Secretly Conforming to the Rule of the Golden Section

[A mathematical formula showing the derivation of the figure 0.618 omitted]

In the 2,500 years since then, this formula has been considered by formative artists as the golden rule of aesthetics. As convincingly testified to by the history of arts, almost all artistic works considered masterpieces, whether created in a casual manner or through intentional effort, have all been close to or in accord with this formula in their basic aesthetic features. People had long marveled at the beauty of the Parthenon Temple of ancient Greece, suspecting it to be the creation of a god. With measurement and calculation, it was found that the relationship between its vertical lines and horizontal lines were entirely in accord with the 1:0.618 ratio.

In his book Vers Une Architecture, the great modern architect Le Corbusier also established his most important theory of “basic design scale” on the basis of the rule of the golden section, a theory which has had profound and extensive influence on architects and architecture in the world. Regrettably, this formula which the Creator may have meant to use for revealing to mankind a rule for all spheres through a demonstration in one sphere has never moved beyond the realm of artistic creation. Except those Muses with extraordinary gifts, almost no one has realized that this golden rule of aesthetics may become, or is, a rule that should also be followed in other spheres.

It was not until 1953 that J. Kieffer, an American, discovered that seeking experiment points according to the rule of the golden section would make it possible to reach the optimal state the most quickly. His discovery was refined by the Chinese mathematician Hua Luogeng and turned into the “optimum seeking method,” or the 0.618 method. The method was popularized in China for a time. As far as we know, such a popularization campaign based on the human-wave tactic produced little effect, but this episode demonstrated the prospect of applying the rule of the golden section in spheres other than the sphere of arts.

In fact, before the emergence of the notion of consciously grasping the rule of the golden section, people had repeatedly applied it to their own spheres of practice on the basis of their instincts. Of course, the military sphere had not been left out. We can easily see the ephemeral marks of the paws of this mysterious beast in the famous amazing campaigns and battles in the history of war. Without looking afar, you will see examples of conforming to this rule everywhere in the military realm. The shadow of 0.618 can be seen in such things ranging from the arc of the cavalry sword to the apex of the flying trajectory of a bullet, shell, or ballistic missile and from the optimum bomb-release altitude and distance for an aircraft in the dive bombing mode to the relationship between the length of the supply line and the turning point in a war.

By casually reading pages of the history of war, you will be certain to be silently amazed by the fact that 0.618, like a golden belt, can be faintly seen in ancient, modern, Chinese, and foreign wars. In the Yanlin battle between Jin and Chu during the Spring and Autumn Period, Duke Li of Jin led a military force in attacking Zheng. The Jin force had a decisive battle with the Chu force at Yanlin. Adopting advice made by Miao Penghuan, a defector from Chu, Duke Li used a portion of his middle army to attack the left army of the Chu force, used another portion to attack the middle army of the Chu force, and used the upper army, lower army, new army, and forces of the lords to attack the right army of the Chu force. The point of attack selected was exactly at the point of the golden section. We mentioned above the Battle of Arbela between Alexander and Darius. The Macedonians selected the juncture of the left flank and the centre of the Persian force as the point of their attack; marvellously, the point was exactly the “golden point” for the entire front.

For hundreds of years, people have found it difficult to understand why the Mongol cavalry of Gengis Khan were, like a hurricane, able to sweep across the Eurasian continent. Such factors as the barbarians’ truculence, cruelty, and cunningness or the mobility of the cavalry did not provide convincing explanations. Perhaps there were other more important reasons ? As can be expected, the rule of the golden section showed its miraculous power again: We can see that the battle formation of the Mongol cavalry was different from the Western traditional phalanx. In regard to their five-row formation, the ratio of heavy cavalry to light cavalry was 2:3, with 2 for armoured heavy cavalry and 3 for fast and mobile light cavalry, that is, another example of the golden section! You have to admire the genius-level understanding achieved by that thinker on horseback. It was natural for a force under the command of such a commander to have more striking power than the European forces that it confronted.

It seems that, while highly gifted in applying the rule of the golden section to religion and arts, Christian Europeans were late in coming to understand the application of this rule to other spheres. The Dutch general Maurice, who had been the first to transform the traditional phalanx by mixing similar numbers of musket-armed soldiers and pike-armed soldiers, failed to realize this point even in the black powder period when muskets were gradually replacing pikes.

It was King Gustavus of Sweden who adjusted this formation of a strong front and weak flanks, thereby turning the Swedish army into an army with the strongest combat power in Europe of that time. What he did was to have an additional 96 musket-armed soldiers in addition to the squadron composed of 216 pike-armed soldiers and 198 musket-armed soldiers. This change gave immediate prominence to the use of firearms, thereby becoming the watershed separating battle formations of the periods of cold weapons and hot weapons. Needless to say, we again saw the shining light of the rule of the golden section in the ratio of 198 plus 96 musket-armed soldiers to 216 pike-armed soldiers.

There is still more. Let us see how it had stubbornly “manifested” itself to give us clear suggestions before we recognized it as something more than a rule of arts. Napoleon attacked Russia in June 1812. In September, after failing to eliminate effective Russian forces in the Borodino battle, he entered Moscow. At that time, Napoleon did not realize that his genius and luck were gradually leaving him bit by bit, and that the peak and turning point of his lifelong career were approaching simultaneously. A month later, the French forces withdrew from Moscow as it snowed heavily. There were three months of victorious advance and two months of declining. It seems that, in terms of the time sequence, the French emperor was standing on the line of the golden section when looking down at the city of Moscow through the burning fire.

In another June 130 years later, Nazi Germany started the Barbarossa Plan against the Soviet Union. For as long as two years, German forces maintained their offensive momentum. It was in August 1943 that German forces turned into Defence at the conclusion of the Castle action and would no longer be able to launch an action that can be called a campaign against the Soviet forces. Perhaps we also have to call the following fact a coincidence: The battle of Stalingrad, which has been considered by all historians of war to be the turning point in the Soviet Patriotic War, happened exactly in the 17^th month of the war, that is, November 1942. This was the “golden point” in the time axis encompassing 26 months during which the German forces turned from booming to declining.

Let us also take a look at the Gulf War. Before the war, military experts estimated that the equipment and personnel of the Republican Guards would basically lose their combat effectiveness when losses resulting from aerial attacks should total or exceed 30 percent. To make Iraqi forces’ losses reach this critical point, US forces extended the bombing time repeatedly. When the Desert Sword was taken out of its sheath, Iraqi forces had lost 38 percent of their 4,280 tanks, 32 percent of their 2,280 armoured vehicles, and 47 percent their 3,100 artillery pieces, and only around 60 percent of the strength of the Iraqi forces was left. Through such cruel data, the mysterious light of 0.618 began to flicker again in the early morning of 24 January 1991. The Desert Storm ground war ended 100 hours later.

Such instances scattered across history have truly been something marvellous. When viewed in isolation, they do look like accidents happening one after another. But the Creator never does anything without a reason. If too many accidents demonstrate the same phenomenon, can you still calmly view them as accidents ? No, at this moment, you have to admit that there is a rule here.

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