Para-bishops and the infinite-dimensional chessboard

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If you have learnt chess, you would have learnt that bishops have a particular handicap that no other piece has: it cannot visit every square on the board. In particular, given that the board is chequered as it is in a FIDE chessboard, a bishop is limited to half the squares on a board: if it started on a white square, it cannot move to a black square, and vice-versa. It is because of this that pieces that cannot see all the squares on a board are called colourbound.

If you venture outside of the FIDE chess world, you will be able to find other pieces with more interesting colourblindness patterns. For example, we have the Dabbabah, which is a piece that jumps two squares orthogonally, jumping over any piece in the way. It's not too difficult to convince yourself that this piece is colourbound twice-over: it can only access ¼ of all squares on a chessboard. We call this 4-way colourbound.

► Convince yourself that for any number n, you can make a chess piece that is n-way colourbound.

The direction that I want to explore today however is going to be a little bit different. It would involve taking the square board and upgrading it to have more and more dimensions until at last we reach the infinite-dimensional chessboard in the title, and seeing how things change as we increase the number of dimensions.

A three-dimensional chessboard

Consider a chessboard in three dimensions. If you like, you can stack them one atop the other; however, if you don't have enough space to literally do it, or imagination power to do it in your head, you can instead think of a list of three numbers: one for x, y and z. For example, a square could be called (0, 0, 0).

Let's step back a little bit and consider a bishop in two dimensions. If a bishop is at (0, 0), then it can move to (1, 1) or (2, 2) or (-3, 3), and so on.

In any case, giving that it starts at the origin, it can move to any square where both coordinates are the same number, ignoring that number's sign.

What does that mean if we plop the bishop down in three dimensions? Well, there is actually a small ambiguity to how that definition we pasted above works. Did we change /all/ the numbers, or just two of them? This question does not matter in two dimensions, but it matters for higher dimensions. Conventionally, the constraint on unspecified coordinates is that they must remain unchanged.

► Try to justify the convention by appealing to how you might define how a Rook moves in a two-dimensional board using the coordinate notation.

This ambiguity is genuine, which is why in three dimensions, there are two "Bishops": one that moves through "face diagonals", called the Bishop, and one that moves through "space diagonals" (also "triagonals"), which is called the Unicorn. The Unicorn was first introduced in a fairly old – but by no means ancient – game called Raumschach.

Raumschach, C-f for "Unicorn" to see its definition.

Because of this, we'll keep the name "Bishop" for all the pieces that change exactly two of their coordinates as they move, and the name "Unicorn" for all the pieces that change all of their coordinates by the same amount as they move. In all cases, the sign of each change is ignored.

Let's examine how the pieces behave in the larger board.

On their native board which is 5 × 5 × 5, a single Unicorn can visit 30 squares on the board. A Bishop on the same board can again visit half the squares on the board, which one can verify fairly easily by considering that on a layer-by-layer basis, a Bishop can only visit half the squares, and which half alternates per layer.

► Verify that the Unicorn is 4-way colourbound as the size of the board increases without limit.

This means that the Unicorn is in some way more limited than the Bishop, and because three-dimensional chess is rather hard for people in our current society to understand the piece is unlikely to be used competently by players and therefore would look even worse.

The Unicorn has no two-dimensional move, leaving it substantially weaker than one would like. It's the weakest piece in the game.

3-D Chess FAQ File, Chess Variant Pages

A somewhat more interesting fact is that the Unicorn actually moves further from its starting square than a Bishop does for every step they move. This is easy to demonstrate, as the Pythagorean theorem works in every dimension. The Bishop, moving one square one way and one square the other, moves a total distance of √(1² + 1²) = √2 squares. The Unicorn, moving one square in each of the three orthogonal direction, therefore moves a total distance of √(1² + 1² + 1²) = √3 squares. To remove ourselves of annoying square roots, we will refer to the total distance moved with its square, which is called the Square of Leap Length, or SOLL.

Now let's move things up a dimension again.

