Plane geometry
References
Guides
The principle ancient guide will be Plato‘s dialogue of ‘Timaeus’; which may be considered a kind of philosophical creation myth. Four translations of Timaeus are used here. The Loeb version is academic and archaic but includes the Greek text. The Penguin version is in modern English but is rather unsympathetic to the traditional sensibility. The Taylor version is more difficult to read but is quite sensitive to the Pythagorean aspects of the dialogue. Of all these the Jowett version is easiest to read while retaining a fair degree of sensitivity to the spirit of the work, however it is not currently in print. Your author has sometimes interpolated between versions, particularly in respect to the use of the term ‘God’. The Loeb version seemed somewhat anachronistic and therefore misleading in its rendering of the Greek term ‘Theos’ as ‘God’, as Plato was not a monotheist in the modern sense. The Penguin version is confused on this point and does not capitalize the term. In Taylor’s version the word ‘Theos’ in Greek is rendered as ‘the Divinity’ and this was felt to be the least confusing and most consistent with ancient usage.
The most notable ancient work on geometry is Euclid‘s ‘Elements’. While the Elements is a masterpiece of lucidity and directness, it may be noted that this very quality is achieved through a merciless editing, a kind of minimalism to the point of transparency. Only functional operations are mentioned in Euclid’s text. Philosophical connections are strictly excluded. These connections, however, are provided by Proclus in his equally lucid ‘Commentary on the First Book of Euclid’.
Hambridge, Huntley, Bragdon, and D’Arcy Thompson are early 20th century commentators on proportion and geometry in art and nature. They are often cited by as references by more recent writers.
Critchlow and Lawlor are both insightful, contemporary writers of your author’s acquaintance. Critchlow, is currently head of the Research Department of the Prince of Wales’s Institute of Architecture and is your author’s academic mentor.
Hambridge, Huntley, Bragdon, and D’Arcy Thompson are early 20th century commentators on proportion and geometry in art and nature. They are often cited by as references by more recent writers.
Critchlow and Lawlor are both insightful, contemporary writers of your author’s acquaintance. Critchlow, is currently head of the Research Department of the Prince of Wales’s Institute of Architecture and is your author’s academic mentor.
Introduction
Order of the Subjects
The next subject in the Quadrivium is Geometry. But it has been said that Music or Harmony should take precedence here. For example, in the TOA (p 56) we read, “…in the text entitled ‘On the Gods’ Pythagoras distinguishes them [the subjects] as follows: ‘Four are the foundations of wisdom – arithmetic, music, geometry, astronomy – ordered 1, 2, 3, 4.’ And Cleinias of Tarentum says ‘These things [numbers] when at rest give rise to arithmetic and geometry, and when moving to harmony and astronomy.'” The reference here is to the stable quality of the Triad, and the active, indefinite quality of the Dyad. Thus the subjects would alternate, Arithmetic – still, Music – moving, Geometry, still, Astronomy – moving.
But another possible reading of the four level Tetractys as metaphor, might be that the Two, as generator of line, that is to say, the extremes, represents Geometry, and the Three as generator of the proportional mean between the extremes is more appropriate to represent Music. Perhaps a deeper and more evenhanded approach might be to say that both space and time are co-equal in the decent from the realm of Ideas (the One, Arithmetic) to the realm of the material (the Four, Astronomy). In this view, space and time are co-creators of each other. More subjectively though, since we come to the perennial path through inquiry into the roots of form we will place Geometry second to Arithmetic.
Approaching Geometry
The term ‘geometry’ comes from the Greek words for earth, geo and measure metry. The subject is said by Herodotus [HIS 110, p 169] to have perhaps originated in Egypt where, each year, the Nile flooded and wiped away the existing land divisions. When the river receded the earth had to be measured out again. The practices involved though, are surely much older; as geometric forms can be observed in the sun and moon, in plants, animals, minerals, and in the movements and proportions of the human body itself. The above anecdote only considers the practical, or applied aspects of Geometry.
Mathematically, geometry is the science of ‘Order in Space’, as Critchlow has called it. If arithmetic may be considered one dimensional, in the sense that it has only magnitude, then geometry may be considered number expressed in two dimensional shape and three dimensional form.
