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 investment. Then priceaction will advance for a period of time as buyers again weighbenefits versus risks.
When studying waves of price action on stocks and commodities,the Golden Mean and its various ratios are often evident. Considerjust how often the Golden Mean can be found in science andnature. It then stands to reason that the Golden Mean can playa role in the emotional behaviour of buyers and sellers acrossfinancial markets. For example, in the human body we have onenose, two eyes and three segments to our limbs. Our arm consistsof the segment from the shoulder to the elbow, the segment fromthe elbow to the wrist and the segment from the wrist to the fingertips. Our leg consists of the segment from the hip to the knee, thesegment from the knee to the ankle and the segment from theankle to the tips of our toes. One, two and three are all relatedto the Golden Mean. As a practical exercise, measure the distancefrom your wrist to your fingertips. Divide this measurement bythe distance from your elbow to your wrist. Examine the resultingnumber and you will find a ratio of the Golden Mean at work.
Figure 1-4 illustrates weekly price action of the TSX CompositeIndex (the Index that tracks the performance of the Toronto StockExchange) from the March 2009 low to the March 2011 peak andthen to the subsequent October 2011 low. The increase from the2009 low to the 2011 peak is a move of 5471 points on a closeto close basis. The decline into the October 2011 lows on a closeto close basis was 2086 points. This decline expressed as a ratioof the increase equates to 0.381. This figure closely approximatesthe figure 0.382 which is the inverse ratio of the root of phi (Φ)raised to the 4th power. As this October 2011 sell-off was takingplace, many traders and investors who were unaware of esotericmathematical phenomena were shifting into panic mode. Inhindsight we now see that this move down was nothing morethan an orderly market retracement in harmony with science andnature.
For traders who regularly use trading platforms that comecomplete with built-in suites of technical indicators, the aboveexample will immediately be recognized as what is referred tocolloquially as a 38% Fibonacci retracement. By the end of thisChapter, I trust you will have a heightened understanding for themathematics behind the value of phi (Φ) and a new respect forhow phi (Φ) can operate in elegant harmony with the markets.
If you do not regularly use such retracements as part of yourtrading regimen, I encourage you to study price charts acrossmany different time frames. Calculate the price move betweentwo points in time and then express any subsequent retracementmove as a ratio and you will very often see phi (Φ) in some form oranother making its presence felt. You may see a ratio involving theinverse of phi (Φ) raised to a power 'n'. You may also see a ratioinvolving the inverse root of phi (Φ) raised to a power 'n'.
Figure 1-5 illustrates some examples of what you may come acrossin your chart studies. In Chapter 5, I will dig deeper into phi ()with some very special chart patterns.
The Golden Sequence
Demonstrating the construct of phi (Φ) to post-Dark-Ages societywas significant in itself. But, Fibonacci brought the concept ofphi (Φ) to the forefront of 13th century mathematical thinkingwhen he demonstrated its sequential properties. By raising phi(Φ) to a sequence of incrementally higher exponential powers, hedemonstrated the construct of what is called the Golden Sequencewhich has unique additive characteristics.
Fibonacci expressed the uniqueness of the Golden Sequence inlayman's terms with his story of two rabbits. In his story, a farmerstarts with a breeding pair of rabbits (one male and one female)in his field. After month 2, the female rabbit produces a pair ofoffspring (one male and one female). She produces another pairof offspring each and every month that follows. A given pair ofoffspring can begin producing pairs of offspring after they aretwo months old. After the first month, the farmer has only onepair of rabbits. After the second month he will have two pairs.After the third month he will have three pairs of rabbits as theoriginal female gives birth. After the fourth month, the originalfemale produces yet another pair and the female born two monthsago produces a pair, giving the farmer five pairs of rabbits. Asan interesting exercise, work this rabbit example further to seefor yourself the beauty and harmony of the Golden Sequence.
This sequence was dubbed the Fibonacci sequence by Frenchmathematician Eduard Lucas in the late 1800s.
Table 1-1 shows the first 27 terms of the sequence which goes 1, 1,2, 3, 5, 8, 13....
Note that a given term of the sequence is the sum of the twopreceding numbers in the sequence. Rounded to the nearestinteger, the expression F(n) = (1.618)n√/5 will also produce thevarious Fibonacci sequence numbers.
Column three of Table 1-1 shows how the result of dividing a giventerm of the sequence by the prior term will converge to 1.618which is phi (Φ).
As a further interesting exercise, column four of the Table takesthe individual digits of a term of the sequence and sums them.The results run the gamut from 1 to 9. Ancient societies regardedthe number nine to be sacred and representative of three trinitiesand also of perfection, balance and order. Note also that the sumof any 10 consecutive numbers is divisible by 11. Every 4th termof the sequence is divisible by 3. Every 5th term is divisible by 5.Every 6th term is divisible by 8. These divisors themselves are thecorresponding terms of the sequence. For any four consecutivenumbers of the sequence, ABC and D, the relation C2-B2 = A x Dholds.
