Culture and Civilisations

Chess and the mathematics of Heaven

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Chess and the mathematics of Heaven

In the past, chess has been employed both as a mathematical metaphor and adapted as a method of penetrating the secrets of the universe. An early and celebrated instance appears in an episode from Dante’s Paradiso, where he interrogates his guide about the number of angels in the heavens. More recently, as we shall see, Professor Michio Kaku, the eminent string theorist, has enlisted chess in his quest for a theory of everything.

We already saw, in my column of May 29, 2021 “Did Roman emperors play chess?” how chess possibly developed from a concatenation of Classical Greek games of skill with Indian games of chance. This happy conjunction developed into Chaturanga (based on the ancient Indian army) and then Shatranj. There are alternative explanations, but they are apocryphal, one such appearing in The Divine Comedy of Dante, himself said to be an avid chessplayer:

Lo incendio lor seguiva ogni scintilla;

Ed eran tante , che il numero loro

Più che il doppiar degli scacchi s’immilla.

Every spark followed its kindler;

And so many were they, that their whole

Number far more thousands counts,

Than ever did the doubling of the chess. 

Paradiso, Canto XXVIII, lines 91-93.

Dante (referring to the number of angels in heaven) alludes to the legend (first mentioned by the Persian poet Abu Al-Qasim Firdausi in The Shahnama, or The Book of Kings, of around 1000 AD) that chess was invented for a King by a magician who demanded as reward, one grain of rice on the first square of the chessboard, doubling thereafter and amounting to 2 to the power of 64 minus 1 grains. This equates to 18,446,744,073,709,551,615, a number so cosmic in itself that I hesitate to pronounce it by name: I calculate that it is eighteen quintillion, four hundred and forty-six quadrillion, seven hundred and forty-four trillion, seventy-three billion, seven hundred and nine million, five hundred and fifty-one thousand and six hundred and fifteen. This would have been quite sufficient to bankrupt several kingdoms. The king’s reaction is not recorded, but my guess is that it would have been something resembling: off with his head!

An example of the astounding nature of exponential growth by doubling the number on the next square.

Shatranj, the early form of chess, swept both east and west. By 800 AD the Chinese already had their own version, where a central river divides the hostile forces; through Korea, the game reached Japan, where it is still played under the guise of Shogi. In the Japanese version, captured pieces do not vanish permanently from the board, they defect to the enemy. Perhaps this indicated that Japanese battles of that period were fought with mercenary armies, which rapidly redefined their loyalties, once defeat had become inevitable.

Chess moved west with even greater momentum. After the explosion of the Arabic expansion of the seventh century, Shatranj, enjoyed a golden period. It reached Europe via the Moorish invaders of Spain, through the Byzantine Empire and thereafter Russia. However, it was during the 8th and 9th centuries AD in the caliphate of Baghdad of the Abbasid Dynasty, that the game truly flourished. (Baghdad was, in fact, to Shatranj what Moscow has been to the modern game). Grandmasters began to appear, the subtlety of whose play (which still survives in brief published fragments) rivalled that even of modern masters. Among such were: Rabrab, Ar-Razi, Al-Adli, As-Suli and Al-Lajlaj (“The Stammerer”).

The profound expertise of these players is evidence to me that Shatranj was the product of an immensely long heritage. Such honed excellence does not burst forth unannounced in a new game after a mere century or so. Even with the improvement in terms of travel, and hence communications, which was to take place in the post-mediaeval world, there was still to be a gap of three centuries between the advent of the Renaissance new chess and the arrival of a master such as Philidor. Yet the Baghdad Caliphate could boast several players, whose relative strength was comparable to that of the great Frenchman.

The diagram below shows the development of the Egyptian hieroglyphics sign of the ‘Bull‘s head’, via Greek, to the Latin letter ‘A’. The ox, the most important component of primitive agricultural economy, naturally provides a basis for the first letter of the alphabet. It can be observed from the table how the horns of the ox have been gradually transmuted into the inverted crossbars of the letter ‘A’. The ox, ‘ aleph’, the most important animal, is also ‘alpha’, the beginning of the system of written symbols.

