du Sautoy on symmetry

An interesting TED talk: Marcus du Sautoy on symmetry -- interesting to watch, lots of pictures; ignore the fact that it starts with the standard slightly overwrought version of Galois' story. If you want a more accurate version, I recommend Amir Alexander's Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics . About halfway through he gives a presentation of a proof that the two groups of order 6 are different. (One, the cyclic group of order six, is abelian; the other, the symmetric group on three elements, is not.) I particularly like the pictures of the Alhambra, on which du Sautoy has overlaid animations showing the effect of rotating them. Presumably they won't let you do this if you actually go there.

What if gravity reversed itself?

A friend of mine, not long ago, noticed that it was snowing up outside her office window. (When I've seen this happening, it's usually due to something like an internal courtyard in a building, so the air currents get sufficiently weird to push the snow upwards. Don't worry, physics isn't broken.)

This raises the (somewhat silly, I must admit) question: what if gravity had actually reversed itself outside my friends' office? Three scenarios are possible:

• gravity instantaneously reverses itself;


• the downward-pointing vector of gravitational acceleration decreases in magnitude, goes through zero, and then becomes an upward-pointing vector;


• the gravitational acceleration vector always has the same magnitude, but swings around through different angles to point up instead of down.

I claim that the third of these is not reasonable. Basically, this is for reasons of symmetry. Imagine gravity that neither points straight down nor straight up. How would gravity know in which of the many possible "sideways" directions to point? There are only two points that could possibly have any effect on the direction of gravitational acceleration, namely the point that you are at and the center of the Earth. The direction of gravity must be invariant under any isometry which fixes those two points -- thus it must point straight towards the center of the Earth or straight away from it.

Apparently I am more attached to gravity being a central force than to its particular strength or even to the fact that it is attractive, not repulsive.

(I'm not sure whether the first or second of the models suggested above is reasonable. This post is silly enough as it is.)

A graphical illustration of different types of attraction

Go look at http://thismight.be/offensive/uploads/2007/09/04/image/attraction.gif -- a chart of different types of attraction, from thismight.be/offensive. (The site doesn't allow external referrers; when I posted the link I thought it just didn't allow inline links to images.)

The chart has an x-axis "physical attraction" and a y-axis "mental attraction", and then the square is divided into regions corresponding to different types of relationships: Zone of Pain, Just Friends, One Night Stands, F Buddy, Awkwardness, Dating Potential, Dating Zone, Marriage Potential, and Null. ("Null" is in the extreme upper right, corresponding to a very high level of both physical and mental attraction; I take it that the creator of this image finds this impossible.

What I immediately noticed is that the chart is basically symmetric across the diagonal, which implies some sort of symmetry between mental attraction and physical attraction. I don't think I believe this, but what sort of hidden symmetry is there in human interaction in general?

Patent for a five-sided dice [sic]

A patent for a "five sided dice" [sic].

The patent was filed by Louis Zocchi, inventor of the Zocchihedron, which is a 100-sided die; the Wikipedia article indicates that the Zocchihedron isn't a fair die.

Will this one be?

Previously I wrote about how one might design asymmetric dice; Mark Dominus claims that "the probability that the hexahedron will land on face F is not proportional to the area of F, but rather to the solid angle subtended by F from the hexahedron's center of gravity." I'm not sure if I believe this. It seems reasonable, because it captures how the die is likely to rotate in the air, but dice bounce when they hit the table, and I'm not convinced that the "bouncing" behavior isn't chaotic.

Anyway, the patent indicates that the die is basically a triangular prism (although with beveled edges), with 1 and 5 on the triangular faces and the pairs (2,3), (2,4), (3,4) on the rectangular faces (thus 2, 3, or 4 will appear "upwards" when the die comes to rest); by symmetry, 1 and 5 should occur with the same frequency, as should 2, 3, and 4. So there is such a die.

Part of the patent reads as follows:

The present invention has been tested for fairness wherein different sizes of dice were included in the test ranging from 13-18 milimeters in thickness.... During initial testing, it was felt that the 14 millimeter thickness was the closest size to providing equally random outcomes for each of the five faces so that each face would occur one-fifth of the time. Specifically, 0,63 rolls were made of the 14 millimeter thickness test dice which yielded 6,152 rolls in which a rectangular silhouette was seen and 4,011 rolls which yielded a triangular silhouette. This means that the two triangular faces came up 4,011/10,163=0.3947 of the time. If the dice was perfectly fair, those faces should come up exactly 04000 of the time. Given the number of rolls, the uncertainty (one standard deviation) was estimated to be 0.0070 which indicates that the experiment detected no significant deviation from fairness.

