Happy Pi Day! 03 14 15

Your humble author, working on pi-cutting skills: West Point, 1987.

Since it is the fourteenth day of the third month (which can be abbreviated 3.14), this day is often playfully referred to as "pi day," in recognition of the fact that the transcendental number pi (which goes on forever and has no discernible pattern to its endlessly-unfolding decimal sequence) begins with 3.14 . . .

And, because this is the year designated AD 2015 (which means that this day in this year can thus be abbreviated 3.14.15), this particular "pi day" is the only one since 1915 that has reflected the "first four digits of pi" after the decimal place (the first 9 digits of pi after the decimal place are 3.141592653 . . .

Mathematically, of course, pi relates the diameter of a circle to its circumference: the circumference of a circle can be expressed as 

π

d, where "d" is the diameter of the circle.

Because of pi's intimate relationship with the shape of the circle -- a shape with great symbolic significance, representative of the heavens above and hence of the world of spirit -- pi has long been treated with reverential honor and respect.

At the ancient monastic academy of West Point, for example, the act of cutting any pie or cake was once treated as a ritual of tremendous importance, as can be witnessed in the photograph above (which was not staged), in which a circular dessert pie is being inscribed with a knife-line across its diameter, while gentle coaching and advice is offered in order to ensure that the ceremony is done precisely in accordance with tradition.

This level of attention to detail is appropriate, because the act of placing a dividing line across the circular space of the pie is symbolically akin to coming into contact with the infinite.

Mathemagician Marty Leeds, who is very attuned to the symbolic significance of number, and especially of the transcendental number pi, explores the profound spiritual significance of pi in his book The Peacock Tales, reflecting:

I was spellbound by pi's infinite digits, its powerful trinity and its mysterious transcendence. [. . .]

[The] cosmic birth was the act of a great try-unity, first manifesting itself within itself and then releasing that potential by dividing itself; and pi is the ratio and geometric symbol given to us to help us remember and understand this story. God manifesting itself in the beginning of time as the sphere of creation is recognized in the circumference of pi, the division of itself recognized in pi's diameter, and the expansion of our universe recognized in pi's ever-unfolding, infinite digits.

In short, the mathematical constant of pi is a representation of the creation of our expanding universe.

15 - 17. [Italics and boldface type in the original].

Later in the same book, Marty dives deeper and deeper into the profound meaning of pi:

In the study of sacred geometry, the circle represented Heaven and the square represented Earth. This motif is seen within the Freemasonic Square and Compasses (the compass being Heaven and the square being Earth), within the ancient Chinese cosmographic concept known as Gai Tian (the square earth is a chariot, the round heaven its canopy) as well as utilized by the Buddhists in the building of the sacred, mound-like or semi-hemispherical structure known as the Buddhist Stupa. It is important to note for our study that the place where God resides, Heaven, is geometrically symbolized by the circle. If one takes a length of string and makes it a triangle, a square, a rectangle, or any other sort of a polygon, the amount of space that string creates will always be smaller than if one makes it a circle. The circle encapsulates the most amount of space with the least amount of effort. Bringing the circle out into three dimensions to make it a sphere will encapsulate the most volume. This geometric fact is one of high symbolic value. The circle is expressing something about the nature of its own being. It is speaking to you. It is representing, within its own essence and qualities, the limit or sphere or creation. [. . .]

The circle also leads us to the first number in existence, zero. If we were to attribute zero to a geometric form, the most obvious form we could equate it to would be the circle. The number (or non-number) zero expresses the concept of nothing. [. . .] The circle, by encapsulating the most space, yet simultaneously being a representation of the zero, or NO THING, condenses all time and all space into one geometric form. Put simply, the circle represents the concept of all or nothing. As far out into the abyss we wish to go, and as close to nothingness as we may conceive, the circle represents both. 125 - 127.

