Extreme Case Sampling on Predictor Variables: A Powerful Qualitative Exploratory Research Method

May 21, 2016

Abstract

Qualitative analysis of extreme cases selected from a sample of thousands of data often provide knowledge discovery in a highly efficient, elegant, and powerful manner. This qualitative exploratory research method is frequently overlooked. It is especially useful in cases where (a) a predictor variable is believed to have a linear or non-linear continuously increasing or decreasing effect on an outcome variable, (b) potential predictor variables are well-defined but potential outcome variables are unclear, difficult to measure, or difficult to assign values to, (c) a “double sampling” procedure can be performed where a minimum of 3 cases to a maximum of 5% of cases from a sample of at least 100 subjects that is representative of some population can be obtained, (d) scores for a predictor variable or composite of predictor variables can be assigned to all of the 100 or more subjects from which either the highest or lowest scores can be selected as the cases to study, (e) there are one or more theories or models that can be evaluated by a person or persons with content expertise, (f) these theories or models have not already been confirmed with more definitive quantitative studies, and (g) the possible relationships between predictor variables and outcome variables may be complex and/or there are a large number of potential predictor variables. Even if all seven of these conditions do not exist, studies of the most extreme cases on the predictor variable, whre the extreme cases are selected frm a sample of thousands of data, may still be a very effective research design.

Introduction: Knowledge Discovery in an Era of Big Data

One task of a research methodologist is to select from a very large toolbox of potential research methods the research methods that are most appropriate for a particular question that the researcher has. The research methodologist carefully considers the nature of the question, the specific goals that the researcher has, and the available data and resources that can be applied to answering the question.

We are now in an age where often there is a large amount of data and there are suspected relationships within the data. These suspected relationships within the data may be based on anecdotal evidence or hunches. The researcher wants to explore the possibility that some of these suspected relationships may actually exist. The researcher may point to a few cases where some particular predictor variable may have a relationship to some particular outcome. Precisely what the outcome is and how to measure the outcome may not be clear.

Consider, for example, this scenario: a graduate student is a believer in astrology. The student’s advisor regards astrology as superstition and nonsense but the advisor agrees to explore the possibility of conducting research to determine if relationships in the data can be found.

The student informs the advising professor that:

1. He has about 5,000 birth data from a birth certificate or birth record for famous people and biographies of these people are readily available online.

2. It is believed that the zodiac sign makes Aries pioneering, Taurus stubborn, etc.

3. The student would like to validate that these relationships exist. The advising professor is now taking an interest in this study because he will be happy to see that these supposed relationships are nonsense. Thus, the student and the professor see that a public good can be derived from the research: the student sees the opportunity to open up a new academic field with a pioneering study that confirms an astrological principle, and the professor see the opportunity to provide strong envidence that can help remove the widespread proliferation of superstitious and unhelpful ideas.

The question for the research methodologist is what research design is most effective in building a model to confirm that these relationships exist.

Extreme Case Sampling Defined

Extreme case sampling is simply selecting the most extreme subjects from a large sample on either the predictor variable or the outcome variable. Thus, extreme case sampling may be a double-sampling method where first sample data is obtained from a population and the most extreme cases from this data are used in the analysis. Alternatively, the extreme cases may be selected from the entire population.

Using our scenario of an astrological study, suppose that a person has both Sun and Moon in Aries. This is more Aries than a person who has only Sun or Moon in Aries. By assigning points to a planet in Aries, we can obtain a composite Aries score for each of the 5,000 people in the database. The person with the highest number of points is the most extreme Aries. The individuals who scores highest in Aries provide provide extremely important information that can determine whether the astrological theory is viable, as described below.

Extreme case sampling on the outcome variable is also possible. If we have a way to measure the personality trait of independence, then the person with the greatest independence, then we can select extreme cases based on this expected outcome variable. In the scenario presented here, obtaining scores based on the predictor variable is very simple whereas obtaining scores based on an outcome variable is likely to be much more difficult. Even if a measurement of the outcome variable can be obtained, there are likely to be advantages to sampling on the predictor variable.

A Good, But Imperfect, Sample is Adequate

Let us suppose that in our example scenario that there are about 5,000 birth data to analyze and the data was collected by astrologers in order to have access to data of celebrities, leading politicans, and other famous people. The data was no collected without regard to supporting or defending a theory so obvious selection bias that favors particular astrological beliefs is not a concern. This scenario is typical of existing databases in that the data is very useful but may not be completely beyond any possible bias because it was not collected in a rigorous way that would virtually make bias impossible.

