Sacred Geometry
The claim that ancient and medieval sacred structures have some special connection to geometry, something that relates to factors considerably more important than the mere practical requirements of the builder’s trade, goes back a very long way. The term “sacred geometry” has come to be used for the ideas underlying these claims. As we’ll see for at least two sets of sacred structures—the Gothic churches and cathedrals of medieval Europe and the Hindu temples of India—the concept of a special sacred geometry rests on solid documentation, and for several others—the temples of ancient Egypt and Greece among them—there is at least some evidence to support the claim.
Unfortunately, like so much of the spiritual legacy of the past, the entire subject of sacred geometry has become mired in a swamp of vague assertions, unprovable claims, and fragments of alternative pop culture tossed into the mix more or less at random. These days, if you pick a book at random that has the words “sacred geometry” in the title, you’re as likely as not to end up reading a nearly random assortment of over-the-top conspiracy theories, speculations about lost continents, and wild claims about the profound spiritual powers of this or that fairly ordinary geometrical construction. There are some excellent books on sacred geometry in print as well,132 to be sure, but the modern literature on the subject is a decidedly mixed bag.
Even among the better books on the subject, consensus on the geometrical patterns that underlie any given sacred structure of the past is often lacking. The Parthenon in Athens, to cite the most blatant example, has been analyzed by sacred geometers repeatedly for well over a century. No two of the resulting analyses have found the same underlying geometrical pattern.133 The human mind’s habit of pattern recognition is strong enough that it’s possible to overlay almost any geometrical pattern over the Parthenon, or any other sacred structure you care to name, and get something that looks like a match. Skeptics have accordingly rejected the entire notion of a sacred geometry underlying the great churches and temples of the past.
A great deal of the difficulty here comes from the fact that geometry can have at least four roles in a religious structure. To begin with, of course, a structure of any size or complexity will need to be built with an eye toward at least the most basic forms of geometry to ensure that walls run straight, doors fit on their hinges, and ceilings stay up. Additionally, the human eye finds geometrical regularity pleasing, and so geometrical designs very often get used to make structures that are esthetically appealing. We can call these two functions “practical geometry” and “esthetic geometry,” respectively, and assign them to the traditional Masonic principles of strength and beauty.
A third aspect of geometry in architecture, which we can call “symbolic geometry,” comes into play when certain forms or patterns are assigned symbolic meaning within a given cultural and religious context. In Europe during the Middle Ages, for example, most churches were built in the shape of a cross, to reflect the symbolism that made the sacrament of the Mass a reenactment of the self-sacrifice of Christ; threefold patterns symbolized the Trinity, fourfold patterns the four apostles, and so on through an entire vocabulary of geometrical symbolism. This dimension of the old operative Masonry may be assigned to the traditional Masonic principle of wisdom.
Finally, the central thesis of this book suggests that there is also a fourth, concealed dimension of geometry that relates to the biological effects on which practitioners of the temple tradition focused their efforts. For reasons that will be explained later in this chapter, we can call this aspect “resonant geometry.” Before we can tease this fourth geometrical function out of the tangle, though, a more basic question needs to be asked and answered. Is there any evidence that geometry played any role in the sacred structures that belong to the temple tradition?
Sacred Geometry in the Hindu Tradition
In point of fact, there’s a great deal of such evidence. The Hindu tradition is particularly rich in geometrical symbolism, for example. Throughout India and wherever else Hindu religious teachings spread, geometry plays an obvious and central role in temple design. The vast temple complex of Angkor Wat in Cambodia is among the best documented examples here. As already noted, its geometries and proportions were worked out so precisely that a modern scholar, Eleanor Mannikka, has been able to decode the symbolism of the entire structure from its measurements.134
There is also documentary evidence along the same lines contained in the Shilpa Shastras, the traditional Hindu manuals for temple construction. These give detailed geometrical procedures for laying out the ground plan of a Hindu mandira and developing the rest of the structure from the ground plan—the same broad procedure as was used by the master masons of medieval Europe, though the details differ. The following procedure from the Manasara Shilpa Shastra, one of the few Shilpa Shastras that has been discussed in detail in any Western language, provides a clear look at the geometrical basis underlying one very important manifestation of the temple tradition.135
To lay out a mandira following the instructions of the Manasara Shilpa Shastra, once the site has been chosen and leveled, a vertical pole is erected in the center of the site, point A, and a circle is drawn around the pole with a radius twice the pole’s height, using a rope looped around the central pole as compass; see figure 1. Over the course of a day, the master builder observes the shadow cast by the pole as it moves from west to east and marks the two points, B and C, where the end of the shadow touches the surrounding circle: a line connecting these two points will run exactly east and west; see figure 2. The builder then uses a rope looped around the pole to draw a smaller circle, measuring the distance from the pole to the east-west line BC; see figure 3. Using circles with the same radius around points B and C, the circle is divided into equal northern and southern halves by a second east-west line; see figure 4.