Pi connects the finite to the infinite, in that it relates a finite linear diameter (or radius, which is half the diameter and which is the distance between the points of a compass used to sweep out a circle, the compass thus being symbolic of the circle which it creates when it is used) to the shape that represents the infinite and the ineffable.

The architects of the Great Pyramid incorporated pi into the very structure of that massive monument, which has a perimeter around the base that is 2π times the height of the pyramid.

Similarly, the monument we know today as the Pyramid of the Sun at Teotihuacan in modern-day Mexico has a base perimeter which is 4π times the height of the pyramid.

These structures can thus be understood to symbolically unite heaven (the circle) with the earth (the square) and to act as symbolic link between the finite and the infinite, between the material realm and the realm of spirit.

In the second passage above from

The Peacock Tales, Marty Leeds also mentions the Square and Compasses of Freemasonry, which can be seen to point to the same connection between heaven and earth, material and spiritual, finite and infinite. The right-angle square tool clearly evokes the square, while the compass device evokes the circle: Heaven and Earth.

Interestingly enough, there is an important part of the heavenly sphere which contains two constellations that can be seen as evoking the very same profound concept of connection between heaven and earth, the infinite and the finite. These two constellations are the Great Square of Pegasus and the constellation known as the Triangulum, or Triangle, which is located close by. 

The Great Square, of course, evokes the square which represents bounded, limited, physical space: the earth (which goes around the sun in an annual circuit that is bounded by the "four pillars" or the "four corners" denoted by the two solstices and the two equinoxes). 

The Triangle evokes the instrument of the compass which is used to sweep out a circle. Hence, the two constellations, providentially located in the sky so close together, evoke the connection between Heaven and Earth, spiritual and material, infinite and finite.

The Triangle is actually also located very near to the critically-important constellation of Aries the Ram, whose brightest stars are also three in number (and which in fact are not quite so bright as those of the Triangle, making the Triangle a very useful tool for locating Aries, as discussed in this previous post).

Below is a screen-shot of the sky showing the position of the Great Square (outlined in green) and the Triangulum (outlined in red), both of which are outlined along with Aries (also outlined in red, only the brightest three stars being connected in this illustration).

As you can see from the above diagram, which I created using the outstanding open-source planetarium app Stellarium (available at stellarium.org), on this particularly significant "pi day," the earth is located on the opposite side of the sun from the portion of infinite space containing the stars which make up the constellations of the Great Square and of the Triangulum. 

This means that in order to "look towards" the Square and the Triangulum on this 3/14/15 day, one must look towards the sun itself. You won't be able to see these constellations in the night sky on this day, because when we are on the "night side" of the earth (the side turned away from the sun) we will be facing an entirely different sector of the heavens.

This seems somehow appropriate and even significant, since the calendar day itself that we designate as 3/14/15 is a date on the solar calendar (and the year AD 2015 itself is a year which commemorates the year of the Lord, the Christ, who can be shown to have strong connections to our sun, the enabler of all life on earth and the single most-essential intermediary through which or through whom we relate to the heavens beyond).

Ultimately, one of the most important messages of this mysterious and transcendental number pi is what it tells us about ourselves: that we ourselves, like pi, at once touch both the finite and the infinite, occupy the interstices of Heaven and Earth, matter and spirit. 

Pi day can help remind us of this truth -- a truth which the exigencies of the physical world often work to obscure. 

It is a perfect day to pause and meditate further on these truths.

And, once we become aware of it, pi can continue to remind us of these concepts more than just once a year -- whenever we encounter a circle in the form of a sacred drum, for example, or consider the infinite vastness of the night sky that opens up within the circle of the horizon.

Happy Pi Day!

For those interested in previous posts which touch on the concept of pi, see also:

• "Black Elk and the sacred circle"
• "Scott Onstott and the metaphor of form"
• "The conjunctions of Jupiter and Saturn"
• "Ross Hamilton's Star Mounds"

Also, in light of the photo above, it is interesting to note that the Chinese character for "knife" is:

刀

A Ghostly Trio of Haunted Posts for Halloween

image: West Point magazine Fall 2011 (link), with added specter from Wikimedia commons (link).