The requirements for exploratory research are not as rigorous as for a hypothesis test. In this exploratory research we are seeking strong evidence that either the astrological hypothesis is valid or not valid. We are not seeking the convincing evidence that a replication of a hypothesis test provides. We are making a large incremental step from anecdotal evidence to a method where most selection bias, and in our sample scenario most likely all seelction bias, is removed by the fact that the researcher does not have the liberty to personally select the data. A database of 5,000 chart data that is available through existing commercial software is used. The scenario that is presented here is typical of many scenarios, such as where databases of student performance, etc. are available in the area of educational psychology. Also, even though our database was not collected using random sampling or systematic sampling, if evidence for the astrological hypothesis is not found through the analysis of extreme Aries individuals, the rejection of the astrologicl hypothesis is very strong because any bias which might exist, even if only remotely possible, would favor the astrological hypothesis.

Thus, if we can demonstrate that Aries does not incline people to be more independent than other people, then we can reject the astrological hypothesis. If certain conditions are met, we can reject (or fail to reject) with good confidence the astrological hypothesis with the available database and without conducting a hypothesis test and without the need for a control group. As we shall see, these conditions are met in the scenario presented in this paper.

An Important Requirement: A Continuously Increasing or Decreasing Graph

Extreme case sampling is a useful research method only if the relationship of the predictor variable and the outcome variable is hypothesized to be continuously increasing or continuously decreasing. This is a critically important requirement and a violation of this requirement results in extreme case sampling being an inappropriate research method.

In our hypothetical scenario, the graduate student informs us that having many planets in the zodiac sign Aries increases the Aries personality trait of independence. It is not critically important whether the relationship is perfectly linear. Perhaps having a fifth planet in Aries as opposed to four planets in Aries makes very little difference but does slightly increase the inclination to be independent. As long as a line graph with the predictor variable along the x axis and the outcome variable along the y axis is expected to gradually go up (or gradually go down) without any peaks and valleys, then the person with the most Aries is also the person expected to be the most independent.

The hypothesized relationship fits into a regression model where perhaps the weights of the individual components of the composite predictor variable score are unknown but all of these components are added together and none are subtracted. For example, perhaps Sun in Aries is more important than Mercury in Aries so a regression equation would assign a greater coefficient to the Sun sign variable than to the Mercury sign variable. We do not need to know with precision what these coefficients are to derive good information from our extreme case sampling study. We can also create alternative composite predictor variable scores using different guesses for coefficients to test different models. Interactions of these components are also possible as long as the interaction of variables is positive. In practical terms with our scenario, if a person has Sun, Moon and what astrologers call the Ascendant in Aries, the person is an extreme Aries. Virtually all astrologers agree with this. Variations of the formula can be created to test variations of the theory. Thus, we are able to identify with very good confidence individuals who are extreme on the predictor variable. Keep in mind that we are conducting exploratory research to determine if a theory is viable. We are not conducting a rigorous hypothesis test because we do not need to invest the great resources needed to determine if the astrologial hypothesis is viable. As long as content experts (i.e., professional astrologers in this scenario) agree that the selected people based on the given criteria are exremely strong in the trait, we can proceed with the extreme case sampling research procedure to provide strong indications of whether the suspected relationships actually exist.

The power of extreme case sampling derives from the tremendous impact that extreme cases have on a correlation. All statisticians are familiar with the huge impact of extreme cases on a correlation. Note that we should not assume that these extreme cases are outliers. We have almost no assumptions about the distribution of the data and the extreme cases may not be outliers in the sense of being clearly separate from the majority of the data which lies near the middle of the graph. Using the data in our scenario to illustrate the point: if the most extreme Aries people are not independent, we can reject the astrological hypothesis. Because of the enormous impact by extreme cases on a correlation, there would need to be an enormous and extraordinarily unlikely tendency of the rest of the data to have a very strong correlation of Aries with an independent personality. Thus, we can reject the astrological hypothesis by analyzing only a few cases and we can generalize from these cases to an entire population.

Again, keep in mind that this is an exploratory research method that can be conducted with extremely few resources in order to make an enormous incremental step in model building beyond what is achieved with anecdotal evidence. The goal is to develop building blocks beyond what is possible with even a massive amount of anecdotal evidence by subjecting the hypothesized relationships to a procedure that removes selection bias of the researcher and evaluates the viability of the proposed relationship.

A Summary of Key Points and Additional Important Considerations

What seemed to perhaps be an intractable question because of the enormous resources needed to address the question has become a relatively easy question to answer given the following critically important characteristics of this scenario:

1. There is an available database of data that provides predictor variable data with very low measurement error. In our scenario, this is a database of about 5,000 famous people with recorded birth times and a large amount of biographical information online. A score for Aries (or Taurus, Gemini, etc.) can be computed by adding the number of planets in the zodiac sign and optionally weighting some planets more than others. Content experts (in this case, professional astrologers) are needed to confirm that this formula is reasonable. In the case of our scenario, the extreme people will have so many planets in the zodiac sign that virtually every astrologer will agree that these are extreme cases.
Note that extreme case sampling is a kind of “double sampling” procedure. First, we have some database that may be an entire population but more often is sampled from some population. Then we sample from this sample by assigning scores to each subject in this population based on our predictor variable and we select some number of extreme cases for our study. The number of cases selected for the study is discussed in item #3 below.