shilpa shastra geometries
Next, with a rope equal in length to the circle’s diameter, two arcs are drawn, one from each of the ends of the second east-west line. A straight line connecting the points where the two arcs cross divides the circle in half with a north-south line; see figure 5. The same operation is then done again, using the points where the north-south line crosses the original circle as centers for the two arcs; see figure 6. When this is finished, there are eight points where arcs intersect each other, and lines connecting them divide the circle exactly into eight; see figure 7. The intersection of the dividing lines and the circles define the corners of a square and the midpoints of the square’s sides, providing the floorplan of the garbhagrha, the Holy of Holies at the heart of a Hindu temple. From there, further arcs and lines expand outward to define the other parts of the temple, and proportions derived from the geometries already laid out are then applied to all other parts of the temple.
The Manasara Shilpa Shastra is a crucial piece of evidence for two reasons. The first is that it demonstrates how at least one major architectural tradition with ties to the temple technology explored in this book did in fact use geometry as a core element of design. The other is that despite the huge cultural differences between classical India and medieval Europe, the geometrical principles that guide temple construction according to the Manasara Shilpa Shastra are identical to one of the two main systems used by the master builders of the Gothic era in the European Middle Ages.
Sacred Geometry in the Gothic Tradition
There is a great deal of surviving information about the role of geometry in another major branch of the temple tradition—the Gothic churches and cathedrals of the high Middle Ages. We know that geometry had a central role in Gothic architecture because the builders and designers of the Gothic era say so repeatedly in their treatises on the subject.136 Examples abound: the thirteenth-century French master builder Villard de Honnecourt filled his surviving sketchbooks with elaborate geometrical analyses of churches and building details; Matthias Roriczer, the fifteenth-century master mason who built Regensburg Cathedral, set out the geometrical way of designing buildings “according to true measure,” and there are also the well-documented conferences held between master builders working on the Cathedral of Milan in 1391 where everyone present treated geometry as the foundation of architecture.
The same link between geometry and architecture remains central to the rituals and traditional teachings of Freemasonry, which equate the craft of building with the art of geometry. The traditional secrecy of the modern Masonic craft, for that matter, was once used to conceal the geometrical methods of building design. Thus the Regensburg ordinances of 1459—a set of rules established by an assembly of German master builders—barred any master from passing on certain geometrical secrets to anyone but a qualified candidate for mastership.137
The medieval master builder used geometry to make all the different parts of a church fit together according to a common scheme of proportions. There were two basic schemes used in Gothic architecture. One called ad quadratum (“by the square”) took the ratio between a square and its diagonal as its basic module, as shown in the diagram of Gothic geometries on page 142—the same geometry that appears in the Manasara Shilpa Shastra. In arithmetic terms, the side and diagonal of a square are related by the ratio of one to the square root of two, which is written 1:
2,
and works out to 1:1.41427 . . . . Like 
, the ratio between a circle’s diameter and its circumference, 2 in decimal terms has an infinite number of digits. If the height or width of a square is equal to 1, the square’s diagonal is equal to 2, and this relationship was used over and over again to provide a building with its proportions.
How would this work out in practice? Once the orientation was determined by the process described in an earlier chapter, a medieval European master builder working in the ad quadratum system would start by laying out a series of squares on the bare ground of the building site using ropes and wooden stakes. One square, for example, might be the floorplan of the sanctuary of the church; another of the same size might mark the floor of the crossing; two short rectangles, one on each side of the crossing, might be the same length as the original square but 1/2 as wide, setting out the north and south transepts; and then the floorplan of the church west of the crossing would likely be a rectangle as wide as the original square but 2 times as long. Meanwhile the heights of the walls would also be based on the original square, multiplied or divided one or more times by 2, so that every measure from the largest to the smallest was related to all the other measures by some multiple of the square root of two.
