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Log Spirals In The Stock Market

From Technical Analysis of Stocks and Commodities Magazine - February, 1999. Reprinted with permission from Technical Analysis of Stocks & Commodities™ magazine. © 1998 Technical Analysis, Inc., (800) 832-4642, http://www.traders.com

This article is also available in PDF format.

The basic mathematical characteristic of the log spiral is that while increasing or decreasing in size, its shape remains constant. This is also true for rectangular spirals, the parameters of which are determined by their related log spirals. Many major market moves share the properties and predetermined progression of these spirals, indicating the close correlation between multiple manifestations of ordered form in Nature and the architecture of markets themselves.

Three important turning points, between 1974 and 1978, have influenced the growth pattern of many subsequent major and minor market moves, up to and including the July 20, 1998 peak and the September and October, 1998, lows for the DJIA and S&P, respectively. Some of the many moves influenced by the 1974-78 activity will be examined in detail in this article. The triangle in Figure 1 connects the 1974 low, S&P, with the 1976 peak and 1978 low, at which points both the DJIA and S&P reversed on the same day. At the 1974 low the DJIA reversed 45 days later than the S&P. The triangle connects Points 1-3. The dotted points indicate those moves which are analyzed in this article.

FIGURE 1: SPIRAL GROWTH PATTERNS EMANATING FROM POINTS 1-3, CONNECTED BY THE TRIANGLE, INFLUENCE THE GROWTH PATTERN OF MANY SUBSEQUENT MAJOR AND MINOR MOVES. The dots represent those moves examined in this article.

DATES FOR CORRESPONDING NUMBERED CHART POINTS IN FIGURE 1

1 S&P, October 4, 1974     1A DJIA, December 9, 1974     2 September 22, 1976     3 March 1, 1978     4 February 13, 1980     5 March 26, 1980     6 S&P, November 24, 1980     6A DJIA, April 27, 1981     7 S&P, August 9, 1982     7A DJIA, August 12, 1982     8 October 10, 1983     8A November 30, 1983     9 July 25, 1984     10 August 25, 1987     11 October 20, 1987     12 July 16, 1990     13 October 11, 1990     14 July 20, 1998     15 DJIA, September 1, 1998     15A S&P, October 8, 1998 When the DJIA and S&P reverse on different days, both dates are used. This is termed a Compound Pivot, explained later in this article.

THE CASE FOR ORDER

Before proceeding to the examples, it is helpful to review two schools of thought regarding the possibility of order in the markets and some properties of log spirals. Technical analysis has made amazing progress since the advent of computers, but this progress has not quieted the debate between random walk proponents and advocates of mathematically ordered markets. Random walk supporters cite the millions of subjective, individual decisions and unpredictable fundamental events as sufficient reasons for the impossibility of ordered markets. The philosophical argument for orderly markets is based on the following:

Figure 3: HALF SECTION VIEW OF A NAUTILUS SHELL. Successive increments of growth are united by a constant common ratio of expansion.

One of the most widely known examples of orderly progression in Nature is the nautilus. Figure 2 shows a half section view of this remarkable mollusk’s shell. As the animal’s growth forces the shell to increase in size, the shape of the shell never changes. The radius increases proportionately as the shell grows longer. Successive increments of growth are united by a constant, common ratio of expansion.

FIGURE 4: LOG SPIRAL OF THE NAUTILUS SHELL. A rectangular spiral, bounded by the Log Spiral, has been added.

Figure 4 approximates the spiral growth curve of the nautilus. Two perpendicular lines, similar to polar axes, have been drawn through the center of the spiral at point X. As the spiral curves around Point X it crosses the axes at a series of points labeled A through J. These points have been connected with a series of straight lines. Note that each line is perpendicular to the preceding and successive lines in the series. Therefore adjoining pairs of lines are the legs of successively larger right triangles, with segments of the axes forming the hypotenuse of each triangle. Example: Triangle EFG has legs EF and FG, and axis segment EG is the hypotenuse.
The axes segments have the same constant growth ratio as the spirals. Therefore the diagram could be turned "inside out" using the axes segments as the rectangular spiral. The values for the previous rectangular spiral segments would now become axes segments. The shape of the log spiral would remain constant.