2. The relationship of the predictor variable and outcome variable is continuously increasing or decreasing.

3. The number of extreme cases that are selected to analyze should be at least three. In many cases the hypothesized relationship can be rejected after analyzing only the three most extreme cases. For example, we find that the three most extreme Aries individuals do not exhibit independence in any identifiable way more than other people. We then reject the astrological hypothesis or change the astrological hypothesis to fit the findings. We might, for example, propose that the independent quality is so subtle, psychological or internal that only a psychological test or interviews can reveal it. Such alternative explanations tremendously erode the original hypothesis. The researcher should set a cut off point for how much data will be analyzed before analyzing the data! Otherwise there is a temptation to cut off after some number of data that fits the hypothesis. The researcher should set the number to at least three. The primary determinant of the cut off point is the time and energy that the researcher is willing to commit to the study. In the given scenario where biographies of the famous people need to be studied and sometimes works that they produced analyzed, a researcher with a limited amount of time may wish to study only the top three cases. Studying more than the top 5% (25 in the case of 5,000 data) is not advised because the greater the number of data analyzed, the less extreme some of this data is.

4. The outcome variable may be difficult to assess. Whether a person is independent or not is difficult to determine. In what ways is the person independent? One of the great benefits of extreme case sampling is that we are able to refine and make much clearer what the outcome is! Extreme case sampling assists us in developing a viable model. In my personal experience conducting dozens of extreme case sampling studies, the typical finding is that a more clearly defined outcome variable emerges from the study. The predictor variable is already clearly defined and now the outcome variable becomes much more clearly defined as well. Future studies and anecdotal evidence are much easier to evaluate because we have a much more clearly defined outcome variable rather than an outcome variable that is very general and ambiguous. Extreme case sampling provides an enormous step towards narrowing the scope of possibilities so that future studies can determine whether the scope narrows down to a clear relationship or to the discovery that there is no relationshp of the variables and therefore the hypothesis can be rejected.

5. There may be various outcome variables that are explored. Extreme case sampling is an exploratory research method that helps us determine what models are viable. We have a choice of whether we want our research to entertain a wide range of possible outcomes and be much more exploratory in nature of whether we have a more specific outcome that we expect. By focusing on a more specific outcome we move more quickly to a confirmation of our hypothesis but we miss the opportunity to notice outcomes that may be a bit different from what we expected. Do we expect Aries to incline towards self-employment, strong personal opinions, aggresssiveness, or any of the above? The wider the net so to speak, the more likely we are to find some relationship but the weaker is our confirmation of a relationship. Regardless of how wide the net is, extreme case sampling enables us to draw much more reliable conclusions about what kinds of relationships may exist between the variables, if any, than through anecdotal evidence gained through convenience sampling. Unlike anecdotal evidence, extreme case sampling also helps develop a body of literature that future researchers can build from.

6. The extreme cases must have high scores or ratings on the outcome variable. If they do not, you must reject the the hyothesis. You may radically alter the hypothesis to make it compatibile with the findings but the hypothesis as stated is rejected with very good confidence.

7. Extreme case sampling is particularly helpful when the relationship of predictor variables and outcome variables is complex or there are many competing theories. If a relationship between variables has already been determined by previous research, then a research method that can draw stronger conclusions, like a hypothesis test, may be appropriate. However, when relationships between variables have not yet been clearly determined, hypothesis tests are “shots in the dark” that usually miss the mark. Extreme case sampling can sort out a tangle of data into promising connections and also remove from consideration many relationships. If Aries has no relationship to independence, then it is extremely unlikely that an independent trait will be found in the top 3 or more extreme Aries people. At best, the researcher will need to narrow done to some particular definition of independence that is tremendously more specific and easier to test than the much more general notion of independent personality that was initially proposed.

Advantages of Sampling on the Predictor Variable

By sampling on the predictor variable, rather than the outcome variable, we are not limited to narrowly defined outcomes. We can consider a wide range of possible outcomes. The potential for knowledge discovery is enhance as we explore possible relationships. The extreme cases based on the predictor variable are believed to have some effect but the exact nature of the effect does not need to be known and unexpected relationships may be discovered. By qualitatively analyzing these extreme cases we have also opened the door to discovering relationships that are not easily quantified. Thus, this research procedure has cast the widest possible net for finding any possible impact from the predictor variable. If no viable outcomes can be proposed from these extreme cases, then we have obtained very strong evidence that the belief that these variables are predictors of any kind of outcome is very strong.

A failure to find any outcomes associated with strong Aries in our hypothetical scenario suggests very strongly that the belief in Aries is superstitious. This conclusion is much stronger than a negative finding from other forms of research which may fail because (a) there are outcomes that were not investigated, (b) the measurement of the outcome suffered from too much measurement error to detect the outcome sufficiently, or any other problems that might arise from the limitations of selecting particular outcomes and attempting to measure them.