The other proportional scheme used by medieval master builders, the ad triangulum (“by the triangle”) system, did the same thing with a different ratio: one to the square root of three, 1:3, which works out to 1:1.73205 … and so on for an infinite number of digits. Where the ad quadratum system unfolds from the square, the ad triangulum system unfolds from the circle: more specifically from two circles drawn so that the center of each one is on the circumference of the other, forming the figure traditionally known as the vesica piscis (“vessel of the fish”). If the distance between the centers of the two circles is equal to 1, as shown in the diagram on page 142, the distance between the places where the two circles intersect—the “points” of the vesica—is equal to 3. Connect the centers of the two circles and one point of the vesica with straight lines, and you have a perfect equilateral triangle, thus the name of the system.
In laying out a church using the ad triangulum system, a medieval master builder would start by laying out a series of overlapping circles on the bare ground of the worksite, again using ropes and stakes. Just as with the ad quadratum church, the various parts of the structure would be proportioned to one another using one or more multiples of the governing ratio—in this case, 1:3—so that every measurement from the smallest to the largest was tied together in a single proportional scheme. Much of the beauty of Gothic architecture comes from the subtle visual harmony this practice creates in a building.
Both the ad quadratum and ad triangulum systems have an additional feature that makes them very well suited to architecture in an age that cared about proportions and harmony. Starting from either system, it’s easy to pass by way of a series of simple geometrical constructions to the famous Golden Section, which geometers denote with the Greek letter Phi, 
. In numerical terms, 
equals 1:1.61803 … and so on, once again, for an infinite number of digits. Psychologists have found in repeated experiments that shapes based on the Golden Section look more pleasing and balanced to the human eye than those based on any other ratio. Medieval architects may not have had access to modern psychological journals, but they knew all about the Golden Section and wove it into their architecture.
Chartres Cathedral, generally considered the greatest example of early Gothic architecture, is a case in point. The original groundplan of Chartres, which was derived from an earlier cathedral on the same site, follows the ad quadratum system. On that solid traditional basis, however, the master builder of Chartres—like so many of the great architects of the Middle Ages, his name has been lost to history—raised a structure that is permeated with Golden Section proportions.138 The proportions of the entire structure combines 2 and 
relationships in a bravura display of geometrical mastery that contributes mightily to the pervasive sense of harmony and unity that strikes so many visitors to Chartres.
Despite their similarities, the division between the ad quadratum and ad triangulum systems ran deep. Masons’ guilds in the Middle Ages used one or the other—never both. The conferences between master builders at the building of Milan Cathedral mentioned above took place because that construction project was large enough to attract masons from southern Germany as well as northern Italy, including partisans of both systems. Scholars today have traced out some of the extensive body of symbolism, philosophy, and traditional lore that went with each system, but there may have been more to it than that; we simply don’t know.
One thing that is well known, though, is that the Scottish masons’ guilds that gave rise to modern speculative Freemasonry were firmly on the ad quadratum side of the great divide. Thus the master of a modern Masonic lodge wears a stonemason’s square around his neck, rather than a compass, and squares and right angles of various kinds play important roles in Masonic symbolism. The 47th proposition of Euclid, another bit of geometry that has a significant role in the symbolism of today’s Freemasonry, also comes out of the ad quadratum system; this proposition is better known as the Pythagorean theorem, and medieval master builders who followed the ad quadratum method used it constantly as a way of tracing right angles on the bare ground of the building site. If you know the Pythagorean theorem, you can use a knotted cord or any other measuring device to get square corners on any scale you need.
Geometries of Resonance
That the Gothic churches of the Middle Ages were designed on a geometrical framework, then, is not in doubt. One of the major questions that has to be faced in tracing the temple tradition through history is just how far back the standard Gothic geometries went. The Temple of Solomon, as we’ve seen, had a much simpler geometry: the debir (Holy of Holies) was a cube thirty feet on a side, and the heikal (main hall) outside it was formed of two such cubes set end to end. Though any number of attempts have been made to find complicated schemes of the ad quadratum or ad triangulum sort in ancient temples, few of these are convincing; most other ancient temples seem to have had proportional schemes that were just as simple as the one that governed Solomon’s famous structure.