CLASSICAL CYCLICAL ANALYSIS - ERMANOMETRY CONSIDERS MARKET TIME SPHERICAL

All markets exhibit orderly growth patterns in both time and price. Ermanometry considers time the dominant influence on market movements, and the examples herein will concentrate on time. There are always multiple growth patterns influencing the market at any given time. In classical cyclical analysis a forty week cycle might be moving up while a forty month cycle is moving down, and the analyst seeks to find the point in time when the majority of long and short term cycles are in sync; due to top or bottom together within a relatively short time span. The analysis of growth patterns is similar in that the analyst looks for that place in time where several growth patterns intersect. This is the only similarity between Ermanometry’s analysis of time and classical cyclical analysis. Classical analysis views market time as always moving forward, left to right on the chart, or plane. A new cycle, of any given length, is expected to begin where the previous cycle ends. Growth pattern analysis in Ermanometry perceives market time as spherical, and multidimensional. Growth segments are contiguous but the essence of their continuity is skewed when portrayed on the plane. Two contiguous segments may be separated by long periods of time on the plane. This phenomenon is similar to the difference in perspective that would arise if a flat map of the earth were wrapped around a globe. The obvious problem is that it would not "fit" and the resulting picture would bear little resemblance to a true picture of the earth. A spherical perspective requires that time be viewed as revolving, and analysis is done "clockwise" and "counterclockwise", so that those growth pattern segments that are not evident on the plane may be considered. As few as three segments may be used for proper analysis. This would equate to 270 degrees or three quarters of one complete revolution of the log spiral. Values for market moves may appear and reappear in different geometrical shapes. A straight line may appear as a circle or triangle, or as a series of non-contiguous straight lines.

IDENTIFYING THE SPIRALS

The starting point for uncovering spiral growth patterns is at major market pivots. "Major" is a relative term and its quantification depends upon the time frame being analyzed. Analysis of short term patterns, 15 minute bars, might require only that the search start at the beginning of the current minor trend. For macro analysis of long term trends it is necessary to begin as far back as available data permit. The examples herein will cover the current long term patterns, and the year 1974 is used as a starting point. After the great depression low in 1932, both the DJIA and S&P reached their then historical nominal price peaks in 1973. "Nominal" as opposed to constant dollar/inflation adjusted historic highs, which were reached by the DJIA in 1966, by the S&P in 1968. The lows in 1974 represent the lowest nominal prices reached after the 1973 peak and many analysts consider 1974 the true beginning of today’s macro bull market. The S&P and DJIA reached their extreme intraday lows 45 trading days apart, and therefore the 1974 bottom is a Compound Pivot: S&P, 10/04/74, DJIA 12/09/74. Both indices made important lows on both days, but though the DJIA made a lower low on 12/09, the S&P held above its 10/04 low. Since the S&P pivot came first, it is the logical starting point for identifying a macro log spiral.

FIGURE 5: SEED SEGMENTS VALUES PLACED ON SEGMENTS (DE) AND (EF) OF RECTANGULAR SPIRAL.

The first move of sufficient importance considered in identifying a macro log spiral is the advance from the S&P 1974 low to the 1976 top. On Figure 1 the heavy line connecting Points 1&2 shows this 497 day move. The second move considered is the decline from 1976 to 1978, Points 2&3, 362 days. The third move is from the 1974 low to the 1978 low, Points 1&3, 859 days. Ermanometry considers the time distance between any two important pivots as a separate and distinct move, even if it encompasses other important moves. Therefore, though the 859 days from low to low ( point 1&3) has already been considered as a 497 day advance and a 362 day decline, the 859 days is treated as a third move.