Success in finding some traits or behaviors that are shared among these thexe extreme cases provides the researchers with ideas, hypotheses, or theories that have the potential to be verified in future research.

Extreme Case Studies that Do Not Use Extreme Case Sampling

Not all extreme case studies use extreme case sampling. For example, an astrologer may have a client who has several planets in Aries. Conclusions derived from the analysis of these anecdotal extreme cases are much weaker than conclusions derived from extreme cases that are sampled from a database. The extreme cases that are anecdotal cases can very easily be influenced by selection bias. The astrology may, for example, may be more responsive to cases that confirm a belief and consciously or unconsciously avoid analyzing extreme Aries cases that contradict the astrological belief. Therefore, the term “extreme case sampling” is preferred to “extreme case study” in describing this methodology because not all extreme case studies meet the criteria for being able to draw strong evidence from the study.

Conclusion

There are particular research questions that extreme case sampling is remarkably well suited for. However, extreme case sampling often gets ignored as an important research method. For example, belief in astrology is pervasive with most surveys indicating that about 1/3 of the population of many countries believing in astrology. Those who believe that astrology is a harmful superstition could use extreme case sampling to demonstrate that the ideas are not valid, and the believers could use extreme case sampling to build viable models. The resources needed for extreme case sampling are readily available in the area of astrology and in many other areas. The scenario presented in this paper is is only one example of a situation where extreme case sampling is a brilliant tool in our toolkit of research methods. Extreme case sampling is often overlooked, and researchers in many fields are likely to find that extreme case sampling needs to be resurrected as an invaluable research method that has gotten buried beneath the other valuable tools in our research methods toolkit.


Two Big Astrology Conferences in early 2017: the Kepler Conference and the NCGR Conference.

September 17, 2015

In early 2017 there are two major astrology conferences on the east coast of the USA:

1. The first annual Kepler Conference is in January, 2017 in Cape Canaveral, Florida. See http://www.TheKeplerConference.com for more information.

2. The “Many Faces of Astrology” NCGR Conference in February, 2017 in Baltimore, Maryland. See http://www.geocosmic.org for more information.

Several people have asked why two astrology conferences are within a month of each other and both on the east coast of the USA?

The answer is that the organizers of the Kepler Conference did not know about the plans of NCGR to have a conference in February, 2017 and the plans had already been set and work begun on having the Kepler Conference in January 2017 when news was received that the NCGR Conference would be only one month later. Of course NCGR did not know about the Kepler Conference because this is a new conference.

Major astrological organizations have traditionally avoided giving conferences at a similar time and this is a postive tradition that shows respect and appreciation for the other organizations, and also has the practical benefit of not lowering attendance to the conferences by people having to choose which one to attend. Of course some NCGR board members and supporterswere concerned and upset that a new conference would be offered so close in time. However, upon learning of how this situation has come about, NCGR was very gracious and understanding.

I am an adviser to NCGR as well as the person who originally came up with a concept for a conference that evolved into becoming the Kepler Conference. I regret that I did not ask NCGR board members what the plans for the next NCGR Conference were. I was so out of touch with NCGR activites that I nearly missed the deadline for the call for papers to speak at the NCGR Conference. Thankfully, the good people at NCGR are very understanding and supportive. I must say that I am very impressed by the generosity of spirit, kindness, and good will of all of the NCGR board members! This is impressive and NCGR members can be proud to participate in an organization that has its eye on the greater good of astrology.

We will be sending out flyers of the NCGR Conference as well as the Kepler Conference. I am very happy to have been selected as a speaker at the NCGR Conference, we will have a booth at the NCGR trade show. I am also very happy to be speaking at the Kepler Conference and to be giving the keynote address.

Both conferences will be excellent events!


Why the name “Kepler Conference”?

September 17, 2015

The dream of the Kepler Conference is the same dream Johannes Kepler had: to discover reliable astrological rules. The search is not easy. Many of the ideas accepted as facts in astrology are most likely not correct. Also, some astrologers believe that astrology lies somewhere outside the realm of science, but the dream of the Kepler Conference is the same dream that Kepler had over 400 years ago. Kepler stated this passionately and eloquently:

“It should not be considered unbelievable that one can retrieve useful knowledge and sacred relics from astrological folly and godlessness. From this filthy mud one can glean even an occasional escargot, oysters or an eel for one’s nutrition; in this enormous heap of worm-castings there are silk-worms to be found; and, finally, out of this foul-smelling dung-heap a diligent hen can scratch up an occasional grain-seed, indeed, even a pearl or a gold nugget.”
Tertius Interveniens by Johannes Kepler

For my entire adult life, I have been obsessed with the same vision that Kepler had. In the 1970’s I attended astrological conferences where others shared this same dream. The dream faded in the astrological community in the 1990’s, as astrology was embraced more as a metaphorical and mythological language, or a form of divination beyond the domain of science.

