graph on vertices has more than edges then it necessarily contains a complete subgraph on vertices. This is tight for every and .
One could ask instead how many edges one must have before there is forced to be a cycle subgraph, where it doesn’t matter what the length of the cycle is. This is actually an easier question, and it is easy to see out that if one has edges there must be a cycle.
It also seems more natural, in that it can be phrased topologically: how many edges must be added to vertices before we are forced to contain an embedded image of the circle?
What is the right two-dimensional analogue of this statement? In particular, is there a constant such that a two-dimensional simplicial complex with vertices and at least two-dimensional faces must contain an embedded sphere ? If so, then this is essentially best possible. By taking a cone over a complete graph on vertices, one constructs a two-complex on vertices with faces and no embedded spheres. Without having thought about it at all, I am not sure how to do better.
In any case, the corresponding question for torus seems more interesting, but for different reasons. In a paper with Eric Babson and Chris Hoffman we looked at the fundamental group of random two-complexes, as defined by Linial and Meshulam, and found the rough threshold for vanishing of the fundamental group. To show that the fundamental group was nontrivial when the number of faces was small required a lot of work — in particular, in order to apply Gromov’s local-to-global method for hyperbolicity, we needed to prove that the space was locally negatively curved, and this meant classifying the homotopy type of subcomplexes up to a large but constant size.
It turned out that the small subcomplexes were all homotopy equivalent to wedges of circles, spheres, and projective planes. In particular, we show that there are not any torus subcomplexes, at least not of bounded size. (Linial may have recently shown that there are not embedded tori, even of size tending to infinity with .) On the other hand, just on the other side of the threshold embedded tori abound in great quantity. It is interesting that something similar happens in the density random groups of Gromov — that the threshold for vanishing of the density random group corresponds to the presence of tori subcomplex in the naturally associated two-complex. It is not clear to me if this is a general phenomenon, coming geometrically from the fact that a torus admits a flat metric.
Some of the great successes of the probabilistic method in combinatorics have been in existence proofs when constructions are hard or impossible to come by. It would be nice to have interesting or extremal topological examples produced this way. Nati’s question suggests an interesting family of extremal problems in topological combinatorics, and it might make sense that in certain cases, random simplicial complexes have nearly maximally many faces for avoiding a particular embedded subspace.
Update: Nati pointed me to the paper Sós, V. T.; Erdos, P.; Brown, W. G., On the existence of triangulated spheres in $3$-graphs, and related problems. Period. Math. Hungar. 3 (1973), no. 3-4, 221–228.
Here it is shown that is the right answer for the sphere. Their lower bound is constructive, based on projective planes over finite fields. Nati said that being initially unaware of this paper, he found a probabilistic proof that works just as well as a lower bound for every fixed 2-manifold. So it seems that the main problem here is to find a matching upper bound for the torus.
I enjoyed this recent arXiv post by Zack Treisman, apparently notes for a summer course for high school students he taught. It leads a beginning mathematician by discovery through the basics of arithmetic of complex numbers, higher dimensional geometry, a word or two on knot theory, and then onto the main topic: the Hopf fibration. I wish someone had told me about these things when I was in high school.
Thanks to several beautiful illustrations by Lun-Yi Tsai and Zack’s charming writing style, it is fun to flip through for mathematicians who are already familiar with these ideas as well. I hope Zack and Lun-Yi make a coffee table book one day.
A simplicial complex is said to be flag if it is the clique complex of its underlying graph. In other words, one starts with the graph and add all simplices of all dimensions that are compatible with this -skeleton. A subcomplex of a flag complex is said to be induced if it is flag, and if whenever vertices and is an edge of , we also have that is an edge of .
Does there exist a flag simplicial complex with countably many vertices, such that the following extension property holds?
[Extension property] For every finite or countably infinite flag simplicial complex and vertex , and for every embedding of as an induced subcomplex , can be extended to an embedding of as an induced subcomplex.
It turns out that such a does exist, and it is unique up to isomorphism (both combinatorially and topologically). Other interesting properties of immediately follow.
- contains homoemorphic copies of every finite and countably infinite simplicial complex as induced subcomplexes.
- The link of every face is homeomorphic to itself.
- The automorphism group of acts transitively on -dimensional faces for every .
- Deleting any finite number of vertices or edges of and the accompanying faces does not change its homeomorphism type.
Here is an easy way to describe . Take countably many vertices, say labeled by the positive integers. Choose a probability such that , and for each pair of integers , connect to by an edge with probability . Do this independently for every edge.
