Castling and the 4th Dimension

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It's possibly the most eccentric rule in chess - moving your king and your rook on the same move so that they (almost) change places with one another. The rules governing the move seem straightforward enough, and they give the chess player enough information about it for the purposes of playing the game. As I will demonstrate in this post, however, there is far more to the move that deserves to be explored than how it is described by the rules of chess - for these rules capture only the surface phenomena of castling, while neglecting to address the underlying processes by which it occurrs. An inquiry into what these processes might be - for that has so far been unclear - suggests that castling in chess may be proof of the existence of the 4th dimension - which is the thesis that I will now proceed to argue.

What initially got me thinking about the underlying nature of castling was a simple question: how exactly is it that the king and rook are able to move past one another - which they must do in order to change places - when the player makes this move? Three possibilities intuitively came to mind, yet with simple reasoning I was able to quickly eliminate all three. The first is that the two pieces simply move through one another, as if one or both of them were ghosts. Yet this is impossible, for in chess it is clearly established that pieces cannot move through one another. If they could, a player could simply take his opponent's rook with his own on the first move - for the rook can move through the two pawns in between. The idea of pieces moving through other pieces even goes against the purpose of castling itself, which is usually to shield the king behind a wall of pawns. These pawns act as a fortress protecting the king - hence the term "castling" - which the opponent must break through in order to attack the king. Yet if pieces could simply move through other pieces, then this wall would be permeable, and in effect nonexistent - in which case there would be reason to castle in the first place. Clearly, the idea of "ghost" pieces cannot explain the movement of the king and rook when castling.

Equally unsatisfactory is the second explanation that came to mind, which is that either the king or the rook jumps over the other piece. It is simple knowledge that in chess the only piece that can jump over other pieces is the knight, aided by his strong, intrepid horse. The rook is a tower rooted to the ground, and the king is burdened by the heavy cross that he bears on his head; thus, it is absurd to expect either of them to jump to any significant height.

A third possible explanation is that the two pieces pass one another side by side, as would two strangers walking in opposite directions on a sidewalk. Yet one will easily observe that in a majority of cases when one castles, most of the pawns directly in front of the king and the rook in question have not been moved. Thus the movement of the two pieces is constricted (by the pawns hemming them in) to the back rank - which, needless to say, is much too narrow for the king and rook to pass side by side. Thus this explanation, too, does not stand up to the test of common sense.

Moreover, all three explanations discussed thus far suffer from a common fatal flaw: they do not explain why the king and rook are able to move at the same time. The notion that pieces can move simultaneously is under normal circumstances a flat-out abomination in the realm of chess, a strictly turn-based game. If pieces could move simultaneously, how could they be captured? As soon as the capturing piece makes its move toward a target, the target can simply move away, and by the time the capturing piece reaches the square that the target originally occupied, the latter would already have escaped. The game of chess would thus completely break down if simultaneous moves were allowed, which is perhaps why they are not - with the exception of castling. What is it, then, that allows castling to be immune from the general law against simultaneous movement?

Before I present the explanation that I discovered for both the physical dilemma of how the pieces can move past one another, and the question of how they can do so while moving simultaneously, I must introduce the topic of dimensions into the discussion. I will begin on a lower-dimensional scale, with a question that mathematicians, physicists and philosophers have pondered for at least the past several centuries: what would a three-dimensional solid look like if it were to appear in a two-dimensional world? An example that is often discussed in relation to this question is a sphere passing through a plane (as in a flat surface, not a Boeing 747), on which there exists a world of two dimensions. When the sphere first comes into contact with the plane, it will appear to an inhabitant of the two-dimensional world as if a dot has suddenly appeared out of thin air. As the sphere continues to move into the plane, the dot will become a circle that gradually increases in size (Of course, the inhabitant of the 2d world will not be able to see the circle, but he can infer it from what he can see, just as humans can infer the shape of a 3d building without having seen all of its sides. But in any case, such details are beside the point of this discussion.) until the point when the sphere is bisected by the plane. From then on, the circle will become smaller and smaller, until it shrinks to a dot and finally vanishes, returning to the mysterious nothingness from whence it originally arrived into the world of our 2d observer.

The point of the preceding discussion is to note that movements in the third dimension may be seen as appearances and disappearances in a 2d world. It is also worth mentioning that these movements are not confined by barriers in the 2d world - which would appear as lines on the plane analogous to walls in our world. While inhabitants of the 2d world cannot traverse these barriers, a solid that can move in 3 dimensions can just as easily intersect the plane on one side of the line as it can on the other, and could even "appear" within a closed box, which in the 2d world cannot be entered from the outside.