On the other hand, a few astrologers, myself included, continued with our consistent and unwavering passion to follow Kepler’s dream. This dream has propelled me on my own quest to 1) put into software as much of the entire universe of astrological thought and astrological techniques as possible, (2) to learn celestial mechanics, (3) to master research methodology and statistics, and (4) to continually study the ever-widening range of astrological books and traditions.

I have noticed that in recent years, this passion for a marriage of astrology and science is reviving in popularity and once again, we see increasing enthusiasm about discovering what astrology really is. The first annual Kepler Conference is an important step forward in this new movement. At the Kepler Conference, we can all learn about new discoveries and breakthroughs in research, while meeting with the people who are doing this important and exciting work.

My obsession with Johannes Kepler’s work permeates every aspect of my life. It is so pervasive and deep that three astrologers, all of whom are very intelligent and not prone to flights of fancy, actually believe that I was Johannes Kepler in a past life! Despite my protests that this is not true, they are nevertheless convinced. I do accept the idea that if there are past lives, I may have known Kepler. I have not mentioned this before in public because of the hubris implied, but I share it now to emphasize that the choice of the word “Kepler” is not just one among many clever words that can be used for an astrological institute or product.

In my mind “Kepler” is the purest and most perfect word for an astrological product or institute. All of us have our favorite colors and some of us have a hero or a favorite sports team. For my entire life, the guru of my professional pursuits has always been Kepler. Kepler is to scientific astrology what Shakespeare was
to Elizabethan theatre, or possibly even what Mary is to the Catholic Church. Therefore we should expect that many astrological organizations will want to identify with the name “Kepler.”

In fact, “Kepler” is already used widely throughout the astrological community. My first software program that I began developing in 1982 is named “Kepler” and the Kepler software continues today to grow in power and flexibility. The number of Kepler users also continues to grow rapidly throughout the world. In the late 1990’s my wife started developing a Kepler School of Astrology, but then, Kepler College opened, so we changed our school’s name to the Avalon School of Astrology. There is also the Kepler Institute in Belgrade, Serbia; another example of an astrological institute that uses the name “Kepler”. There has even been a “Kepler Conference” held periodically in England.

Some people have suggested that using the name “Kepler” for the Kepler Conference is inappropriate and confusing. I agree that there is some confusion, what with several astrological organizations and products already called Kepler. However, I stand by our decision in naming the Kepler Conference.

I was one of four founders of the Kepler Conference. Now I am indirectly involved as a supporter of the conference and a voluntary advser to the conference coordinator. I fully support and agree with the conference coordinator’s decision to retain the name of the conference. If I was the conference coordinator, I would also keep the name of the conference.

We are following in Kepler’s footsteps. We are attempting, in our own way, however humble or limited it may be, to scratch up a pearl or a gold nugget, to use Kepler’s words. I suggest that all of us should get accustomed to the fact that Kepler, like Shakespeare and Mother Mary, will be enshrined in a great many institutes.

I predict that future generations will come to realize how extraordinarily prophetic Kepler was, for shining a light down a road that few could comprehend until centuries after his passing. “Kepler” is the most appropriate name; especially for any organization or product that is dedicated to the highest standards in critical analysis, research, technical detail, or the pursuit of clear and dramatic discoveries. “Kepler” is the name that perfectly expresses the mission and goals of the Kepler Conference.


The Kepler Conference, January 20 -22, 2017, Cape Canaveral, Florida

July 7, 2015

The Avalon Astrology Conference has been expanded into a larger and more exciting conference and has a new name: The Kepler Conference. The concept for the conference, as described in my blog below, is still the same, but the conference is being organized and developed to be more comprehensive, more professional, and more exciting! For more information about The Kepler Conference please visit http://www.thekeplerconference.com. As of the date I am writing this on July 7, 2017, the website is still under construction but the Call for Papers page is up: http://www.thekeplerconference.com/call-for-papers


October 14, 2014

The Basis of Sign Rulers in Prime Numbers

David Cochrane

October, 2014

The Symmetry of Prime Numbers in Zodiac Signs and Hellenistic Bounds

In ancient western astrology each zodiac sign was divided into 5 subsections, which today are referred to as “bounds” by Hellenistic astrologers and as “terms” by classical astrologers. One of the five visible planets, Mercury, Venus, Mars, Jupiter, or Saturn, was assigned to each bound. The bounds are of seemingly arbitrary size and there appears to be no clear rule for which planet is assigned to which bound.

In the paper “The Basis of Hellenistic Bounds and Medieval Terms in Prime Numbers” at http://www.astrosoftware.com/HellenisticBoundsAndPrimeNumbers.pdf I propose a formula based on the symmetry of prime numbers within each zodiac sign that determines how malefic planets are assigned to bounds. This formula is based on the idea that a lack of symmetry in the distribution of prime numbers indicates a malefic bound. With four relatively simple rules we are able to assign the bounds of Mars and Saturn and thus what at first appears to be almost random assignments of planets to bounds begins to take on a pattern based on clear rules. Whether the hypothesized rules given in that paper are correct is not yet verified. We can, however, continue to pursue the idea of prime numbers and particularly symmetries of prime numbers as a possible basis for ancient astrological ideas.

