I took an experimental approach. I used three cubes immersed in containers filled with water.

The first was a hollow plastic cube about 1.25 inches on a side, left over from a game called Instant Insanity. While not uniform, strictly speaking, it differs very little from a cubical volume of air. It weighs about a tenth of what a similar volume of water would weigh.

The second was a similarly sized piece cut from a rectilinear block of margarine. Its density is very close to that of water; it floats, but roughly nine-tenths submerged.

The third was a cubical block of wood 3.5 inches on a side, which I sawed off from the end of a (misnamed) 4x4 beam. Its density is around half that of water.

The plastic toy is quite precisely cubical. The other two are close to, but not perfect, cubes.

To sum up my observations, the margarine cube floated face up. It seemed to have no preference for one face over another, maintaining its orientation no matter which face was uppermost, as one would expect with an (almost) perfectly symmetric cube. The other two floated corner up, and they likewise seemed to have no preference for one corner over another.

These observations suggest that cubes of high specific gravity float face up, while those of low specific gravity float corner up. But two samples are not enough to conclude this with certainty. It's possible that in some range of densities cubes float edge up. Ideally I would like to test ten or more cubes with densities equally distributed between those of my first and second samples, but I don't have objects made of appropriate materials.

I read here https://orca3d.com/wp-content/uploa... where it makes the claim that a cube with half the density of water can float with one of the faces up, but "none of these flotation conditions are actually stable...if the cube was disturbed, it would rotate to the following condition, which maximizes the waterplane inertia," showing a corner up position.

Based on my (very scanty) tests, I would say that cubes with high specific gravity float with a face uppermost and cubes of low specific gravity float with a corner uppermost. I don't know where the dividing line is. I would guess that it's at 0.5.

Do you not trust the simulator shown on the article you linked, http://datagenetics.com/blog/june22... where you can step through the density ratio rho and it shows how the cube will float?

This shows that the cube floats face up below rho = about 0.20, it floats corner up at a 45 degree angle above rho = about 0.29, and between those values it floats at an angle between 0 and 45.

<This shows that the cube floats face up below rho = about 0.20, it floats corner up at a 45 degree angle above rho = about 0.29, and between those values it floats at an angle between 0 and 45.

For example at rho = 0.25, it looks like the face makes an angle of about 30 degrees with the water surface.>

The block of wood I tested looked as if it might be a few degrees off from exactly corner up. But as I said, it wasn't a perfectly cubical body, so I couldn't draw any conclusion from that.

Hear , hear !

<John barleycorn> is a racist , abusive troll.

He does not know much of maths - no depth. Perhaps some breadth of knowledge but no depth.

Yes , that is what <John barleycorn> is about - flame wars. A racist , abusive troll.

No, I respect his mathematical ability. His problem is emotional illness: <A psychological disorder characterized by irrational and uncontrollable fears, persistent anxiety, or extreme hostility.> https://www.dictionary.com/browse/e....

Oh well, John Nash was severely disturbed too. But he eventually recovered. https://www.nytimes.com/2015/05/25/..>

Hopefully , <John barleycorn> will recover too and instead of engaging in flame wars non stop engage in productive conversations in a harmonious way.

https://www.chessproblem.net/viewto...

Is there actually any way to know the difference, if you reside on the real torus?

<Is there actually any way to know the difference, if you reside on the real torus?>

Imagine that a chessboard is adjacent to four more on the north, south, east, and west sides, and imagine that they in turn are similarly surrounded, and that this pattern is replicated an infinite number of times, thereby tiling the whole infinite plane. On each board the same position is set up, and whenever a move is made the same move is made on all the replicas. If the move carries a piece across a boundary of a particular board onto an adjacent board, the corresponding piece on all the other boards executes the same maneuver, and one of them re-enters that first board.

This situation is isomorphic to what we have in the problem. If you focus on one single board, those boundary-crossing moves will look as if they are re-entrant. Another way to express it is to say that the board's boundary conditions are doubly periodic.

It's also isomorphic to the surface of a solid torus. Take a physical torus (say, a hollow plastic quoit -- a doughnut would be messier) and cut it across the minor diameter. Straighten out the resulting tube and slit it longitudinally. You now have a flat piece of plastic which is rectangular, approximately square. Color it with 64 squares.

That is the chessboard in the problem.

A toroidal surface is really two-dimensional. (That's how it can be isomorphic to a rectangle.) Only two-dimensional beings can live in a two-dimensional world. They can't tell whether they're on the inside or the outside of the torus, because there is only one surface.

Suppose you draw a line on (for a true two-dimensional surface it would be more proper to say "in") between the d file and the e file of a chessboard.

If you join the first and eighth ranks first, that line becomes a minor circumference; if you join the a and h files first it becomes a major circumference.

If you first join the first rank with the eight, then a1 is joined to a8, b1 to b8,... h1 to h8. Then you have to join the sides. It seems to me that a2 is joined to a7, a3 to a6, and a4 to a5.

If you first join the a-file with the h-file, then a1 is joined to h1, a2 to h2,... a8 to h8. Then you have to join the top and bottom. It seems to me that b1 is joined to g1, c1 to f1, and d1 to e1.

I flat-out don't understand that.

The second join should connect the a-file with the h-file. Thus a1 will abut h1, a2 will abut h2, and so on. But you don't mention the h-file at all.

<If you first join the a-file with the h-file, then a1 is joined to h1, a2 to h2,... a8 to h8. Then you have to join the top and bottom. It seems to me that b1 is joined to g1, c1 to f1, and d1 to e1.>

Now your first join puts the a-file next to the h-file, so the second join should put the first rank next to the eighth. But you are talking only about *first-rank* squares (b1, c1, ..., g1).

The ratio between the minor and major circumferences is variable. A torus has major circumferences of different lengths. A major circumference can be drawn along the doughnut hole or around the part of the surface farthest from the hole. And of course the doughnut can be skinny in some places and fat in others, so the minor diameter need not be the same all the way around.

Not only that, but the minor/major ratio can be any number between 0 and infinity. Imagine a torus with ripples in the surface. If the ripples are all parallel to the major circumference, that makes the distance around minor circumferences greater. And if the ripples are parallel to minor circumferences, that makes the distance around major circumferences greater. With enough ripples, either distance can be made arbitrarily great.