The important annual holiday of Halloween is upon us, and if you have not already done so, you may wish to go back to the post from a few days prior which discusses the spiritual meaning and symbolism of this annual autumnal festival.

Newer visitors to this blog may not know it, but there have been several ghostly posts from the past which explored the phenomenon of phantoms. Here are some links to a few of them for your All-Hallow's Eve enjoyment!

1. "Scary Ghost Story (West Point)" October 31, 2011.

Perhaps the spookiest of them all, especially because I can well remember some of the first-hand or second-hand stories I heard from other cadets while at West Point of the eerie experiences that would sometimes take place at night in "the Divisions" -- specifically the 47th Division (pictured above). Some of these included awakening in the middle of the night to the feeling of someone or something pressing down on their chest with enough force to make it difficult to breathe.

2. "The Cheltenham Ghost" December 03, 2012.

Account of the apparition seen by numerous witnesses over a period of several years during the 1880s at a house in Cheltenham, England. Guaranteed to send shivers up your spine.

3. "Is the Cavalier Hotel really haunted?" July 12, 2012.

I've been to this hotel myself (a couple of times), and it certainly feels haunted! There have been numerous accounts recorded over the years, which you can explore and evaluate for yourself.

Bonus selection: "Don't miss the intriguing interview with Sheldon Norberg on New Dimensions Radio" November 05, 2011.

This post features an amazing interview from 2011 with Sheldon Norberg, but unfortunately the public link to that interview is no longer free. However, you can still listen to it (and download it) for just $1.99 at the New Dimensions Radio website here, and if this is a subject that interests you, it is well worth the price! You can also check out a short TV feature from 2011 in which Sheldon Norberg discusses his work here.

Also, along with its discussion of the concept of houses which seem to have some supernatural presences attached to them, this same post discusses some of the other haunted locations on the old post of West Point, including an old house on "Colonels' Row" which I was told on very good authority has a very strong presence in the upper rooms of the house.

Enjoy! 

Scary ghost story (West Point)

If you're looking for a spine-tingling ghost story for Halloween this year, look no further than the current issue of West Point magazine, published by the West Point Association of Graduates.

In a story entitled "Ghosts of West Point Past," found on pages 20 through 23, it describes the well-known story of paranormal events in the "47th Division," which every cadet hears from his classmates or upperclassmen ("The Divisions" are an older section of barracks, officially called "Scott Barracks" after General Winfield Scott, located on the west side of North Area facing east, and so named because they are designed with a single entrance leading to a series of vertical floors connected by a staircase with landings and rooms on each landing, so that each vertically-connected section is effectively divided-off from the others to its left and right; each division is numbered, and there is an additional wing folded around the back and not facing North Area, with numbers above 50, known as the "Lost 50s" -- see map detail below, from the West Point area map available online here).

This particular article, however, contains the first-hand account from Cadet John Feeley, West Point Class of 1973, which I had never actually heard before.

I will have to tell the Association of Graduates to stop sending me their quarterly magazine if it continues to contain such frightening material.

The possibility of the existence of spirits, of course, must be vehemently denied by conventional Darwinism. There is no Darwinian explanation involving gene mutation or natural selection that would explain the process of evolving something called a spirit (or at least none that most academic biologists would admit). Spirit beings have no place in a completely materialistic or "naturalistic" Darwinian worldview (see the previous discussion entitled "Supernatural or Extraterrestrial" for related musings on this topic).

The story above provides one possible explanation for the occurrence described by the cadets in October 1972: a case of sleep paralysis accompanied by hypnogogic hallucinations. This explanation raises the epistemological question of how we know anything. In other words, if our minds experience something, does that mean that it is real? If our minds can register sensations as if we were perceiving them with our sensory organs (including our eyes or the nerves in our hands and fingers), how do we know there is anything actually originating those sensory signals?