Extending the Symmetry Within Each Zodiac Sign to the Entire Zodiac

If the symmetry of prime numbers within each zodiac sign were noticed and given importance by some astrologers in ancient times, then it may have also occurred to someone to see if the symmetries extend beyond a zodiac sign. For example, in Table 1 we see that 105 degrees is the center point of Cancer and that the prime number pair 103 and 107 is symmetrical around 105. also the prime number pair 101 and 109 and the prime number pair 97 and 113 are symmetrical around 105. Continuing into the adjacent zodiac signs we find that 89 is the next lowest prime number and 89 = 105 – 16. To obtain the symmetric point of 89 we add 16 to 105 and obtain 121. The number 121 is not symmetric so fail to find a symmetric pair. However, the next lowest and prime number is 83, which equals 105 – 22, and the symmetric point at 127 (105 + 22) is a prime number, and thus 83 and 127 are also symmetric around the center point of Cancer.

Table 1. Symmetries within each zodiac sign.

Pair: First Second Third Fourth
Distance from center: -14 -8 -4 -2 +2 +4 +8 +14
Degree of zodiac sign: 1 7 11 13 17 19 23 29
Aries 1 7 11 13 17 19 23 29
Taurus 31 37 41 43 47 53 59
Gemini 61 67 71 73 79 83 89
Cancer 97 101 103 107 109 113
Leo 127 131 137 139 149
Virgo 151 157 163 167 173 179
Libra 181 191 193 197 199
Scorpio 211 223 227 229 233 239
Sagittarius 241 251 257 263 269
Capricorn 271 277 281 283 293
Aquarius 307 311 313 317
Pisces 331 337 347 349 353 359

 

If a large number of prime numbers throughout the entire circle are symmetric around some of these center points, then these center points would, in a sense, be “super centers of symmetry” because they are not only symmetric points of prime numbers within a zodiac sign but for the entire zodiac.

Keep in mind that prime numbers were regarded as very important by some mathematicians in ancient Hellenistic times at around the same time that the complex rules of astrology were either invented or revived (some time in the range of about 700 BC – 200 BC). The possibility that the fascination with astrology and the fascination with geometry and prime numbers would influence each other can reasonably assumed to be very likely.

The Super Symmetric Center Points and Ruling Planets of Signs

There are two ways that ancient astrologers could have identified zodiac degrees that have a large number of prime numbers equidistant from these zodiac degrees: through a formula or by manually calculating the distance of prime number from every degree of the zodiac from 0 to 159.

A formula to determine symmetric points is Xn – (Xn / 2) where X = the result of multiplying the sequence of at least two consecutive prime numbers beginning with 2 and n is any positive integer. For example, for X = 2 x 3, the sequence is 3, 9, 15, etc. and this is the well-known sequence of symmetry of prime numbers at every 6 digits. For X = 2 x 3 x 5, the formula produces the sequence 15, 45, 75, etc. and this is the sequence of center points at the center of each zodiac sign. For X = 2 x 3 x 5 x 7 the sequence is 105, 315, 535, etc. and has a length of 210. The number 210 as a zodiac degree is 0 degrees of Scorpio and as discussed below, the 0 Taurus-Scorpio axis is part of the fixed cross axis that the ruling planets of signs are based on.

Manually calculating the distance of every prime number up to 360 from each degree of the zodiac would have been very time-consuming in ancient times. I wrote a computer program to produce this information. I expected that 105 (15 degrees Cancer) and 315 (15 degrees Aquarius) would be the strongest center points of prime numbers based on the formula described above, but there is the possibility that other points could be as strong or even stronger. Note that in a circle the opposite points are also center points so we are always working with an axis, so it would be the 15 Cancer-Capricorn and 15 Leo-Aquarius axis that I expect to be strongest. This possibility arises especially because the prime numbers are placed around a circle so the distance is measured in a modulus 360 function, which could affect the results. What I discovered was somewhat surprising and very interesting. The strongest symmetry, as described in more detail below, is 0 Taurus-Scorpio. The 15 Cancer-Capricorn axis and 15 Leo-Aquarius axis were among the strongest center points but 0 Taurus-Scorpio is the strongest.

The computer program that I wrote produced three sets of results. The first result includes the number 1 as a prime number. In modern mathematics the number 1 is not considered to be a prime number but it is possible that some mathematicians in ancient times regarded number 1 as a prime number. In the second set of results I removed the number 1 as a prime number. In the third set of results I also removed the numbers 2, 3, and 5 because, as explained in the paper on prime numbers and Hellenistic bounds, the numbers 2, 3, and 5 are the basis of the zodiac and do not appear in the symmetry of numbers within each zodiac sign. The results of all three analyses was extremely similar so the conclusions are clear regardless of which of these three variations of the analysis are done. The results are summarized in Table 2.