What’s more, the great Gothic cathedrals didn’t always follow out their own geometry exactly. Chartres Cathedral, arguably the greatest of the early Gothic structures, is a case in point. Its design, as noted above, displays an extraordinary mastery of medieval sacred geometry—and yet the three bays (spaces between pillars) on the west end of the nave are squeezed together by a total of some eight feet.139 Why? None of the theories discussing Chartres’s geometry have managed to explain this.
If, as this book proposes, the design of buildings in the temple tradition had a practical as well as a symbolic significance, the question is easy to answer. As noted in Chapter Six, at least one of the functions of ancient temples and medieval churches seems to have been related to some form of electromagnetic energy, and as we’ll see, that function may have been triggered into activity by sound waves. All waves—electromagnetic, sonic, or any other kind—are affected by the phenomenon known as resonance, and the effects of resonance provide a straightforward explanation to the role of geometry, and of the curious variations from geometry just mentioned, in the temple tradition.
One easy way to experience resonance in action is to have two tuning forks that sound the same note. If you strike one of them on a hard surface so that it chimes, and then hold the other one to your ear, you’ll find that the unstruck tuning fork is chiming as well. Why? Each note represents a particular wavelength of sound, and a tuning fork is designed so that it vibrates at exactly that note; as the sound waves from the struck tuning fork beat against the unstruck one, they set it vibrating at the same rate, and so it gives off its note.
Electromagnetic waves have exactly the same effect, though you need something a little more elaborate than a tuning fork to demonstrate it. When you tune a radio receiver to the wavelength of a station you want to listen to, what you’re doing is changing the “note” at which electricity goes back and forth in a circuit inside the receiver, and the radio waves coming in from the antenna have exactly the same effect on the tuned circuit that sound waves from one tuning fork have on another of the same note. The tuned circuit resonates with radio waves of the wavelength you want, and the signal carried by those waves is then picked up by other circuits, amplified, and sent to your loudspeaker or headphones for you to hear.
Among amateur-radio hobbyists, working with resonance becomes a fine art. If you want to talk with someone on the other side of the world using an amateur radio rig, you need to make sure that the antenna over which you’re broadcasting your signal is in resonance with the radio-frequency alternating current you’re pumping into it, so that as much as possible of the energy in the transmitter output goes zooming out into the atmosphere. The goal is what’s called a standing wave: that is, each pulse of electricity reaches the base of the antenna, zips to the far end, and comes back to the base, taking just long enough for the whole trip that it arrives at the base at the exact moment that the next pulse hits. An antenna trimmed just right, so that you get what’s called a standing wave ratio of 1:1—that is, a perfect standing wave—can take a relatively weak signal and take it astonishing distances; the further from 1:1 your standing wave ratio is, the less efficiently your antenna broadcasts your signal.
All this may seem unrelated to the architecture of Chartres Cathedral or the temple tradition in general, but there may be a direct connection. Making an antenna get as close as possible to a 1:1 standing wave ratio can’t be done entirely by formula because radio waves are affected by a galaxy of subtle factors, not all of which can be worked out in advance. Amateur radio operators who make their own antennas thus measure them out by formula, leaving some extra at the far end, and then test the result, cutting off a fraction of an inch at a time until the standing-wave meter hits 1:1.
My thesis is that this is exactly what led the builders of Chartres to lop off, in effect, eight feet from the west end of the nave of the cathedral. Like all the Gothic cathedrals, Chartres was built in sections, and the west end of the nave was the last part to be completed; long before it was done, the master builder in charge would have been able to work out—quite possibly using direct methods of the kind mentioned in Chapter Six—the point at which the nave should end so as to establish a standing wave, or something like one, in the nave of the cathedral.
As a way of building a resonance chamber, Gothic architecture is extremely complex, and the great Gothic cathedrals—with their pillars, pointed arches, and side chapels—add further layers of complexity. The Temple of Solomon, by contrast, was a simple cubical structure, and so its builders would have had no difficulty calculating the resonance and making the structure fit a straightforward system of measurements. Most ancient temples followed this simpler and more functional approach. The complexities of the Gothic churches were profoundly shaped by the theological and symbolic ideas of the age that created them, which marked a crucial point in the process where those ideas obscured and then replaced the traditional rules of thumb that made the effect at the heart of the temple tradition work.