Shakespeare probed epistemological questions like these (and several much deeper epistemological questions related to them, such as how we know who we are and how solid this idea of "our identity" really is) in most of his works, including perhaps most famously Hamlet, where such questions are also initiated by the disturbing experience of seeing a ghost.

Reflections after a 20-year reunion

As we have noted in previous discussions, the authors of the 1997 text The Fourth Turning argue for a more cyclical approach to studying history in contrast to the very linear and progressive approach that dominates much of what is ingrained in our thinking from grade school upwards ("progressive" in that we are generally given a picture in which progress is seen as a given and the whole sweep of human history is painted as a slow and fairly steady climb from primitive to modern, with us at the top).

I am sympathetic to this argument because I have argued that the linear approach to history may well turn out to be both wrong and dangerous (see for instance the discussions in this previous post and this previous post), and because I believe this linear approach is a byproduct of the cult-like Darwinian mentality which crept into nearly every branch of academic inquiry beginning in the late 1800s and tainted almost every model that was adopted from that time forward (including the geological models that we take for granted today).

Note that the authors of the Fourth Turning do not assert that they are against Darwinism or that they believe there were extremely advanced ancient civilizations -- these are positions asserted in this blog and in the Mathisen Corollary book -- but the work that they have done detailing cycles and asserting the validity of a more cyclical approach to history and the dangers in the dominant linear and progressive approach opens up valuable new perspectives for those who are interested in the subjects treated in the Mathisen Corollary.

In their book, authors William Strauss and Neil Howe explain that there is a connection between the four generational patterns that they find recurring throughout the ages and what they call the "saecular rhythm of alternating Crises and Awakenings" that they find repeating roughly every 100 years (Strauss and Howe, 70).

The authors note that roughly every hundred years, a pattern of four stages tends to play itself out between Crisis and Awakening, with a post-Crisis and post-Awakening period after each Crisis and Awakening. They observe that every fifty years or so, a world-defining Crisis tends to erupt, in which the direction and even the very existence of a civilization is challenged. They identify World War II as the most recent of these, and note that based on their research we are about due for another one (which they predicted would arise between 2005 and 2025).

Conversely, they identify the opposite of a Crisis as an "Awakening," and identify the period of the 1960s and 1970s as the most recent of these (we have discussed some of the musical artists who were influential during that period, and also noted that the concept of the "Age of Aquarius" is of course directly related to the precession of the equinoxes, which is explained in numerous previous posts such as this one).

To complete the four cycles, the authors identify two "shadow" periods after a Crisis and after an Awakening. After each Crisis there is a post-Crisis "high" which tends to set the stage for the next Awakening, and after each Awakening there is a post-Awakening "unraveling" which tends to set up a Crisis.

Connected to this cyclical pendulum swing between Crisis and Awakening, according to Strauss and Howe, is a cycle of generations which are profoundly impacted by their experience and stage of life during the Crisis or the Awakening. For example, one generation is below the age of twenty during a Crisis, and will be shaped by watching it from the general vantage point of childhood -- think of those too young to fight during World War II, for instance (of course, many fought as young as sixteen during that conflict, but these are general age groupings, not hard and fast lines). Another group was in their prime years of young adulthood (roughly from 21 to 41) during World War II and were thus shaped quite differently by the experience. Those between the ages of 42 and 62 are generally at a different stage of life -- they are generally directing the efforts of those young adults in the generation behind them, since those between 42 and 62 are in the years in which they generally hold the reins of power.

During the years after the Crisis was resolved, these generations would continue to exhibit the traits that were stamped upon them during the Crisis itself, so that those who had been between the ages of 21 and 41 during the Crisis generally exhibited patterns that the authors of the Fourth Turning call the Hero generation, while the generation that was basically in childhood during the Crisis exhibit a pattern that they call the Artist generation.

