CHAPTER 1

The house which King Solomon built for the Lord was sixtycubits long, twenty cubits wide and thirty cubits high. Thevestibule in front of the nave was twenty cubits long equalto the width of the house and ten cubits deep in front ofthe house. 1 Kings 6: 2-3, The Bible

The Golden Mean

The Dark Ages

For thousands of years, mankind has been fascinated with themathematical relationships between numbers. Some of thesenumerical relationships are so intriguing that over time they havecome to be regarded as sacred and treasured. But the treasureddocuments underpinning these mathematical relationships nearlycame to be lost in 410 AD when the Roman Empire collapsedafter Alaric and his Goths sacked the city of Rome. The legions ofsoldiers and elegant architecture of Rome gave way to a tide ofbarbarians and the Dark Ages set in across the lands. Pursuit ofknowledge gave way to pursuit of hostility.

Enlightenment

In 600 AD, a society known as the Arabs left Mecca (located inmodern day Saudi Arabia) under inspiration from their leaderMohammed. They ransacked places like Damascus, Jerusalemand Alexandria taking with them not only the usual spoils ofwar but also knowledge in the form of old Greek manuscripts.To this ancient Greek knowledge, they added the arithmetic andastrological knowledge of the ancient Hindus. By 650 AD, Baghdadhad grown to become the cultural epi-center of the East.

Meanwhile, in the West, the remnants of the former RomanEmpire came to be reconfigured into the Kingdom of Franciawhich encompassed much of modern day Europe. In 768 AD, anenergetic, charismatic leader by the name of Charlemagne wascrowned Emperor of Francia and immediately revived the pursuitof learning and knowledge. The Dark Ages that had spread acrossthe former Roman Empire were finally over.

The pursuit of knowledge was given further impetus in 1000 ADwith the election of Pope Sylvester II who revived interest in theseven liberal arts (grammar, rhetoric, logic, arithmetic, geometry,music and astronomy) across the Christian World. Between 1000AD and 1100 AD, East met West as Arabic knowledge melded withChristian desire for learning. Ancient Greek mathematical workslike Euclid's Elements were translated into Latin along with variousother ancient works.

Filius Bonacci and phi

In 1170 AD a son was born in Pisa to an Italian merchant and hiswife. The merchant's name was Bonacci and the son was namedLeonardo. In the language of the day, 'filius' meant 'son of' andbefore long young Leonardo became known as Filius Bonacciwhich became shortened to Fibonacci. Leonardo spent muchof his youth in Barbary (modern day Spain) where his fatheroperated the Customs House. Leonardo had the great fortune togain exposure to much of the old Greek and Arabic mathematicalknowledge while spending time in Barbary. In 1202, he publishedthe now famous Liber Abaci in which he demonstrated how tosolve quadratic equations. Leonardo also became proficient inPythagorean mathematics and Euclidean geometry. One of thegeometrical constructs Leonardo focused on was the GoldenMean.

Figure 1-1 shows a rectangle divided into two parts (part 'a' andpart 'b').

There is only one point where this rectangle can be divided intoparts 'a' and 'b' such that the following ratio holds:

y /x = x/(y-x)

To solve this equation one of the variables must be eliminated. So,let x=1. The expression above then becomes:

y = 1 / (y - 1)

Multiplying both halves of this expression by (y-1) yields thequadratic expression:

y² - y - 1 = 0

The one and only viable solution to this quadratic expression is:

(1 + √5) / 2

Solving this expression yields 1.618, the Golden Mean, also calledITLφITL and denoted by the symbol Φ.

Phi (Φ) is elegant and mystical.

Of the mathematical relations known to Greek and Arabicmathematicians, phi (Φ) was probably the most powerful. Hence,its esteemed status as a sacred mathematical term.

The number 1.618 (phi or Φ) derives its name from 5th centuryGreek sculptor Phidas who used it in creating the proportions ofthe 9 meter high Athena Parthenos statue and the 13 meter highstatue of Zeus in BC 430. But, as English explorer Howard Vysenoted in 1837, the ancient Egyptians also understood the conceptof phi (Φ) long before the Greeks. Vyse observed that the angleof inclination of the pyramid of Cheops is 51 degrees, 51 minutes.Vyse calculated the trigonometric relation 'tangent' of 51 degrees,51 minutes and arrived at 1.273 which he noted to be the squareroot of phi (Φ).

When contemplating phi (Φ), it is interesting to study the inverseratios of it. Figure 1-2 presents some of these ratios. Notice howthese numbers increase by the multiple of 1.618. That is, 0.146 x1.618 = 0.236 and so on.

It is likewise interesting to study the inverse of the square rootsof phi as shown in Figure 1-3. Notice that these numbers increaseby a multiple of 1.272 which is the square root of phi (Φ). That is,0.236 x 1.272 = 0.300 and so on.

Using the Inverse Ratios as part of a Trading Strategy

Stock prices, commodity prices and index values move in distinctwaves. In a rising market, buyers—driven by emotion—bidprices higher until the short run marginal benefit of owning theinvestment is outweighed by the short run marginal economicrisk of buying it. Price action then will recede for a period of timebefore staging another advance. In a falling market, shareholderssell until price reaches a point where emotion changes and morebuyers than sellers are attracted to the...