Only two moves, which will be termed "seed segments" when referring to log and rectangular spirals, can be used to initially identify growth patterns, because a constant ratio of growth must be established. The 497 and 362 day segments will be used for the first example:

Values For Rectangular Spiral Segments
AB = 139.88
BC = 192.05
CD = 263.67
DE = 362.00
EF = 497.00
FG = 682.35
GH = 936.81
HI =1286.17
IJ =1765.83

EXAMPLES

PROJECTING THE 1982 LOW

CD 263.67 DE 362.00 EF 497.00 Total 1122.67 FIGURE 6: POINTS 1, 2, & 3, ARE TERMED "SEED POINTS". All seed points are valid points from which to project forward with values derived from the spirals. When a projected point does in fact become an important pivot, that new pivot is considered part of the "spiral family", and becomes a valid point from which to project forward. Figure 6 illustrates how the sum of the 2 Seed Segments, DE & EF added to the segment 90º counterclockwise, CD, totals 1122.67 days. This is the number of days from point 3, the 1978 low, to the 1982 low. Three perpendicular lines, or 270º, become a single straight line which is the four year move. 1982, Point 7 is now a valid point from which to project forward with spiral segments. 1982 is a Compound Pivot with the indices reaching their extremes 3 days apart.

PROJECTING THE 1982 LOW

CD 263.67 DE 362.00 EF 497.00 Total 1122.67 FIGURE 6: POINTS 1, 2, & 3, ARE TERMED "SEED POINTS". All seed points are valid points from which to project forward with values derived from the spirals. When a projected point does in fact become an important pivot, that new pivot is considered part of the "spiral family", and becomes a valid point from which to project forward. Figure 6 illustrates how the sum of the 2 Seed Segments, DE & EF added to the segment 90º counterclockwise, CD, totals 1122.67 days. This is the number of days from point 3, the 1978 low, to the 1982 low. Three perpendicular lines, or 270º, become a single straight line which is the four year move. 1982, Point 7 is now a valid point from which to project forward with spiral segments. 1982 is a Compound Pivot with the indices reaching their extremes 3 days apart.

PROJECTING THE 1984 LOW

DE 362 EF 497 FG 682.35 GH 936.81 Total 2478.16 FIGURE 7: 2 SEED SEGMENTS AND 180º CLOCKWISE ADD UP TO THE ENTIRE 10 YEAR MOVE, 1974-84. In figure 5, 3 perpendicular lines "straightened themselves out" to become an important move. In Figure 6, 4 perpendicular lines, one complete revolution of the spiral, equate to an important 10 year move.

PROJECTING THE 1980 S&P PEAK

XE: 292 XF: 401.75 Total: 693.75 FIGURE 8: INTRODUCING THE USE OF AXIS SEGMENTS FOR PROJECTIONS. The Pythagorean Theorem can be used to easily calculate the value of each axis segment. Because of the "reversing inside out" character of the structure and identical growth ratio, axis segments are additional tools for projections. The 2 axis segments summed in FIGURE 3 represent the advance from the 1978 low to the S&P peak in 1980, Point 6. Please refer to the table of dates for the numbered points on the chart. Points 6 & 6A represent the peaks for the S&P in 1980, and the DJIA in 1981, 103 days apart. These points are the extreme highs for the respective indices after the 1980 low, and together they comprise a Compound Pivot.

PROJECTING THE DJIA 1981 PEAK

FH 1159 FIGURE 9: USING A SINGLE AXIS SEGMENT FOR PROJECTIONS. The 1159 day time distance value for FH exactly equals the number of days from the 1976 peak (both indices) to Point 6A, the 1981 DJIA peak. Note: Points designated with an "A" do not appear separately on the chart, and therefore the 1159 day line connects Points 2 and 6, instead of 2 & 6A. Please refer to the table of dates for the dotted points at the beginning of this article. Points designated with an "A" are parts of Compound Pivots.

PROJECTING THE 1987 PEAK

XD 213 XE 292.6 XF 401.73 XG 551.53 XH 757.22 XI 1039.6 Total 3255.68 FIGURE 10: SIX AXIS SEGMENTS CONSTITUTE THE ENTIRE ADVANCE FROM 1974 TO 1987. The error in this example is 1.32 days, within the allowed tolerance of 2 days.