This is sometimes called the Rado graph, and because it is unique up to isomorphism (and in particular because it does not depend on ) it is sometimes also called the random graph. It is also possible to construct the Rado graph purely combinatorially, without resorting to probability. The I have in mind is of course just the clique complex of the Rado graph.
We can filter the complex by setting to be the induced subcomplex on all vertices with labels , and this allows us to ask more refined questions. (Now the choice of affects the asymptotics, so we assume .) From the perspective of homotopy theory, is not a particularly interesting complex; it is contractible. (This is an exercise, one should check this if it is not obvious!) However, has interesting topology.
As , the probability that is contractible is going to . It was recently shown that has asymptotically almost surely (a.a.s.) at least nontrivial homology groups, concentrated around dimension . For comparison, the dimension of is .
I think one can probably show using the techniques from this paper that there is a.a.s. no nontrivial homology above dimension or below dimension . It is still not clear (at least to me) what happens between dimensions and . It seems that a naive Morse theory argument can give that the expected dimension of homology is small in this range, but to show that it is zero would take a more refined Morse function. Perhaps a good topic for another post would be “Morse theory in probability.”
Another question: given a non-contractible induced subcomplex (say an embedded -dimensional sphere) on a set of vertices, how many vertices should one expect to add to before becomes contractible in the larger induced subcomplex? For example, it seems that once you have added about vertices, it is reasonably likely that one of these vertices induces a cone over , but is it possible that the subcomplex becomes contractible with far fewer vertices added?
I recently posted an article to the arXiv, on a problem suggested by Persi Diaconis. “Hard discs in a box” is a classical setting in statistical physics. One very simple model of matter is as non-overlapping (or hard) discs of radius , bouncing around inside a unit square. This is a very well studied model in computational statistical physics, although not much seems to be known so far mathematically. In particular, the statistical physicists observe experimentally that there is a phase transition of some kind when .
The details of what exactly it means that there is a phase transition will have to be left for another post. But what is relevant for us here is how they sample a uniformly random configuration of hard discs, via the Metropolis algorithm. (Please see the recent survey article by Persi Diaconis in the Bulletin of the AMS.) Basically, one starts with the discs in a reasonable starting configuration, say evenly spaced along some lattice, and then one applies Metropolis, which means the following. Choose one of the discs uniformly randomly, and make a small perturbation of its position. If this results in overlapping discs, or going out of the square, reject the move, otherwise accept it.
Under certain hypotheses on your space, one can show that this Markov chain converges to the uniform distribution on the space of all possible configurations. But to know this, one must know that the Markov chain is irreducible, and in particular that the configuration space is path connected.
Diaconis, Lebeau, and Michel observed that there is some constant such that if , the discs can always be freely rearranged, and in particular that the Metropolis Markov chain is irreducible. The real point of their article was to put useful upper bounds on the mixing time of Metropolis in general, and then they applied it to the hard discs setting as a motivating example. It would be nice if the same holds even in the range of the phase transition; in other words we would like to replace in the above statement with .
However, their result is essentially best possible. In my recent preprint, I gave examples of stable configurations (meaning that no single disc can move) with . Here is a picture that should give some feeling for the main idea behind the construction.
There is still quite a lot to do here. As Persi points out in his survey article, these configuration spaces of hard discs are naturally endowed with a topological structure, as well as probability measure. In his survey article, he says, “very, very little is known about the topology of these spaces.”
At a workshop on, “Topological complexity of random sets,” this past week at the American Institute of Mathematics, Yuliy Baryshnikov, Peter Bubenik, and I were able to establish a connection to classical algebraic topology. In particular, if , then the configuration space is not only path connected, but it is homotopy equivalent to the classical configuration space of labeled points in the plane. It is fairly clear that this happens for some sufficiently small radius, but as far as we know, no one had put any quantitative bounds on it yet.
In another article, in preparation, I discuss the combinatorial aspects of these configuration spaces, and give an example of what I call a “transitionary” configuration, as a first attempt to give a heuristic explanation of the phase transition near . The disc packing example illustrated above might be more interesting to discrete geometers than to probabilists, since one would like to believe that examples like this are rather rare, in a measure theoretic sense. Nevertheless, we believe that deeply understanding the various geometric and topological features of these spaces, and how they change as varies, might be a first step in explaining the phase transition mathematically.
