Horsey Geometry

Please, put a knight on a central square and place 8 pawns on all its neighbouring squares, center all 9 pieces well.

In German we say 'Springer' ("jumper") for knight. Is it paradoxical to let the Springer jump into the 8 directions it could trot?

The knight move is supposed to be as old as the game. So is the rook move. Diagonal moves were given in the moves of king, pawns, vizier and elephants. Geometrically, the knight move, from the beginning, was a complementary move to moves following diagonals or horizontals or verticals of squares.

Board and Grid

Now, if it would, the Springer could really just walk over there- no need to jump. In contrast to others the knight is not blocked by pieces on neighbouring squares. On our picture, it could trot over to its destination on a straight.

And in a grid of 8x8x8 cubicles (in a would-be 3-d-chess cube) it often couldn't help but taking the direct, unobstructed line. Within a cube, analogous to on a board, the knight's directions were given by the gaps left by the pieces that could surround it in all 26 cubicles immediately neighbouring its own. From the point of view of our knight, all neighbouring pieces would hover centered in cubicles on verticals, horizontals and diagonals of cubicles. Looking through the gaps between the pieces, it could see those desired closest cubicles not on verticals, horizontals or diagonals, where it could dart to.

Our central knight on a board can move following 8 straight threads (in a 3-d grid I have trouble imagining the correct number) just without following files, ranks and diagonals.

On Disparity

On conception of the game, for disparity in move geometry, it was a logical step to have a piece that does it wholly differently than the others do. Differences made up the set. In today's rules they are distributed like this: The rooks have the verticals and horizontals, the bishops have the diagonals, the queen has both of them, the king has both but is restricted in scope, the pawns, idiosyncratically, are going straight forward, taking diagonally, while being restricted in scope and direction.

Another Ballpark

The knight, by not moving on diagonals or verticals or horizontals of squares at all, seems to be playing quite in another ballpark. If we took that complementary definition to its extremes, the knight would instantly become the strongest piece: Placed anywhere on the rim, on an empty board, it would have 42 squares at its disposal (63-7-7-7), placed in the centre it could reach 36 squares (63-7-7-7-6), if the only constraint was not to move on diagonals, lines or ranks.

Of course, calculation would be a nightmare ;) But okay: Other moves could have been picked from this pool. (-That was done, for example, on the bigger board of the 12x12 Grant Acedrex, defining several knight-like pieces, but no knight).

Now, what lifts out 'our' knight move from the lot of possible moves on this picture?

The Median

On Ng1-f3, Nf3 is situated exactly between the diagonal g1-a7 and the line g1-g8. On Ng1-e2, Ne2 is situated exactly between the rank a1-h1 and the diagonal g1-a7. So, 'our' knight move lies on the exact median of vertical and diagonal, or horizontal and diagonal.

That is a nice feature, because, while the way to any square not on its own file, rank or diagonals is unobstructed by pieces on neighbouring squares, if only they are centered pin-pointedly enough, the way to the squares lying on the medians is the best example of unobstructed availability, and the squares are the places most distinguished from, most far away from play on diagonals and ranks and files.

If we accept this reasoning, we may be happy to see that only the yellow polka dots on our picture above meet this criterium.. and then wonder, why the knight was not conceived as a long-range piece, like the rook, or did not turn into one, like bishop and queen.

Long-Range Knights

Using a best example for an unobstructed straight between the vertical d1-d8 and the diagonal c1-h6, a Nd2 could check a king on g8- or pin a Qf6 to it (g8 lying on the median, i.e. right between d8 and a would-be j8). And this Nd2 check or pin could then be answered by a move e5-e4, for example.

Looking at such lines would put to rest thinking of the knight move as L-shaped, I guess. And not 'jumping', 'up-to-triple jumping' would be an apt description.

Its quality would add 64x4=256 squares to the 336 squares which an ordinary, short-range knight can reach from all 64 squares on an empty board. It would be a different game, maybe worth a try.

How come our knight started and made it, instead?

from Tschaturanga, with pieces crossing plains, rivers, mountains and seas, to alla rabiosa

Looking back on tschaturanga (which in Sanskrit means 'having four limbs' and, with regard to the army, invoked its four divisions infantry, cavalry, elephantry and chariotry) we see that the ratha, the rook- the chariot- was indeed the only piece of long range on the ashtapada (the uncheckered 8x8 board). The bishops were elephants that jumped to the second square diagonally, the queens were viziers that only moved one square diagonally, the pawns had no option for a two square advance and did only promote to a vizier. Our simple knight move for the ashva- the horse- fitted into that constellation, in that it was short range, it was different, and strong enough.

Spreading over the world, the game (if it was ever one) bore hundreds, if not thousands of local variants. It would be interesting to know how many coming from tschaturanga kept the ashva move. My bet is many. Two of the biggest variants of the family chess belongs to, shogi and xiangqi, only have sort of a one-quarter- and a three-quarter-knight, respectively, indicating both: That the knight move was neither necessary for this kind of board game, nor was it far-fetched.

When, ages later, bishop and queen appeared on the scene, accelerating the game immensely- so out of bounds that it was widely greeted as 'mad queen's chess'- again, the knight proved strong enough to take up the fight in its littlest or simplest form: It needs nothing more than one of the closest squares not on a rank, file or diagonal to be a full member of the band.

The short-range move is the first solution for moving to squares sharing neither rank, file or diagonal with the square of origin, and for moving to squares precisely as little on a rank or file as on a diagonal. Distinguished from the get-go, working out within the set.. Beauty of a simplest form, our knight move.

That knight happening to be equal to the more recent bishop and with what this equality brings to the game, it is all but illusory we see either piece change again on an 8x8 board.

For the old game, they picked the earliest form making the difference, a full fruit ripe forever ;)

Hop

On a real board, pieces' shapes and square sizes are not made to leave gaps big enough for a knight to move through, and that is because we can simply make use of the extra dimension and jump. Look how files and ranks are alleys, diagonals are just fine lanes. The medians the knights take can never be as wide, because of the geometrical relationship to the diagonals, files and ranks.

Thus, jumping. It could be grasshoppers, mountain goats or kangaroos for that, but it ain't natural obstacles that must be overcome but groups of friends fighting one another, so it's horses.

So, all fall into place: That complementary move- and the size of board and pieces that would best become play on verticals, horizontals and diagonals- and the use of the horse, having it jump on the- grace a lui- not too crowded, not too lonely board to where the complementary piece logically just needed be pushed, but whinnies!