Table 2. Number of Prime Numbers that are part of a Pair of Symmetric Number

 

Symmetric Point Include Number 1(Total = 73) Exclude Number 1
(Total = 72)
Exclude 1, 2, 3,  and 5
(Total = 69)
Conclusions
0 Taurus-Scorpio 58 56 56 0 Tau-Sco Highest
15 Aries-Libra 54 52 52 15 Degrees Cardinal, Mutable, Fixed  except 15 Taurus-Scorpio are Second Highest
15 Cancer-Capricorn 52 52 52
15 Virgo-Pisces 52 52 52
15 Gemini-Sagittarius 52 50 50
15 Leo-Aquarius 52 50 50
15 Taurus-Scorpio 50 48 48 15 Taurus-Scorpio and 0 Degrees of all other signs except Aries-Libra are Third Highest
0 Cancer-Capricorn 50 48 48
0 Leo-Aquarius 50 48 48
0 Gemini-Sagittarius 48 48 48
0 Virgo-Pisces 48 48 48
12 Gemini-Sagittarius 46 46 44 0 Aries-Libra and 12 Degrees Mutable are Fourth Highest
0 Aries-Libra 46 44 44
12 Virgo-Pisces 44 44 44

 

If, as suggested by the evidence from the analysis of Hellenistic terms, that some ancient astrologers used the symmetry and lack of symmetry of prime numbers to determine the effect of different areas of the zodiac, then they may have noticed that the 0 Taurus-Scorpio axis is the strongest point of symmetry. The strongest symmetrical points fall roughly into these four groups:

  • (1) 0 Taurus-Scorpio
  • (2) 15 Degrees of Zodiac Signs
  • (3) 0 Degrees of Zodiac Signs except Aries-Libra
  • (4) 0 Aries-Libra and 12 Degrees of Mutable Signs

Interestingly, the 0 Taurus-Scorpio axis is used to divide the outer planet rulers of signs from the lights and inner plantes. In Figure 1 we see that the planets Mars,, Jupiter, and Saturn are on one side of the 0 Taurus-Scorpio axis and the Sun, Moon, Mercury, and Venus are on the other side. In ancient times astronomy was visual, without the aid of telescopes and often with a geocentric model of the solar system. The planets Mercury and Venus always stay close to the Sun. The Moon is visually the same size as the Moon and is a kind of mirror of the Sun during the night.  Mars, Jupiter, and Saturn are really the most obvious wandering stars; they can wander anywhere along the ecliptic plane without being tethered to the Sun. Mercury and Venus, on the other hand, are always tethered to the Sun, as if on a leash and unable to wander far from the Sun. The Moon wanders from the Sun but is clearly distinct from the much smaller wandering stars of Mars, Jupiter, and Sun and appears closely associated by being visually the same as the Sun and is often thought of as the other celestial light.  In short, the 0 Taurus-Scorpio axis divides the outer wandering planets from all other plants.
Figure 1. Ruling Planets of the Zodiac Signs.

SignRulers

The cross of the fixed signs is a cross upon which the ruling planets of the zodiac signs are arranged. They are placed symmetrically around the 0 Leo-Aquarius axis with the Lights, Mercury, Venus, Mars, Jupiter, and Saturn equidstiant from this axis, and with the outer wandering planets separated from the other wandering objects by the 0 Taurus-Scorpio axis. This is a simple and elegant design.

However, for decades I have found this elegant design rather mysterious. Why is it “lop-sided” rather than centered around the cardinal signs. Why is it centered around the fixed cross rather than cardinal cross? The answer may be that the design is influenced by symmetrical patterns of prime numbers just as the placement of malefic planets in the Hellenistic bounds may also be influenced by symmetrical patterns of prime numbers.

Note that both bounds and ruling planets involve the assignment of a planet to some area of the zodiac. In the case of ruling planets of zodiac signs, the 5 visible planet and the 2 Lights are given ownership of a zodiac sign. In the case of the bounds, only the 5 visible planets are given ownership of a bound. The concept is similar and a similar principle, symmetries of prime numbers, may have strongly influenced the pattern of planets.

Reflections, Considerations, and Conclusions

During the time of the revival or invention of astrology somewhere in the time period of roughly 700 BC to 200 BC Platonic thinking was one of the more dominant intellectual perspectives and philosophies. From the divine numbers of Pythagoras to the archetypal concepts of Plato to the idealized lines and circles of Euclidian geometry, the pervasive influence of pure abstract concepts is evident. Some of the astrologers of the times may have looked for beautiful patterns and divine designs in the heavens. Even today the word “heavens” can suggest the starry sky or a divine paradise. The physical sky and celestial paradise can at times be conflated to some extent and a pure mathematics could naturally be associated with the astrological doctrines. The symmetries of prime numbers may have been perceived as part of a natural order and a celestial design.