Harmonic and beautiful indeed.
The Spiral Calendar™
While researching this book, I came across many unique applicationsof the Golden Sequence to trading, some of which tie directly toastrology. Perhaps the most elegant such treatment is called the SpiralCalendar which was developed and trademarked by former exchangefloor trader Christopher Carolan. Carolan's resulting book entitledthe Spiral Calendar is a riveting read as he demonstrates the use ofMoons as an effective way of measuring time across the history of thefinancial markets. I highly recommend obtaining a copy of his book.
To demonstrate the power of the Spiral Calendar, Table 1-2 liststhe first 27 terms of the Golden Sequence. The column third fromleft shows the square root of each of the various Golden Sequenceterms. The Spiral Calendar methodology assigns a value of Moonsto these square root terms. That is, one Moon cycle is taken as 29.5days. Multiplying 29.5 by the square root term yields the valuesin the column 4th from left. (ie the root of the third term of theGolden Sequence is 1.41. 29.5 times 1.41 gives 41.8 days).
Trading the Markets using the Spiral Calendar℣Technique
Picking up on the example of the TSX Composite Index illustratedin Figure 1-4, the calendar day count from the March 2009 low tothe March 2011 high totals 728 days. Looking at the 15th term ofthe Golden Sequence in Table 1-2, one can see it is 610. The squareroot of 610 is 24.7 Moons or 729.4 days according to Carolan'sSpiral Calendar approach. Clearly then, the TSX Composite Indexwas behaving in harmony with science and nature as it reached itspeak in March 2011. The time from the March 2011 peak to theOctober 2011 low totalled 209 calendar days. A look again at Table1-2 shows the 10th term of the Golden Sequence to be 55. Thesquare root of 55 is 7.42 Moons or 219 days according to Carolan'sSpiral Calendar approach. Once again, the TSX Composite Indexwas behaving in very close harmony with science and nature as itdeclined into its October 2011 low.
CHAPTER 2
Astrology is a language. If you understand this language,the sky speaks to you.
Dane Rudhyar-astrologer
The Power of Planetary Transits
There are many ways to apply astrology to the trading of thefinancial markets. Some traders look for significant price pointsto align with Full Moons or New Moons. Some look for variousaspects between planets to likewise align to significant swings inprice. In my recent book The Bull, the Bear and the Planets, I offermany examples to show the reader the power of astrology.
But, there is one astrological technique that continues to amazeme with its awesome power. Each time I use this approach, I findmyself pausing to reflect on the harmony of the markets withscience and nature. What I refer to is the plotting of planetarytransit lines on top of price charts for stocks and commodities.
The Wheel of 24
In order to overlay transit lines on price charts, one must becomecomfortable with the notion of expressing degrees of planetarymotion (as determined from the zodiac wheel) in terms of price. Atfirst blush, the visceral instinct is to say that degrees of motion arein no way related to price. But, thanks to the works of W.D. Gannthere is a way to convert degrees of motion to price and it is calledthe Wheel of 24.
Figure 2-1 illustrates the Wheel of 24. The image for this Figurehas been gleaned from a 1980s publication by trader and authorJeanne Long. Long took the basic concept of the Wheel of 24 asposited by Gann and trademarked it as her Universal Clock.
The Wheel of 24 (Universal Clock) is divided into an innerring and an outer ring. The inner ring displays the numbers 1through 360, with these numbers advancing by an increment of1 in a counter-clockwise direction. These numbers comprise 15revolutions of the twenty four wheel segments. The figure of 360is also a very close approximation to the 365 days of our calendaryear. The outer ring presents price data. From the 90 degreeposition where the number 1 appears, move right until you hitthe outer ring. This is where the first entry of the price data willappear. Generally, the price data entries will be structured so thatfrom lowest to highest the outer ring data takes into account theentire price range of the stock or commodity over the past 365days. The price data advances by a suitably chosen increment sothat in about 8 or so revolutions of the Wheel of 24, the desiredprice range can be expressed.
You may find yourself pausing at this point to deeply ponderthe Wheel of 24 as it is a unique concept for many traders andinvestors that can stretch the mind and the sensibilities.
With price data placed on the Wheel, the next step is to use theWheel to help create planetary transit lines. In Gann's day, datawould have to be entered on a blank copy of a Wheel sketched ona piece of paper—a time consuming exercise indeed. Thankfullytechnology now makes working with this Wheel a bit easier. Thereare software programs available that provide you with nice neatcircular printouts of this Wheel. However, these software programsretail for up to $3000. I have avoided buying these expensiveprograms, opting instead for the simplicity of a Microsoft Excelspreadsheet.