Philosophically or qualitatively geometry, far from being confined to the earth, is a discipline capable of advancing consciousness to the realm of Form itself. In Plato’s Republic, Socrates tells us [REP 527], “…no one with even an elementary knowledge of geometry will dispute that it’s a science quite the reverse of what is implied by the terms its practitioners use. … They talk about ‘squaring’ and ‘applying’ and ‘adding’ and so on, as if they were doing something and their reasoning had a practical end, and the subject were not, in fact, pursued for the sake of knowledge. … And we must further agree … that the objects of that knowledge are eternal and not liable to change and decay. … Then it will tend to draw the mind to the truth and direct the philosopher’s reason upwards …”
Lawlor [SG p 6], adds that geometry is “…an approach to the way in which the universe is ordered and sustained. Geometric diagrams can be contemplated as still moments revealing a continuous, timeless, universal action generally hidden from our sensory perception. Thus a seemingly common mathematical activity can become a discipline for intellectual and spiritual insight.” and [SG p 8] “Functioning then at the archetypal level, Geometry and Number describe fundamental, causal energies in their interwoven, eternal dance.”
Like the study of number, geometric study is not just a mental activity, but works in three worlds, ideal, intellectual and material, simultaneously. Just as arithmetic was taught with concrete objects, so geometric constructions need to be ‘performed’ in time like ritual dances. For purposes of remembering the eternal, the performance must be concentrated and mindful; the means economical. Ancient geometers in Egypt needed only their aprons, the Greeks used just a stick and a string; for us today a compass, straightedge and pencil should suffice.
Lawlor puts the invitation to this kind of study most lyrically [SG p 14], “As geometers, equipped only with compass and straight-edge, we enter the two dimensional world of representational form. A link is forged between the most concrete (form and measure) and the most abstract realms of thought. By seeking the invariable relationships by which forms are governed and interconnected we bring ourselves into resonance with universal order. By reenacting the genesis of these forms we seek to know the principles of evolution. And by thus raising our own patterns of thought to these archetypal levels, we invite the force of these levels to penetrate our mind and thinking. Our intuition is enlivened, and perhaps, as Plato says, the soul’s eye might be purified and kindled afresh ‘for it is by it alone that we contemplate truth’.”
Plane Geometry
The Point
Geometry, as number in space, begins with the One. As we have seen in the study of Arithmetic, the One as Monad contains the all. As point-Monad It is the source and potential of every thing, the seed of multiplicity. In attempting to understand the point as representative of the One we must acknowledge that it is essentially paradoxical. The point is dimensionless but is the beginning of all measurement. It is formless, but is the source of all forms. We have no way of knowing from where the point emerges, figure 1a.
As Critchlow describes it [IP p 7], “The nature of origins or the creation point of a subject is grounded in mystery. The nature of a point – the simple, self evident origin of geometry – is one such mystery: is it possible that a point has no dimension, except that it be a metaphysical point, and how can it occupy place if space has not yet been created from its unfolding? Clearly there has to be a precise differentiation between physical and metaphysical, between idea and expression, yet both are embraced by one reality.” A little later on he adds [IP p 9], “The manifestation of an action object or thought (if it can be defined) necessitates a point of origin or departure, in relation both to the manifestation itself and to the person who is conscious of its emergence. The point of emergence does not necessarily reveal its causation either in the field of its emergence or in the mind of the viewer. In the mind the point represents a unitary focus of conscious awareness; in the physical world it represents a focal event in a field which was previously uninterrupted.”