Thanks to Jim McBride for pointing me to this archive, and in particular this delightful paper by Sir Alfred Kempe. Next time I am telling a combinatorics class about Kempe’s 1879 proof of the 4-color theorem, I will have to mention this article. I wasn’t aware that he was also well known for his work on linkages.
Speaking of big cubes, some people I know in Seattle are building the Groovik’s Cube, a Rubik’s Cube, illuminated by 648 superbright computer controlled LEDs, to be assembled and installed at Burning Man this year. I think what they’re doing is technically impressive and really fun. (Next year, Barry, we’ll do a . No, wait, a five-dimensional cube.)
This looks neat:
7 x7 x 7 Rubik’s Cube. Will report back with a review after it comes in the mail…
I haven’t bothered to check their claim of a ludicrously large number of possible positions () yet. But I remember the packaging on the original Rubik’s Cubes advertising something like, “millions and millions of positions possible,” which is true, and might be (quantitatively speaking) one of the biggest understatements in human history. I think the right answer for the is roughly ? I remember some people telling me that it was actually physically impossible to build bigger than , but I guess surprisingly it’s not.
What I’d really like to see is a nice physical implementation of the four-dimensional cube.
Update: Got the V-Cube 5, 6, and 7 in the mail, and they are great. Very cleverly made, they turn well and make a satisfying “click” when aligned. A piece might pop out if you turn it funny, but it clicks back in pretty easily.
Besides being a fun toy, and perhaps the most popular puzzle in human history, the Rubik’s Cube is an interesting mathematical example. It provides a nice example of a nonabelian group, and in another article I may discuss some features of this group structure. This expository article is about an experiment, where I made Rubik’s cubes with two or three colors, instead of six. In particular, I want to mention an interesting observation made by Dave Rosoff about one of the specially colored cubes: It turns out to be foolproof, in the sense that no matter how one breaks it apart and reassembles the pieces, it is still solvable by twisting the sides. It is well known that this is not a property that stock Rubik’s Cubes have.
The first observation about the physical construction of the cube is that it is made out of 21 smaller plastic pieces: a middle piece with 6 independently spinning center tiles, 12 corner cubies, and 8 edge cubies. The frame includes 6 of the stickers. Each corner cubie has 3, and each edge cubie has 2. (And this accounts for all of the stickers. These stickers are on a solid piece of plastic, so they always stay next to each other, no matter matter how much you scramble it by twisting the sides. (I remember getting really mad as a kid, after figuring out that someone else had messed around with the stickers on my cube. Still a pet peeve, taking stickers off a cube is like fingernails on a chalkboard for me.)
So anyway, it’s impossible to get two yellow stickers on the same edge cubie, for example, because that would make the cube impossible to solve: you couldn’t ever get the two stickers onto the same side. But this is not the only thing that can’t happen. Let’s restrict ourselves from now on to just the positions you can get to by taking the cube apart into plastic pieces, and putting them back together. If you take it apart, and put it back together randomly, will you necessarily be able to solve it by only twisting sides? (Of course you can always solve it by taking it back apart and putting it back together solved!) I knew, empirically, as a kid that you might not be able to solve it if you put it together haphazardly. Someone told me in high school that if you put it together “randomly,” your chances that it was solvable were exactly 1 in 12, and explained roughly why: it is impossible to flip an edge (gives a factor of 2), rotate a corner (a factor of 3), or to switch any two cubies (another factor of 2).
This seemed plausible at the time, but it wasn’t until a graduate course in algebra that I could finally make mathematical sense of this. In fact, one of the problems on the take-home final was to prove that it’s impossible to flip any edge (leaving the rest of the cube untouched!), through any series of twists. I thought about it for a day or two, and was extremely satisfied to finally figure it out, how to prove something that I had known in my heart to be true since I was a kid. It is a fun puzzle, and one can write out a proof that doesn’t really use group theory in any essential way (although to be fair, group theory does provide a convenient language, and concise ways of thinking about things). For example, it is possible to describe a number 0 or 1 to every position, such that the number doesn’t change when you twist any side (i.e. it is invariant). Then provided that the property prescribes a 0 to a solved cube and a 1 to a cube with a flipped edge, the invariance gives that the flipped edge cube is unsolvable.
Several years ago, I got inspired to try different colorings of a Rubik’s Cube, just allowing some of the sides to have the same color. I was picky about how I wanted to do it, however. Each color class should “look the same,” up to a relabeling. A more precise way to say this is that every permutation of the colors is indistinguishable from some isometry. (Isometries of the cube are its symmetries: reflections, and rotations, and compositions of these. There are 48 in total.)