The name aleph is derived from the West Semitic word for “ox” (as in the Biblical Hebrew word ‘Eleph’ (אֶלֶף) ‘ox’, and the shape of the letter derives from a Proto-Sinaitic glyph that may have been based on an Egyptian hieroglyph, which depicts an ox’s head. Aleph (or alef or alif, transliterated) is the first letter of the Semitic abjads, as well as Phoenician in the form of ʾālep; Hebrew ʾālef; Aramaic ʾālap; Syriac ʾālap̄; and Arabic alif. It also appears as South Arabianʾälef.

The elephant, apart from being the world‘s largest land-dwelling mammal, also appears as a chess piece in past forms of chess and also currently Chinese Chess, known as Xiangqi. In the course of the game’s long history the elephant has undergone an interesting transformation from pachyderm to prelate. Alexander the Great and his troops encountered Indian war elephants on their campaigns, and the Greeks, witnessing elephants for the first time, and observing their key role, in both oriental warfare and agriculture, named them “aleph-hind”, the “Indian ox”. (The Spanish conquistadors of the 16th century committed a similar error with llamas, calling them “native sheep”). The elephant, of course, appeared as a unit on the primitive Indian chessboard, the symbol of warfare in miniature.

Only in Russia and some Slav countries now, does the word “elephant”(“slon”) persist. Elsewhere, as chess progressed westwards, it was the “l” and “f” of aLePH-hind which survived and it is these sounds which have gone on to signify the Bishop/Elephant chess piece. One sees traces of this in the Arabic term, fil; in the Middle English, aufin; in the Spanish, alfil; in the Italian, alfiere; in the German, läufer; Dutch, loper; the Serbo-Croatian, lovac; and perhaps even in the French le fou.

It is evident that the various incarnations of the “bishop” in European culture placed greater emphasis on preserving the ancient and original phonetics, than the meaning. The Dutch and Germans see this diagonal piece as a “runner”; the Yugoslavs as a “hunter”; the French as a “jester”; while the Spanish and Italian words, closest of all to “aleph-hind”, mean only “the chess piece” and nothing more. The English bishop is furthest, indeed, quite remote from its elephantine original. Nevertheless, when one interprets a medieval court and its most powerful figures as the respective sides of a chessboard battle, then inclusion of the clergy makes perfect sense. More so, indeed, than jesters, runners, hunters and so on.

Table of names of chess pieces


English also has an exception in its word “rook”, where most European languages go for “tower” or “castle” (tour, Turm, torre). Doubtless, a key derivation here is from the alternative Italian word for tower, “Rocco”. In other European languages one sees the roots of “rook” in the word for castling (roque, arrocco, enroque). The Russian word for rook, “boat”, appears to be absolutely unique and has no connection with warfare. It is, indeed, amusing to see Russians referring to chess pieces, which evidently represent castles, as boats. “Rook” in English, as I pointed out last week in my column “White Queen – Black Queen”, may also refer to “Rukh”, the Ancient Persian word for war chariot.

By the year 1000 AD, at the time of the composition of Firdausi’s Shahnama, chess was widely known and popular throughout Europe. Nevertheless, the great technical expertise and the lust for knowledge of the Arabs was gradually being extinguished. With the decline of Baghdad, the writings, games and accumulated chess wisdom of As-Suli and his colleagues dispersed and vanished. For the following centuries, chess became, through the medium of tricky composed problems, part of the repertoire of itinerant entertainers. Many of these, though, were ignorant of the strategy, tactics and even the rules, of the game itself. Still, the popularity of chess, at the common level, can be gauged, for example, from the Isle of Lewis Chessmen. This was a cache of carved walrus ivory Scandinavian pieces dating from the 12th century. The vast horde of disparate pieces, now chiefly in the British Museum, suggests that this was the stock of a merchant supplying chessmen to numerous clients.