PROJECTING THE 1987 LOW

BC 192.05 CD 263.67 DE 362 EF 497 Total 1314.72 FIGURE 11: FOUR SPIRAL SEGMENTS CONNECT THE 1982 LOW WITH THE 1987 CRASH LOW. Note that segments CD, DE, EF connected the 1978 low to the 1982 low. (Refer to Figure 6) The addition of one more segment counter clockwise, BC, 90º, and the total added to the 1982 low, completes the entire move from 1974 to 1987, low to low.

PROJECTING THE 1990 PEAK

GH 936.81 HI 1286.17 IJ 1765.83 Total 3988.8 FIGURE 12: USING THE LARGER SPIRAL SEGMENTS PROJECTS MORE DISTANT PIVOTS. The projection missed the actual pivot day by 1.8 days. However, as in all growth pattern analysis, multiple spirals, from many different time frames, were used to confirm this projection.

PROJECTING THE 1998 PEAK

CD 263.67 DE 362 EF 497 FG 682.35 GH 936.81 HI 1286.17 Total 4028 FIGURE 13: THE MOST RECENT PEAK PROJECTED BY THE SPIRAL SEGMENTS. The first 3 spiral segments totaled the move illustrated in Figure 6. The next 2 segments, FG and GH, total 1619 days, the move from 1978 to 1984. In Figure 6 this move is shown emanating from the 1974 low, because the seed segments were included. The addition of the successive spiral segment, HI, brings the totals to 4028 which is the exact number of days from 08/12/82 to 07/20/98.

PROJECTING THE 1978 LOW

3603.11/3.14159 = 1146.9 round to 1147 3603.11- 139.8 (AB) = 3463.3 3463.3/3.14159 = 1102.4 round to 1102 FIGURE 14: THE DIFFERENT GEOMETRICAL SHAPES OF MARKET MOVES "REINCARNATED". This example is provided to illustrate two principles of Ermanometry that have already been mentioned: 1. The DJIA & S&P are considered one market and the interaction between the two indices is very important in projecting future pivots. 2. Ermanometry considers market time to be spherical, and difficult to graphically portray on a plane. Figure 14, note that the time distance from the 1970 low to the S&P low in 1974 is 1102 days, and 1147 days to the DJIA low in 1974. Parallel vectors on the Rectangular Spiral are added together, which merely means that every fourth segment is used. If we consider that these parallel vectors actually represent the circumference of a circle that has been splintered into 5 straight lines, and we proceed to calculate the diameter of such a circle by dividing the circumference by pi: 3603.11/3.14159 = 1146.9 rounds to 1147 Remember that the basis for the spirals was the 1974 S&P low. 1147 is the time distance from the 1970 low to the 1974 DJIA low. It is possible to conclude that the 1147 day diameter gave rise to a circle that is "visible" in the segments of the Rectangular Spiral. Removing segment AB from the preceding 3603.11 "circumference": 3603.11 - 139.8 = 3463.31 If once again these segments are considered segments of the circumference of a circle, and we calculate the diameter: 3463.31/3.14159 = 1102.4 rounds to 1102 The 1102 represents the time distance from the 1970 low to the 1974 S&P low. The "diameters" representing the moves from the 1970 low to both of the 1974 Compound Pivot pivots, resulted in circles that are seen in the spiral segments. Market time continually "reincarnates" itself in related geometrical shapes. Upon reaching the 1978 low, the analyst would consider the appearance of permutations of 2 previous market moves in spirals anchored by the 1978 low, as confirmation of an important low. CALCULATING A NEW COMFIRMING SPIRAL WITH THE SAME MARKET MOVES   The two seed segments used thus far are 497 and 362. These segments are the advance from 1974 to 1976, and the decline from 1976 to 1978. A second set of seed segments is now used to calculate a new spiral.The first seed segment is the advance of 497 days, used previously. The second seed segment is 859, the sum of the 497 day advance and the 362 day decline, and therefore the time distance from the 1974 low to the 1978 low. Step 1: Calculate the ratio between 859 and 497: 859/497 = 1.72837 Inverse 1/1.72837 = 0.57858 Step 2: Place the seed segment values, 497 and 859, on segments EF and FG of the rectangular spiral. Step 3: The value for the remaining segments of the Rectangular Spiral may be calculated using the constant growth/decay ratios derived in Step 1. AB = 55.69 BC = 96.26 CD = 166.37 DE = 287.55 EF = 497 Seed Segment FG = 859 Seed segment GH = 1484.67 HI = 2566.35 IJ = 4435.6 Important Note: The constant ratio of growth, 1.72837, is 26 percent larger than the 1.37293 for the spirals already analyzed. This more rapid growth causes the spiral to expand and decrease so rapidly that it cannot be drawn to scale without quickly going "off the page". The decreasing spiral becomes so small that it is unreadable. Therefore, the graphic portrayal of the 1.37293 growth ratio will also be used to illustrate this new set of spirals. REMEMBER, IT IS NOT TO SCALE. The principles remain unchanged.