The Twilight of Sacred Geometry
Modern people routinely have a difficult time of it when they try to understand the ideas of the Middle Ages. A core reason for the trouble is the way that medieval thinkers so constantly fused things that most people today see as naturally separate. The first three kinds of geometry discussed at the beginning of this chapter provide a good example of this habit of fusion. In the writings of the master builders and Catholic theologians of the Gothic era, the practical, esthetic, and symbolic sides of geometry weren’t three different things. In the medieval way of thinking, it was the fact that geometry symbolized divine realities that caused a geometrically designed church to be beautiful and sturdy.
This didn’t always work in practice. The history of the high Middle Ages is punctuated with accounts of beautifully designed Gothic churches that collapsed and had to be rebuilt to a more solid design because the geometries that made them beautiful didn’t necessarily make them stay up. Over the course of the Gothic era, the increasing technical mastery of the builders made such disasters less common, but the entire medieval way of thought made it harder to tease out the competing requirements of wisdom, beauty, and strength from a geometrical construction—or from its architectural expression.
The same thing will have been even more true of the fourth function, the resonant geometry needed to generate the effects at the heart of the temple tradition. That tradition was a secret teaching known to certain people—primarily, as we’ll see, members of monastic orders and the guilds of operative masons who worked closely with them in the construction of monastery churches. Those who knew the secret knew that it worked, but they had no way of knowing how or why it worked—just that churches built in certain places, oriented in certain directions, using certain geometries, built out of certain materials, and in which certain ceremonial actions were done reliably got certain effects on local agriculture that to medieval minds must have looked like miracles performed by the saints or by God Himself. To see the resonant geometry that helped produce those effects as something distinct from the practical, esthetic, and symbolic aspects of geometry was a leap considerably greater than most medieval thinkers were capable of making.
Several pieces of evidence suggest, in fact, that the medieval monks and master builders who knew about the temple tradition followed it strictly by rote, without any understanding of the principles behind it. The most important of these is the way that so many famous churches of the Middle Ages, when they were destroyed by fire—a constant risk when candles and incense played important roles in religious services and flammable materials had a significant part in every kind of building—were rebuilt on exactly the same ground plan as before, no matter what oddities of design this required.
One classic example was Glastonbury Abbey, one of the great monastic centers of medieval England. Glastonbury’s great fire took place in 1184, and when the monks got to work rebuilding the abbey church, the growth of the abbey required a much bigger church than the one that had burned. The monks duly laid out a great church, but they built it to the east of the site of the old church and then built a Lady Chapel on the original site to exactly the same proportions as the little Anglo-Saxon church that had served the first monks of Glastonbury. The result was a Lady Chapel sitting incongruously off by itself to the west of the main abbey church until a later generation of monks connected the two with a porch. All this suggests that the monks of Glastonbury were in no way confident of their ability to design a new structure that would have the same effect as the old one, and so they had to copy the structure they knew as exactly as possible in order to get the desired result.
A set of rules of thumb recalled by rote can last for many centuries, but it has its vulnerabilities, and one of the worst of these comes with the arrival of a new generation that thinks, incorrectly, that it understands the underlying principles. That eventually played an important role in obscuring and then erasing the geometries of the temple tradition. With the coming of the Renaissance, the old customs that left building design to the master builders of the guilds of operative masons came to an end over much of Europe. Instead, architects trained in the new secular, humanistic ideas of the era took over the design of churches.
For such men, the practical and esthetic sides of geometry were the ones that mattered, with a little symbolism thrown in here and there if the client wanted it. Impatient of the old rule-of-thumb traditions and unaware that anything but superstition might be behind them, they built churches that are among the masterpieces of European architecture but lack the effects of the older tradition. At the same time, the elimination or transformation of the old monastic traditions as a result of drastic changes in European society closed off the other route by which the temple tradition and its distinctive geometries could have come down intact to the modern world. Obscure references to the powers of geometry in a variety of ancient and medieval writings, and the curious scraps of geometry in modern Freemasonry, were the only fragments left of the geometrical side of the tradition.
132 Among the better examples are Critchlow 1970, Ghyka 1977, Lawlor 1982, Lundy 2002, and Michell 2009.
133 See Padovan 1999, 80–98.
134 Mannikka 1996.
135 I have based the following on the discussion in Critchlow 1982, 29–32.
136 The following examples are from Simson 1962, 13–20.
137 Hiscock 2000, 186–195.
138 See Greene 1989 for a reconstruction, based on archeological data, of this process.
139 Simson 1962, 207.