Of sangaku and wind farms

By Christopher Monckton of Brenchley

The Edo period in Japanese history ran from 1603-1867. For most of that period, 1639-1854, the nation kept itself deliberately aloof from all Western influence. Japan was closed to foreigners. Meanwhile, the aristocratic samurai class developed an intriguing and elegant method of demonstrating piety in their Shinto temples.

They would carve geometric problems, typically involving circles, on timber shingles, paint them colorfully, inscribe them in kanbun (Chinese characters with diacritical marks to distinguish the Japanese usage), and affix them to the temple walls as an inspiration to subsequent visitors.

The use of kanbun by the samurai was akin to the use of Greek by Roman orators or of Latin by the English aristocracy. The sangaku tablets, therefore, were near-exclusively of samurai origin and were comprehensible to their caste alone, and of course to the delighted deities.

Contemplating the monstrosity of the Navitus Bay wind farm proposal, I was struck by the irregularity of the suggested area for the wind farm (Fig. 1).

Figure 1. The proposed site for the Navitus wind array.

Looking at the shape of the proposed site, it is at once clear that the array is as close to the shore as the developers dare to make it, partly to reduce construction and maintenance costs and partly to minimize the formidable transmission losses in the undersea cable. But the edges of the array are defined more by sight-lines from the shore than by any other consideration.

How would a mathematician go about moving the array further out to sea and compensating for the transmission losses by minimizing the shadowing effect by which turbines take each other’s wind, regardless of the wind direction, as well as minimizing the visual impact?

The prevailing wind is south-westerly, but stretching a line of wind turbines north-west to south-east to capture it would represent an unacceptable hazard to coastal shipping.

One answer is to arrange the turbines in a circle, which provides minimal visual impact. But how to dispose the turbines within the circle to minimize wind shadowing still further?

Fig. 2 shows possible solution for 192 turbines, just two fewer than the Navitus array, and occupying approximately the same area of 68 square miles (the “8” was inadvertently omitted from my recent posting on the array – mea culpa).

Figure 2. An array of 192 wind turbines, each with a 505-ft span and separated from its neighbors by five spans, intended to minimize both visual impact and wind shadow.

Assuming that each turbine’s span is 505 feet, and that turbines must be five spans apart, the diameter of the array is approximately 9 miles 3 furlongs.

Before I reveal what all this has to do with sangaku tablets and their geometric problems based on properties of circles, this method of arranging the wind turbines can also be used to create an ingenious board – the mathematician’s version of the chessboard, if you like – where equal numbers of playing-spaces (let us call them “forts”) guard each straight row (let us call it a “street”), and an equal number of streets meet at each fort.

In the board shown in Fig. 3, there are 44 forts on 77 streets, with exactly four forts guarding each street and exactly seven streets meeting at each fort.

Figure 3. The mathematician’s chessboard

Some 30 years ago I successfully marketed a board game based on a layout similar to this one. The rules I came up with are below this posting. Gentle readers, your challenge – should you choose to accept it – is to devise as many sets of new rules as you can for this board. Then I shall relaunch the game with multiple sets of rules. One board, many games.

The criteria for your rules as follows. The game must be one of pure skill, with no element of chance. It must involve simple pieces placed on the board. The rules must be short, simple and quick to learn, yet the strategy should be complex enough to make the game both exciting for all and intellectually satisfying for good minds. Try out your rules before you send them in. If they’re any good, people will become hooked by the game. That’s how we’ll get the big sales, and you’ll get your royalties.

Now for the connection to the Japanese sangaku tablets. The extraordinary and surprising feature of the layout is that, although the forts are connected by straight streets, it is circles that determine the exact positioning of the nodes coincident with the forts. This is a startling and – as far as I know – hitherto-unsuspected property of circles. The samurai would have appreciated that.

Figure 4. The surprising relation between a network of straight lines and the intersections of a set of circles rotated at equal intervals about a common point on their circumferences.