During the mediaeval period, chess was mentioned in courtly tales and both Carolingian and Arthurian romances. A notable Arthurian reference can be found in the Welsh compilation, the Mabinogion. After a lengthy oral tradition, the stories which go to make up this national epic were finally written down in the 13th century. In the story of “The Dream of Rhonabwy”, King Arthur himself contests the battle board game “Gwyddbwyll” against the Welsh hero Owein. In the background fierce warfare rages between Arthur‘s pages and the black ravens of Owein. The symbolism argues that Arthur and Owein are playing chess.

In the Middle Ages chess, as symbolism, flourished; but as a science, as a serious game of reason and strategy, it was now running out of energy. The intellectual fire, the mental acquisitiveness of the great Arab practitioners, had been exhausted. As it was the Greeks who gave the initial impetus to chess, so it is to the Renaissance that we must look for the regeneration of the game.

During the late 15th century the surprisingly rapid process had been initiated, whereby chess emerged from its slow, tortuous, mediaeval form. Suddenly “castling” was introduced – castling involves a player’s king and either of the player’s original rooks. It is the only move in chess in which a player involves two friendly pieces in the same move, and it is the only move, aside from the knight’s leap, where a piece can be said to “jump over” another. The point is to offer greater protection for the king. Furthermore, “pawns” gained the privilege of having the ability to move two squares forwards at their first turn, and the “queen”!was transformed at a stroke from a waddling cripple (the Arabic Vizier) which could only move one diagonal square at a time, to a unit of a devastating ferocity, with the power to traverse the board in one sweep. Perhaps the sudden access of strength of this piece helps to explain the joyous adventures and excursions with the Queen, which can often be observed in the games of players of the new chess in the 15th, 16th, and early 17th centuries.

One of the early versions of chess which has survived and flourishes, as noted above, is the Japanese variant Shogi, the game of the generals. A key piece is a diagonal moving unit, more or less equivalent to our “bishop”. The Japanese piece is called the “Kaku” and, coincidentally, Professor Michio Kaku (who is Professor of Theoretical Physics at City College, New York) has, like Dante, used chess to help explain the workings of Heaven and the mathematics of the cosmos.

According to an enthralling interview in The Guardian by Andrew Anthony, Professor Kaku is a leading proponent of string theory and also a celebrated populariser of science, with multiple TV appearances and several bestselling books behind him. Michio Kaku’s latest book, The God Equation , aims to combine Einstein’s General Relativity with Quantum Theory, to create an all-encompassing “theory of everything” about the nature of the universe. Professor Kaku said that we actually have the theory but not yet in its final form. It involves String Theory, theoretical physics, fiendishly difficult mathematics and mind-bending abstraction, which the general public might find difficult to grasp.

I now quote Professor Kaku directly: “I think the public is curious as to what the consequences of this theory could be. The universe in some sense is like a chess game and for 2,000 years we’ve been trying to figure out how the pawns move. And now we’re beginning to understand how the queen moves and how you get a checkmate. The destiny of science is to become like grandmasters, to solve this puzzle that we call the universe. There are outstanding questions that the public wants to have answers for. For example, time travel, other dimensions, wormholes. What happened before the big bang? What’s on the other side of a black hole? None of these questions can be answered within the framework of Einstein’s theory. You have to go beyond Einstein into quantum theory.

So… one has to ask, are we in 2021 really any further forward in our understanding of the mysteries of the cosmos, creation and the heavens, than when Dante, in 1320, asked how many angels there are in the firmament of Heaven and used the chessboard formula to come up with his answer?

Here are three games where the bishops played a key role: The first game between the World Champion-to-be, Emanuel Lasker and Johann Hermann Bauer in Amsterdam 1889. The second game pits against each other, two rivals from the early twentieth century: Aron Nimzowitsch vs Dr. Siegbert Tarrasch in St. Petersburg 1914. This game was the high point for Tarrasch in his feud with Nimzowitsch, as can be seen in my column of May 22, 2021, “Freudian feud: Tarrasch versus Nimzowitsch”. The third episcopal game is between double World Champion Alexander Alekhine (World Champion: 1927–1935 and 1937–1946) and John Drewitt in Portsmouth 1923.

Raymond Keene’s latest book “Fifty Shades of Ray: Chess in the year of the Coronavirus”, containing some of his best pieces from The Article, is now available from  Amazon , and  Blackwells .

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