After the Crisis is resolved, there will be a saecular "high" period for those who are flush with victory, and the children of this period will tend to grow into a generational pattern that Strauss and Howe call the Prophet generation -- this is the generation that will come of age during the next Awakening period, and corresponds to the Baby Boom generation that was born after World War II. During the Awakening period (two turnings after the Crisis), those who were the Hero generation in their 20s and 30s in the Crisis have moved into what Strauss and Howe call "elderhood" (roughly ages 63 - 83) and a new generation that is twenty and younger during the Awakening (old enough to observe it without participating in it directly) arises which Strauss and Howe call the Nomad generation (corresponding to "Generation X" after the Baby Boomers).

When this Lost Generation begins to enter young adulthood (ages 21 to 41) and the generation that was in young adulthood during the Awakening moves into positions of power and world leadership, there is a period that Strauss and Howe call an "unraveling" which precedes the next Crisis. According to Strauss and Howe, this was the period that we have been moving out of and the end of the unraveling is characterized by intense discontent, distrust of institutions, pessimism and lack of confidence -- all of which describes the present time quite accurately.

In the image above, I have attempted to illustrate the interconnected cycle-patterns that Strauss and Howe describe using an image of an old "Monstrance clock" or "mirror clock" from the 1500s, which had two interconnected dials to indicate the time. We can think of the upper dial in this illustration as showing the "saecular" cycles that Strauss and Howe identify (they use the term and the spelling to indicate a period of roughly a hundred years, and note that this concept goes back to the ancients). We see that this dial moves from a Crisis to a post-Crisis period (Strauss and Howe call this period a "high") and then into an Awakening followed by a post-Awakening period (Strauss and Howe call this an "unraveling").

Meanwhile, each of us has a dial that corresponds to our own stage of life, during which we generally fall into different roles as we move from childhood (ages 0 - 20) to young adulthood (21 - 41) to midlife (42 - 62) to what Strauss and Howe call "elderhood" (ages 63 - 83). If we were in childhood during an Awakening, we will be in young adulthood during the post-Awakening period (which ends with a great unraveling and ushers in a Crisis). Those who were in young adulthood during this unraveling move into positions of greatest power during the Crisis itself (the generals in World War II were mainly in young adulthood during the Roaring Twenties and early 1930s). Those who were in childhood during the post-Awakening (or "unraveling") become the Hero generation of the Crisis, when they are in the period of young adulthood from about 21 to 41.

We can see what Strauss and Howe mean more clearly by thinking of specific individuals who have lived through these different periods during the past 100 years of history, such as John F. Kennedy, who was in the Hero generation that served as the fighters during World War II (ages 21 to 41 during that conflict) and then moved into positions of primary leadership during the post-Crisis "high."

I recently had an opportunity to reflect on this entire model of cycles as I attended the twentieth reunion of the graduating West Point Class of 1991 (which explains the lack of posts over the past several days). In the brass clock pictured above, a twenty-year reunion for a West Point class falls pretty close to the point at which the marker is moving from young adulthood (ages 21 - 41) to midlife (ages 42 - 62) in the stages identified by Strauss and Howe. Returning to the familiar landscape of West Point provides the chance to "turn the dial back" twenty years and consider things that were said and done at the entry to young adulthood, and to perceive just how different one is at 41 or 42 from the person who was doing and saying those things at the age of 21 or 22.

Meanwhile, it is sobering to consider just how the upper dial has moved from where it stood in the years my class was at West Point (from 1987 to 1991). At that time, the cycle that Strauss and Howe believe is ending now was just getting underway. The first members of the Baby Boom generation were just preparing to enter the positions of greatest power on the world stage (for example, Bill Clinton became the first member of the Baby Boom generation to become the President of the United States during the election of 1992).

According to Strauss and Howe, the same twenty years that have moved my West Point class to the beginning of a new stage of life have also moved the upper dial to the brink of a new saecular turning, which they predict will be a Crisis.