If we try to tell a ‘likely story’ about the coming into being of the point we will need to answer questions of who, why, what, where and when. In our story there is not yet a where or when because time and space come out of the movement of the point which has only just emerged. As to the who and why, every revealed tradition has its own answer. Plato has Timaeus [28c] answer, “Now to discover the Maker and Father of this universe is a task indeed, and having discovered Him, to declare Him unto all men were a thing impossible.” From the other side of the world, Lao Tzu, the great ‘Old Man’ of China, the father of Taoism, seconds the notion, saying [TTC 1 p 27], “The Tao that can be named is not the eternal Tao”. Human beings, it seems, are not usually set up to directly communicate the nature or motives of the Ultimate Oneness. Plato’s Timaeus (29d) leaves us with this consolation, “Wherefore, Socrates, if in our treatment of a great host of matters, regarding the Gods and the generation of the Universe we prove unable to give accounts that are always in all respects self consistent and exact, don’t be surprised; rather we should be content if we can furnish accounts that are inferior to none in likelihood, remembering that both I who speak and you who judge are but human creatures, so that it becomes us to accept the likely account of these matters and forebear to search beyond it.” So, we are left with only the ‘what’ of creation, and that is the story which follows.
The coming into being of the point is shown in figure 1a. Since the point is dimensionless, we represent [that is re-present] it in geometric study as an aesthetically appreciable point, a dot.
The Line
We have seen that for perception to take place, for us to become conscious of something, movement or change must occur. The point, as Monad, in order to become known to us, must, as Lawlor says [SG p 23], “…divide itself from within itself…” to become the Two. But the Dyad immediately gives rise to the Triad because two elements cannot physically manifest without a relationship . In geometry this relationship between points is called line. Line is the path traced by the Dyadic points as they separate . The minimum path of separation is a ‘straight’ line. This is the first dimension – length, figure 1b.
Critchlow adds [IP p 9], “If the manifestation of the point is indicative of a departure from its source, then direction is implied. Direction in space is qualitative and hence the first departure or line path from the point is qualitative. Homogeneous space is a contradiction in terms – if it is to be measured – the proof of this lying in the need to relate measurement to quantitative space: a measurement is only possible between two ends of a line-path: hence direction must precede measurement.”
Plane and Circle
The Dyad is made up of polar opposites. If one point is considered active the other may to be considered passive. Passive implies stillness, active implies motion. If one point remains still or fixed, while the other is active or mobile, their relationship remaining constant, these combined qualities may be said to give rise to the circle, figure 1c-d, Thus, growing directly out of the Dyad, the circle may be considered the first or primary plane, two dimensional figure. The circle has three essential aspects: 1, centre, 2, domain or area, and 3, boundary or edge. These are analogous to the cyclic Triad of coming into being, being, and departure from being. All points on the boundary are equidistant from the centre, making the circle a model of balance and harmony. The center, as point, is both existent and non-existent in that it controls the form of the line traced by being fixed in relation to the movement of the active point, but the centre is not itself a part of the circle as it has no dimension. In this sense it is the ‘soul’ or essence, or perhaps, the ‘seed’ of the circle – guiding its outward manifestation without any physical presence of its own.
Center, then, as Monadic point, may be said to give rise to the circle as a projection of itself; so the circle can be considered as a new Monad. As Critchlow puts it [IP p 150], “The circle, as representing the active point of the compasses returning to its own point of departure, is a good symbol of the timeless whole, while the moving point itself represents the passing of time as it describes the circle.”
This reference to the compass brings us to another important aspect of geometric work. It must be physically performed by the student if the qualitative aspects are to be deeply understood. Drawing the circle with the compass is a good example. The tool itself is shaped like the letter ‘A’, the first letter, the beginning. In Greek and Hebrew this letter is also the number one. In the author’s own study of Yoga, at the school of Swami Satchitananda in New York, Om, the primal or first sound, was taught as being pronounced ‘ah-oh-mm’, corresponding to the chakras of the navel, heart and throat, thereby repeating the Triadic process of creation, starting with the ‘A’. The compass is constructed with a pin or small ring at the apex, representing the point of emergence. This point has two extensions, the legs, one passive, with a pinpoint, and the other active, with a lead or marker.
To construct the circle you must first place the pinpoint at the location of the center. The geometer, as creator, must choose the location of the center. The distance between the legs, which the geometer also chooses, will be the radius of the circle, the like the scope of creation. When the legs are open they form an angle at the apex. Like an ‘angel’, this ‘angle’ is analogous to the bringer of the message or Idea from the One. The 17th century alchemist Robert Fludd, in his ‘Mystical Corpus’, suggests that the radius is analogous to the love of the Creator; the greater the radius, the further from the source of love.