At this point, the construction of the analogy is complete, and it is time to ask the question: What does all of this talk about solids and 2d worlds have to do with castling? I will now explain. In castling, the king and rook have to pass through a seemingly insurmountable barrier - namely each other. The pieces cannot move through or jump over each other, and they cannot pass side by side; thus, it seems that there is no possible way that the move can be completed - at least not in the third dimension. As the 2d world analogy demonstrates, movements in a higher dimension are not beholden to constraints that restrict movement in lower dimensions. Thus it seems that the solution to the problem of castling is that (at least) one of the pieces involved - either the king or the rook - has access to the 4th dimension, and uses the power to move in that dimension to make castling possible. As far as the 3d world is concerned, that piece disappears from its original square and re-enters the world at the square that it occupies after castling, and between the diappearance and reappearance the other piece (assuming only one piece can access the 4th dimension) makes a regular 3-dimensional move to its destination square.

Many leading physicists today believe that the 4th dimension is what we think of as time. If that is indeed the case, then the 4th dimension explanation also accounts for why the king and rook can move simutaneously in castling. If one of these pieces can access the 4th dimension, then it would be able to manipulate time, just as humans can manipulate 3-dimensional objects. Thus, while pieces that exist solely in the third dimension must obey the law against simultaneous movement, the king/rook transcends this law, moving through time without regard to chronological constraints that apply only to lower dimensions.

In closing, I must note that while the 4th dimension theory successfully explains why castling is possible, it raises other questions that for the moment I am unable to answer. For one, it is unclear whether the king or the rook (or both) is the piece with the power to move in the 4th dimension. Judging from common sense, it seems likely that it is the king, given that the king is the most important piece in the game (whereas the rook, other than being fairly powerful, is otherwise indistinguised), and can castle with either one of two rooks (so if it is the rook, then the power must be posessed by two pieces, whereas if it is the king it only needs to be be posessed by one). Nevertheless, at present there does not seem to be a way to answer the question with any degree of certainty. Also, it cannot be explained why the king/rook only uses its powers to castle, and behaves as a piece that exists only in the third dimension on all other occasions. To achieve a greater understanding of how castling works in chess, these and perhaps other questions should be investigated by physicists, chess players and other professionals in relevant fields. In the meantime, we can marvel at the apparent fact that a simple, seemingly innocuous rule in chess can prove something as profound and far-reaching in its significance as the existence of the 4th dimension.

The fourth dimension already exists....it is called "time".

See my second to last paragraph.

Macer, you are by far the best chess writer on this site, possibly on all sites. Your posts are informative, imaginative, and fun to read. I am writing this on Sunday, but I'll post it at the exact moment I castle to my queen side in a game, if it works this should show it posted this past Friday around 10am est. Keep up the great posts!

I would like to point out that there is no simultaneity in castling-the rules require you to move the king first

Also, when playing on a physical board a fourth dimension might be necessary, but it is clearly not the case when using using a 2D digital board (like those used to play here), as 3 dimensions would be enough.

+

This is great! It even makes sense... kind of

Chess is a 2d game though, so maybe the 2d pieces simply use the 3rd dimension i.e. hop over each other. Yeah, I think that's it.

They would be captured... more than 1 at a time I suppose

That's chess on chess.com

You missed out the possibility of tunnelling under the board (refuted because there are no holes in it) and teleporting, wrong because there are no kings or towers in star trek.

TLDR

OTB chess is also 2d. The pieces are represented by 3d icons, and the board has some thickness, certainly, but the game itself is 2d.

Macer:  you are getting posts pointing out the samething.  Chess is  a 2d game.  I dealt with this problem when I tried to parallel 2d chess with 4d physics.  It is actually better to think of  it as 3d physics.  The time dimension has to do with symmetric and asymmetric  (change, motion, movement, etc.)  In order to keep the moves of  the pieces and the pawns 2d it  is necessary to use symmetric change.  This means that the pieces  and the pawns never leave  the  surface of the board.  In other words, in 2d there is only right/left and forward/back movement.  There is no up/down movement in 2d.  I solved the 2d problem where the pieces and pawns never leave the surface of the board, but it is too difficult to describe in words. 

If I could get help with setting up a diagram, you know a picture is worth a thousand words.

There is a rather interesting possibility nobody has mentioned yet, and it is that a wormhole lets the rook move without having to hop over or move through the mighty king.

It takes less time to read Flatland than the original post.

@ macer75 I could see castling when on mescaline would inspire this revealation.

I think it's more like defensive linemen pulling off a "stunt" in American football, where the outside lineman attacks inside and the inside lineman attacks outside.  As huge as those men are, they manage to move past each other without invoking a 4th physical dimension.