PROJECTING THE 1982 LOW

EF/FG (497/859)  =  FG/GH (859/1484.67) FIGURE 15: THE 1982 LOW IS CONFIRMED BY A SECOND SPIRAL, CONNECTING THE 1976 PEAK TO 08/09/82. A simple three term continuous proportion, A is to B as B is to C, using the constant ratio of growth and the 2 seed segments: Segment EF/Segment FG (497/859) = Segment FG/Segment GH (859/1484.67) 1484 days (segment GH) is the time distance from Point 2, 1976 peak and terminus of seed segment 497, to Point 7, the 08/09/82 low. Please note that this confirms the projection for the same day illustrated in Figure 8!

PROJECTING THE 1987 LOW

CD 166.37 DE 287.55 EF 497 FG 859 GH 1484.67 Total 3294.6 Round to 3295 FIGURE 16: RECONFIRMING THE 1987 CRASH LOW. Figure 11 has already projected the 10/20/87 time frame. The 5 spiral segments in Figure 15 repeat this projection with a total 3295 days. The actual number of days from the 1974 low to the 1987 low is 3296 days.

PROJECTING THE 1998 PEAK

AB 55.69 BC 96.26 CD 166.37 DE 287.55 EF 497 FG 859 GH 1484.67 HI 2566.25 Total 6012.5 FIGURE 17: RECONFIRMATION OF THE 1998 PEAK. From the S&P 1974 low to July 20, 1998, is 6012 days, the total of the 8 spiral segments in Figure 17. This peak was also projected in Figure 13, illustrating the 4028 day move from August 12, 1982.

PROJECTING THE 1978 PEAK

AB 55.69 BC 96.25 CD 166.36 DE 287.55 EF 497 Total 1102.85 FIGURE 18: USING SPIRAL SEGMENTS AS A CONFIRMING TOOL. In Figure 13 the 1102 day time distance from the 1970 low to the S&P 1974 was considered the diameter of a circle, and reincarnated as the circumference of a circle on spiral segments. In this example, the 1102 days appear as 5 contiguous spiral segments. The appearance of the 1970-1974 1102 day move in a spiral created by subsequent moves (1974-76-78) is a confirmation that the 1978 low day could be a very major pivot day. Figure 17 would be used in conjunction with the illustrated example in Figure 15 as additional confirmation.

PROJECTING THE 09-01-98 DJIA LOW

FG 859 GH 1484.67 FH 1715.26 Total 4058.95 Round to 4059 FIGURE 19: INTRODUCING THE PERIMETER OF A RIGHT TRIANGLE FOR PROJECTIONS. Figure 8 introduced the use of axis segments. Figure 13 introduced the concept of market moves appearing in different geometrical forms. A very common form is the right triangle. Consider right triangle FGH, Figure 19. The legs are segments of the rectangular spiral, FG a seed segment, and GH the contiguous segment and also the market move from 1976 down to 1982. The hypotenuse is axis segment FH and its value is easily computed with the Pythagorean Theorem. The perimeter of right triangle FGH is 4059, the number of days from 08/12/82 to 09/01/98, the current DJIA low.