The pinpoint should be placed deliberately but should not damage the surface. The compass is held by one hand and guided with the other, while both are coordinated by the eyes. To increase understanding and accuracy it may be useful to time the insertion with the moment between inhalation and exhalation. Whether you are you standing or sitting, the work surface is best located at about the level of the heart. In short, it is a whole body activity.
If you begin drawing at the topmost part of the circle, which might be thought of as the macrocosm, figure 1c, the downstroke represents coming into being or the descent of grace into form, referred to as the Arc of Decent by Ardalan & Bakhtiar [SU p 7]. As you approach the lowest part of the circle, notice that you will have to readjust the position of your fingers to maintain control of the compass. This is the material world of action, the microcosm. Continuing, you make the upstroke, the Arc of Ascent, form returning by grace to Unity, figure 1d. Finally, you return to and rejoin the starting place.
We may see seven stages in this process. The geometer, as creator, has: 1, caused the center to emerge, and to be located, 2, set the radius or scope of the circle of creation, 3, proceeded to bring form out of thought into manifestation by beginning the downstroke of the compass, 4, maintained form by adjusting the hand to continue the action of the compass, 5, returned to unity in the upstroke and 6, completed by enclosure a whole entity, a new Monad. The geometer may now, 7, rest and contemplate the creation.
The Paradox of Pi
Another essential paradox of the circle is Pi (the Greek letter ∏), that is, the relationship, or ratio of the length of the circle’s diameter to its circumference. In any given circle, this ratio is mathematically ‘indeterminate’. It can be approximated as 22/7 or 3.14… but even though it is right in front of our eyes, it cannot be communicated exactly by our rational faculty. It is not a question of advances in mathematics. The evidence indicates that all the computers in the world running forever, would not be able to do it. Pi has by now been calculated not just to millions of digits but to millions of pages of digits. It may be suggested that in all the ages to come, the limit of Pi will not be reached. This is because Pi is a ‘transcendental’ number; that is, it transcends our ability to rationally express it. The paradox of Pi is a good metaphor for the limits of human reasoning. Pi represents a kind of ‘event horizon’ beyond which the rational faculty alone cannot see. Like existence itself, we can experience it but we can’t fully express it. This paradox of Pi can be seen as a confirmation of Plato’s admonition not to try to go beyond a ‘likely story’ in our explanations of the universe, and his statements in the Seventh Letter [341] that ultimate truth “is not something that can be put into words like other branches of knowledge.”
The Regular Polygons
The Vesica Piscis
The circle, as we have seen, constitutes its own cycle of creation. It is a Monad, ‘a world in itself’. This Monad contains within itself its own opposite, and so divides itself from within itself to create the Dyad. Geometrically, this may be represented by giving rise to another circle, shown in figure 2a-b. But this new circle can only be experienced if it moves away from its twin.
Following Critchlow’s description [IP p 14]: “As that which is departing or ‘externalizing’ from the original circle proceeds on its course, a series of significant arcs are formed. The most significant of these in our present context is the half-way point, the last position at which the departing circle has contact with the centre of its origin. This position holds essential symbolic value inasmuch as it represents a union of an origin and a manifestation where the centres of both coincide with the peripheries of each, and the amount manifest of the departing circle exactly reflects the amount remaining of the original one. Also the departing circle has moved away by exactly the same amount as the original circle had ‘moved away’ from its hidden centre point in order to establish itself.”
Lawlor is even more direct [SG, p 32], “As the initial circle (Unity) projects itself outward in a perfect reflection of itself there is an area of overlap defined by the two centres and the intersection of the two circumferences. This area and shape is known as the Vesica Piscis.”
The Latin term means ‘bladder of the fish’, or in Lawlor’s words [SG p 31], “a bladder (Vesica) which when filled with air would be in the form of a fish (Piscis)….”