In any case, the number of coincidences in which the arrangement of the prime numbers coincides with astrological concepts is growing. Both the bounds and the ruling planets of signs appear to have a relationship to prime numbers.  We may at last be slowly approaching an understanding of the origin of some of the most ancient astrological ideas and we may be making progress in understanding mysteries such why ruling planets are arranged symmetrically and in a balanced arrangement around the fixed cross.

Given the hypothesis presented here that prime numbers influenced the ordering of the ruling plants of signs, we may wonder why the ruling planets are symmetrical around the 0 Leo-Aquarius axis rather than the 0 Taurus-Scorpio axis. I suspect that the ideas that had developed regarding the astrological significance each zodiac sign fit better with the centering around the 0 Leo-Aquarius axis. The symmetry of prime numbers was likely not the only idea that influenced the doctrine of ruling planets of zodiac signs. Also possible is that ancient astrologers were aware of the formula for determining center points of prime numbers and this would have brought attention to the Cancer-Capricorn and Leo-Aquarius polarities, and these form the central area of the symmetric pattern of planetary rulers.

I am not aware of any extant literature which corroborates the idea that prime numbers influenced the development of astrological ideas. The ideas presented here are based solely on observations of relationships of prime numbers and astrological ideas. These observations may be either coincidence or possibly be a retracing of the thoughts of some ancient astrologers. Further research, study and translations of ancient astrological literature will help clarify these issues. My personal impression is that the close association of the distribution of prime numbers with the astrological doctrines of bounds and ruling planets are too great to be a coincidence. It appears likely to me that some of the ancient astrologers did formulate ideas on bounds and rulerships at least partly from these observations of the distribution of prime numbers. However, without strong evidence from primary sources to confirm this suggestion, the thesis presented in this paper must be regarded as conjecture rather than fact.


Catching the Trane: The Musical Genius of John and Alice Coltrane

July 24, 2011

John Coltrane was born with Mercury opposition the Mars/Jupiter midpint and sesquiquadrate to both planets. Extraordinarily, the angle of Mercury to Mars and to Jupiter is almost exactly 134.4 degrees, a 28/75 aspect as well as a 3/8 aspect, which confers focused creativity that is capable of extraordinary accomplishments. From this configuration, I determined that Coltrane must simply be a genius who was given a saxophone. A quick read of his biography at wikipedia.com confirms this; the man studied Plato and Aristotle. A don’t think most black kids born in Hamlet, NC in 1926 read very much of Plato. His creative genius led him to an interest in Pythagorean thought, he studied harmony and music theory to a very sophisticated level; he did not just concoct these melodies without technical training and understanding.

Recently I’ve been listening to Alice Coltrane and jokingly thinking to myself that she might be a bigger genius than her husband. But that is a bit like comparing Einstein, Neils Bohr, Kepler, and Newton; take your pick. Not having a birth time, I just entered 12 noon and looked at Mercury, Mars, and Jupiter, thinking that they might be in a 5th harmonic or 25th harmonic pattern but a birth time would be need to confirm a 25th harmonic. My reasoning is that they might have truly been made for each other. Alice periodically performed late in her life using her married name of Alice Coltrane. That tells me how deep her love and appreciation of her husband was; a lady of that level of intelligence and sensitivity is not likely to do anything without some deep thought and consideration, in addition to the minor fact that they were married.

Alice’s Mercury is quintile Mars, tridecile Jupiter, and Mercury at 12 noon is tightly square the Mars/Jupiter midpoint. The configuration was in effect for that day. This configuration is one that I have mentioned in lectures many times. It is a special configuration but it won’t necessarily make an Alice Coltrane. It is almost guaranteed to give superlative creative intelligence, however. Alice did grow up in a family that provided her the tools to grow and develop. Of course, without a birth time we are unable to analyze planetary configurations at the time of Alice’s birth in any detail.

I read that the Coltranes are bured in Pinelawn cemetery in Farmingdale, New York. In the unlikely event that I am ever in the area, I’ll put a flower on both graves and wish them well. Namaste my brilliant friends. Many thanks for the music and the inspiration.

Finally, note that the above analysis is based on sophisticated mathematical models that I have developed for astrological analysis. The form of astrological analysis that I employ is almost unique, and unlike most astrologers, I remain skeptical of all astrological models including my own because they are yet to be confirmed through research and if correct, the models, in m opinion, can be validated. Most astrologers, however, have a very different philosophical and epistemological perspective on how astrology can be known and improved.


Attunement to the Cosmos

September 4, 2009

Surrounded always in a vast expanse of bliss, preoccupied by minutiae we may see it not until perchance the heartbeat and breath are still or carried by a simultaneously ancient and new eternal pulse, whence we feel what previously only dimly was perceived – streaming beattitudes and the joyous intelligent presence patiently awaiting our awakening from a deep sleep and rejoicing in the long awaited arrival home.


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