PROJECTING THE 1982 LOW

AB 55.69 BC 96.25 CD 166.36 DE 287.55 EF 497 FG 859 Total 1961.85 FIGURE 20: INTRODUCING THE USE OF BALANCE POINTS. A Balance Point is that day which lies exactly in the middle of two "end" pivots of a Compound Pivot. Since the Compound Pivot at the 1974 low was 45 days wide, the Balance Point would be 22.5 days after the S&P pivot, and 22.5 days before the DJIA pivot. The time distance from the Balance Point, at Point 1, to the 1982 Low, 08/12/74, is 1961.5 days. Please note that this confirms the projections in Figures 6 and 15. Balance Points are used extensively in all of the Ermanometry algorithms.

PROJECTING THE 10/05/92 PIVOT

FIGURE 21: A SPIRAL SEGMENT, HI, ADDED TO THE 1982 LOW, 6A, PROJECTS AN IMPORTANT LOW IN 1992.   Earlier this article stated that many major and minor market moves were influenced by the 1974-78 market moves. This is the first example of a "minor" move, although it is minor only in relation to the other examples. A short term trader would have considered the 10/05/92 turn a major event. This pivot is 2566 days from the 08/12/82 low.

BEYOND THE BASICS

The twenty examples shown thus far are merely the tip of the iceberg of future turbulence created by the market action of 1974-78. There is a great deal more to this iceberg that causes the titanic market to continually change course, at projected pivot days. Perhaps this is a lame analogy since the "iceberg" cannot sink the market! Additional exploration can be accomplished by calculating the square root of the growth ratio between 497 and 859. The growth ratio we have been using for these two seed segments is 1.72837, and the square root of the ratio is 1.31467, which we will now use as a growth ratio, to get deeper "inside" the spiral.

497 x 1.31467 = 653.3934 : 653.3934 x 1.31467 = 859

Only the original seed segments are placed on the spiral. (Note: The spiral diagram is NOT TO SCALE.) The axes segments can be calculated, as explained previously.

USING DISCONTINUOUS FOUR TERM PROPORTION

EF 497 GH 859 EG 820.9 FH 1079.2 Total 3256.1 FIGURE 22: SUMMING THE PERIMETERS OF TWO OPPOSING SIMILAR TRIANGLES. Only the opposing triangles have been calculated and the sum of their perimeters equals 3256. The number of days from the 1974 low to the 1987 peak is 3257 days, as also projected in Figure 10.

PROJECTING THE 10/05/92 PIVOT

EX  497 XG 859 EF 820.9 FG 1079.2 Total 3256.1 FIGURE 23: RECTANGULAR SEGMENTS AND AXIS SEGMENTS CHANGE PLACES.   Earlier in this article it was stated that since the growth ratio for all segments of the diagram was the same, they could be interchanged without changing the shape of the spiral. Figure 23 is an example of switching the segments from Figure 21. Now the sum of the perimeter of triangle EFG equals 3256. Of course this type of exercise is redundant, but it has been presented to illustrate different approaches to analyzing spiral segments.

CONCLUDING COMMENTS

The markets may be appreciated for their harmonious and disciplined movements, and even as objects of beauty. However, that admiration will bring only esthetic rewards. A true belief in perfectly patterned markets should spur the reader to uncover even more evidence of order and to profit from it. It must be stressed that Ermanometry uses many other algorithms for pattern analysis, and not only do they have to confirm growth analysis projections, but spirals from different time frames must intersect at future pivot points to increase the potential of any specific date.

Articles Section

Table Of Contents

• Precise Projections with Basic Geometry
• Compound Pivots and Market Symmetry
• Log Spirals In The Stock Market
• The Quest for Accurate Data: Why it is Important
• Atop Mountains of Data - Atlantan is Unlocking The Markets Secrets

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