The actual standard deviation is more like √((10163)(.4)(.6)/10163 = 0.0049, meaning the results were a bit over one standard deviation from fairness; by the usual standards of statistics, though, it's still in a 95% confidence interval (i. e. within 1.96 standard deviations).

Eventually, it seems these will be manufactured at a thickness of 13.6 millimeters (which would prefer the triangular faces slightly more than the 14-millimeter thickness) but it is then stated that

It is believed that the dice may be ultimately manufactured in a range of size from 13 to 15 millimeters depending on the type of material they are to be used on.

It seems like a lot of trouble to have to have different dice for different purposes, which the inventor seems to think would be needed for fairness. (Perhaps this has something to do with the "bounciness".) There's a standard shape for a ten-sided die which could easily be used for this purpose (just label opposite sides with the same number), and from purely symmetrical grounds it's fair. I've been informed that rolling a ten-sided or twenty-sided (icosahedral) die and reducing mod 5 is standard among people who play role-playing games.

the design of asymmetric dice

Mark Dominus says that "For some reason I've been trying to construct a die whose faces come up with probabilities 1/21, 2/21, 3/21, 4/21, 5/21, and 6/21 respectively." He has not been able to do this exactly, although it wouldn't be too hard to come up with an approximate solution given his claim that "the probability that the hexahedron will land on face F is not proportional to the area of F, but rather to the solid angle subtended by F from the hexahedron's center of gravity."

This sounds right -- if you throw a die in the air then it'll rotate around its center of gravity, and you expect that whichever face is pointing "down" when it hits a table for the first time is the face which will eventually settle as "down". (Note that because the faces of a general polyhedron are not necessarily parallel, I'm considering the "down" face, not the "up" face.) But I'm not entirely sure if it's true, because if the die comes close to landing on an edge it might take a funny bounce; I suspect the system is a little more chaotic than Mark's representation of it, and the only real way to solve the problem is experimentally. Unfortunately, the "experimental" part would require tossing a die a very large number of times, and would be incredibly boring. Stereotypically, one gets one's grad students to do that. But I'm a grad student and I wouldn't do it.

What makes Mark's problem difficult is the lack of symmetry; each face has to be different. Let's say, hypothetically, that I wanted to make a die which had two faces which come up with one frequency p and four with another frequency q; clearly 2p + 4q = 1. If q = 0 this is just a coin; if q = 1/4 it's some kind of four-sided stick; my guess is that we can smoothly vary q from 0 to 1/4 by taking a wide variety of rectangular boxes with two of the three dimensions the same.

Similarly, if we want a die whose sides come up with frequencies p, p, q, q, r, r we should be able to do it by taking rectangular boxes with all three dimensions different.

If we're willing to go away from the rectangular box, but have a shape which is still topologically a cube (by this I mean all six sides are quadrilaterals, meeting three at each vertex) I think we can get dice where four faces have the same frequency and each of the other two has a different frequency by using a truncated square pyramid; clearly the four trapezoidal sides each come up with the same frequency, by symmetry. There are two degrees of freedom in constructing such a truncated pyramid (say the ratio between the height and the side length of the base, and the ratio between the side length of the top and the side length of the base) so there's no obvious "dimensionality" reason why this wouldn't work. (A degenerate case of this has five of the six faces having the same frequency.)

And it's probably possible to get three faces with one frequency and three with another by "stretching" the three faces around a single corner, although I'm having trouble picturing how exactly one would do that.

But all of these constructions take into account the symmetry of the cube; I'm assuming that faces that "look the same" are landed on with the same frequency. Mark can't take advantage of that symmetry in his problem.

I also suspect that designing exotic dice like these is difficult, or perhaps impossible, because the probabilities might not be "stable"; they might depend on how hard one throws the die, which defeats the purpose of dice, which is to be a source of randomness that can't be controlled by the person throwing them. Wikipedia offers some oddly shaped dice, though, and seems to claim there exists a fair seven-sided die in the shape of a pentagonal prism, which suggests that stability isn't as much of an issue as I suspect.