Below, as promised, are my own rules for the game of Battle. I hope this harmless diversion involving “puzzling things in life” will have given pleasure.

Rules of engagement

The objective is to drive all opposing forces from the battlefield.

The battlefield has 44 forts guarding 77 streets. Four forts guard each street. Seven streets meet at each fort. At the outset, the city is empty of all forces.

Alliances: Blue with Green, Red with Yellow may fight as pairs of allies, or armies may fight singly.

The attack: Each army attacks in turn: Blue, Red, Green, Yellow, etc. Each army’s sortie is in two phases: first, expulsion (wherever possible), then occupation.

Suppose that it is Blue, allied with Green, whose turn is to attack.

1. Expulsion

On every street, street by street, every soldier Blue outnumbers is expelled (at the beginning of the battle, of course, there are no soldiers to expel) –

Ø 2 or 3 Blues on a single street expel 1 Red, Green or Yellow soldier on that street.

Ø 1 Blue stands off against 1 Red, Green or Yellow soldier on the same street.

Ø 2 Blues stands off against 2 Red, Green or Yellow soldiers on the same street. But …

Ø 2 Blues expel 2 soldiers of different colours (e.g. Red & Green) on the same street.

Ø If expulsion exposes soldiers of other colours on cross streets, Blue expels them too.

Ø Blue must expel even outnumbered Green allies on any street (this is friendly fire)

Ø Blue returns all expelled soldiers to their commanders for later re-use.

Some examples:

2. Occupation

When Blue has expelled all possible soldiers, a Blue soldier is stationed at any empty fort, ending the attack. Occupying a fort marks the end of Blue’s sortie.

Victory: An army whose last soldier is expelled retires defeated, though its ally, if there is one, may fight on. The army or alliance that drives all enemy soldiers out of the city wins.

Some hints on basic strategy

Strategy: You will learn the rules of engagement in five minutes. But the strategy, especially with two armies in each of two alliances, is surprisingly complex.

Know the terrain: The battlefield gives equal weight to each fort and each street.

Force deployment is critical. At the beginning, there are no soldiers to expel. Deploy soldiers so as to maximize their power to attack opposing armies, to minimize the risk that they themselves will be attacked, to avoid friendly fire against allied armies, and to command as much of the field as possible.

Attack is the best form of defense. The Rules of Engagement favor aggressive tactics and penalize defensiveness.

Superior force wins. On any street, outnumbered soldiers are expelled.

Choice of position is key. Look for – and eventually learn to create – chances to attack on two or more streets by the deployment of a single well-placed soldier.

Alliances can be made or broken. Armies can fight either as standalone units or as allied forces. With three armies in play, for instance, two armies can form a temporary alliance to target the third. This game can be particularly challenging, as the two trailing forces form temporary alliances to try to fight back against the dominant army.

Unity is strength: On any street, two soldiers of the attacking army defeat and expel two soldiers from different armies, but stand off against two soldiers of one army. This rule reflects the fact that a single, united force is often stronger than divided forces. It also makes the classic game between two Generals each commanding two allied armies a particularly fascinating challenge.

Friendly fire must be avoided if possible. Under the Rules of Engagement, the attacking army must always expel all soldiers from armies that it outnumbers on any street, even if those expelled are from an allied army.

Rout: In Battle, the disastrous knock-on effects of losing one or two vital units in the field are simulated. During an army’s sortie, if it expels a soldier from one street, opposing soldiers on other streets intersecting with the fort previously occupied by the expelled soldier may become outnumbered too. If so, they, too, are expelled before the attacking army stations its soldier at any vacant fort. Beware! A defeat can quickly become a rout.

Attrition: The army or alliance in the lead can pressure its opponents to concede victory at any time.

Blitzkrieg: The game can be played against the clock, with a time limit on each move. Or the whole game can be ended at a fixed time, when the army or alliance with the most soldiers in the field is declared victorious.