A four-dimensional chessboard

Now that we have four dimensions, we have a completely new orthogonal direction that a piece can take and therefore we have to resolve the ambiguity again: does a Unicorn move in three directions, or all directions? Because we have to answer this question again and again, let's do it once and for all with a two-parameter name:

So now we have three pieces of interest:

It's not too hard to figure out that their SOLLs are, respectively, 2, 3 and 4. Furthermore, as we have learnt previously, their colourboundedness is fixed no matter how many dimensions they are in. That means that a 4-Bishop is still 2-way colourbound, and a (3, 4)-Bishop is 4-way colourbound.

What about a 4-Unicorn though? Here we are still able to reason from first principles. Consider starting from (0, 0, 0, 0) as we always have. Now have the piece move to the following squares:

(1, 1, 1, 1)

(2, 0, 0, 0)

This means that in two moves, it can move two squares orthogonally. It means that on every 2 × 2 × 2 × 2 hypercube, it is capable of visiting precisely two squares. This means that the 4-Unicorn is 8-way colourbound.

► Generalise this to show the colourboundedness of an n-Unicorn.

Keep in mind though that as we increase the number of dimensions of the board, things get a little bit dicier. Not only is the relative player skill decreasing, the pieces are now also vastly dependent on how large the board is. We've been thinking about their power in an essentially infinitely large chess board, so in that light it's not that hard to understand that as n increases, the corresponding n-Unicorns become increasingly weaker. But in a smaller board, things might be different. The most common 4D chess board – such as anything can be called "common" in this niche area – is 4 × 4 × 4 × 4 which is already 256 cells, exactly the same size as four chessboards put together. However, the way they are bunched up together amplifies the pieces' powers immensely, and in small enough boards it may turn out that the 4-Unicorn's ability to fork many different pieces in many different ways may outperform its general inability to get to other places after all.

And on to infinity

The above arguments generalise to 5-, 6-, 7- or even higher-dimensional boards, which by now you probably have run out of ability to visualise. We're now going to make the conceptual leap to an ∞-dimensional chessboard, which is probably one of the hardest things to imagine.

In fact, part of the reason why I have been using a list of numbers rather than algebraic notation to notate how pieces move is to assist you with "visualising" how this chessboard looks like. Instead of actually imagining all the individual squares of this board, think of them instead as being an infinite list of integers, so as always let's put our test piece on (0, 0, 0, 0, ...) and see what happens.

The ∞-Unicorn is particularly interesting for us because it reaches further than any other chess piece we have mentioned in this article. In fact, one step of the ∞-Unicorn would bring it infinitely far away, but using the same trick we have demonstrated in the previous section, we have found out that it is in fact ∞-way colourbound. That is, you can fit an infinite amount of ∞-Unicorns in this board and they will never be able to touch each other! Yet, it's not like it's unable to visit any square other than the square it started with.

This is a very important thing that you learn when dealing with very high dimensions: diagonals become very long. So long that they can bring you to another world.

The infinite-dimensional chessboard is a very scary place to be. One of the strangest facts of ℝ^∞ is that there exists a projection within the space that is exactly identical to ℝ^∞ itself. Think about it this way: if you have a point (a, b, c, d, ...), you can project it all down to one dimension lower by shifting all the coordinates over by one, so the example point here is moved to (0, a, b, c, d, ...).

This adds some additional complications. Now that we have infinite number of dimensions, "changing all the coordinates by some amount" is not the same as "changing an infinite number of coordinates by some amount". We have the ∞-Unicorn, but now consider this new piece that we can call the Almiraj: it is a piece that must change an infinite number, *but not all*, of the coordinates in the manner I specified above. In a sense, this is a "pictorial ∞-Unicorn" – in certain projections, it is in fact a ∞-Unicorn, but not in the whole picture.

The Almiraj as described here is significantly more powerful than the ∞-Unicorn, because while it can move in no more directions than an ∞-Unicorn can (countably infinitely many), it is much more flexible in which directions it can move to.

► However, no matter how it moves, it still needs two moves to get to squares that any given n-para-m-Bishops can reach in one move. See if you can convince yourself why.

Things are getting a bit complicated for me to actually think through, so I'll just leave you here to think about what kind of moves an infinite-dimensional chessboard can allow you. Here are some pieces that I'll just mention in passing:

🗼 gemini://