It turns out that this only gives a few possibilities. The first is the usual coloring by 6-colors. Although there are several ways to put 6 colors on the faces of a cube, for our purposes here there is really only one 6-color cube. There is also the “Zen cube,” with only one color. (“Always changing, but always the same.”) But there are a few intermediate possibilities that are interesting. First, with two colors, once can two-color the faces of a cube in essentially two different ways. Note that since we want each color class to look the same, each color gets three sides. The three sides of a color class either all three meet at a corner, or they don’t. And these are the only two possibilities, after taking into account all of the symmetries.
So I bought some blank cubes and stickers, and made all four of the mathematically interesting possibilities. (I keep meaning to make a nice Zen cube, perhaps more interesting philosophically than mathematically, but I still haven’t gotten around to it. ) My friend Dave Rosoff and I played around with all of these, and found them somewhat entertaining. A first surprise was that they seem harder to solve than a regular Rubik’s cube. Seems like it should be easier, in terms of various metrics: the number of indistinguishable positions being much smaller, or equiavalently, the number of mechanical positions which are indistinguishable from the “true” solved position being much bigger. However in practice, what happens for many experience cube solvers, is that they get into positions that they don’t recognize at the end: the same-colored stickers seem to mask your true position. Nevertheless, an experienced solver can handle all four of these cubes without too much difficulty.
When playing around with them, occasionally a cubie would pop out fall to the floor. The thing to do is to just pop the piece back in arbitrarily, and the solve it as far as you can. It is usually an edge that pops, so the probability of having a solvable cube is 1/2. And if not, one can tell at the end, and then remedies the situation by flipping any edge back. Dave noticed something special about one of these four cubes, I think just from enough experience with solving it: no matter which piece got popped out, when he popped it back in randomly, the cube still seemed to be solvable. He thought it seemed too frequent to be a coincidence, so after a while of chatting about it, we convinced ourselves that this cube indeed had a special property: if one takes it apart into its 21 pieces, and reassembles it arbitrarily, it is always solvable by twists, a property we described as, “foolproof.” We talked it about it for a while longer, and convinced ourselves that this is the only foolproof cube, up to symmetry and permuting colors. (Well, the Zen cube is foolproof too.)
So which of these four cubes is foolproof? This puzzle yields to a few basic insights, and does not actually require making models of each type of cube, although I would encourage anyone to do so who has extra blank cubes and stickers around, or who wants a neat Cube variant for their collection.
One of my research interests is what you might call non-monotone graph properties.
A seminal result of Erdős and Rényi is that if , then the random graph is a.a.s. disconnected, while if then is a.a.s. connected. This can be made more precise; for example, if with , then as . Connectivity is an example of a monotone graph property, meaning a set of graphs either closed under edge deletion or edge contraction. Other examples would be triangle-free, k-colorable, and less than five components. The fact that every monotone graph property has a sharp threshold is a celebrated theorem of Friedgut and Kalai.
More generally, a graph property is a way of assigning numbers to finite graphs that is invariant under graph isomorphism, and increases or decreases with edge deletion or addition. Examples would be the number of triangle subgraphs, chromatic number, and number of connected components.
Much of random graph theory is concerned with monotone properties of random graphs. But it is not hard to think of examples of non-monotone properties. For example, let the number of induced four-cycle subgraphs. Another example would be the number of complete -subgraphs that are not contained in any -subgraph. When is very large, , then a.a.s. every three vertices share some common neighbor, so every -subgraph is contained in a -subgraph and . Similarly, when is very small, , then a.a.s. there are no subgraphs, and . For intermediate values of , , the expected value of is .
So the expected value of the graph property is unimodal in edge density, and we still have threshold behavior. Can Friedgut and Kalai’s result be extended to a more general setting that includes these cases?
This is a simple and perhaps natural combinatorial example, but my original motivation was topological. I wrote a paper about topological properties of random clique complexes, available on the arXiv, which turn out to be non-monotone generalizations of the original Erdős-Rényi theorem. I will continue to write more about this and other related examples sometime soon, and as I have time, but in the meantime if anyone knows any nice examples of non-monotone graph properties, please leave a comment.
One more thought for now. It seems in many of the most natural examples we have of non-monotone properties , the expected value of over the probability space is basically a unimodal function in the underlying parameter . Can you give any natural examples of bimodal or multimodal graph properties? (Pathological examples?)