In figure 2b we see the two circles and their relationship. The upper one can be considered the realm of the heavenly, or Ideal. The new lower one, the realm of the material, a reflection of the heavenly. Their overlap, the Vesica, is the realm of the created, or of formation. It is the intellectual world, in which the concrete is shaped by the ideal.
As Lawlor puts it [SG p 32], “One of the ways to view the Vesica Piscis is as a representation of the intermediate realm which partakes of both the unchanging and the changing principles, the eternal and the ephemeral. Human consciousness thus functions as the mediator, balancing the two complementary poles of consciousness.” Going further, he adds [SG p 33], “The overlapping circles – an excellent representation of a cell , or any unity in the midst of becoming dual – form a fish shaped central area which is one source of the symbolic reference to Christ as a fish. Christ, as a universal function, is symbolically this region which joins together heaven and earth, above and below, creator and creation.”
The Vesica and Multiplicity
The Vesica in its ‘formative’ role is the birth giver to multiplicity. It can also be thought of as the ‘yoni’ of the Vedic tradition, a symbolic vagina, through which the material world is born. In the constructions that follow, the primary regular polygons of three, four and five sides are generated directly from, or ‘born out of’, the Vesica. Other polygons are generated from the Vesica and some combination of the properties of the primary polygons. The constructions used are designed to serve in the illustration of this metaphor and are not the only, or even the most efficient, methods of obtaining the polygons; as the saying goes ‘Paths are many, truth is one’. The properties of the numbers involved are discussed in the session on Arithmetic.
Twoness, the Line. The Vesica is defined by the two centers of its parent circles and by the two points of intersection of their boundaries. If the centers are connected by a line, this mat be taken to represent a forward – back axis. By connecting the points of boundary intersection a left-right axis is generated, figure 2c. We may see here that the cardinal points of earth measure, North, South, East, and West, arise directly from the Vesica. The two axes meet at a new center. Where they cross they create the ‘right’ angle. It is ‘right’ in the sense of being ‘upright’ or balanced between the poles of the Dyad. It is like the arithmetic mean, equal in magnitude between the two extremes.
Threeness, the Triangle. The Dyad, as we have seen in the Arithmetic session, is in constant fluctuation. Stability or structure requires a Triad, so the minimum number of edges a regular polygon or any polygon can have is three. The triangle may be considered symbolic of fixed structure because it cannot change shape without also changing the length of one of its sides. No other polygon has this property. The triangle created in figure 2d, is known as an equilateral triangle because all three of its sides are the same length, and each of its vertices are equal in angle at 60 degrees. There are two other types of triangles, isosceles, with two equal sides and scalene with unequal sides.
It is fair to consider the equilateral as first among them because of its closer adherence to Unity. In the TOA [p 114] we read, “The first triangle is the equilateral, which has in a sense a single line and angle – I say it is single, because its sides and angles are equal, and what is equal is always indivisible and uniform. The second triangle is the half square, which has a single distinction of lines and angles, and so is seen in terms of the dyad. The third is the half triangle (i.e. half an equilateral triangle): it is altogether unequal in each respect, so from all points of view its number is three.”
To construct the first or primary polygon, the triangle, shown in figure 3a, we start with the center to center line, AB. This line will be the standard unit side or baseline of each polygon in this series. The apex of the triangle, point C, is where the two circle boundaries meet. Connecting the two centers and the boundary intersection completes the construction.
Fourness, the Square. Next is the four sided figure, the square or tetragon, shown in figure 3b. This is created by starting at point C and rotating the two arms of the triangle in opposite directions along the circumferences of the two circles which form the Vesica. The rotation stops when the two sides are at ‘right’ angles to the base, at points D and E. These points are found by, in effect, shifting creating two new Vesicas. Starting from F draw a circle with radius AB whose circumference goes through points A and B. Where circle F intersects circles A and B , at points G and H draw two more circles also of radius AB. By projecting a line through IA, locate point D; and project JB to E. Draw square ABDE.
The Vesica’s ability to create the double Dyad, the right angle cross, the linear manifestation of fourness, has been used here to give birth to the square, the planar manifestation of fourness, through a doubling of the Vesica. It may be said that as the square is composed of four equal lines and four equal angles, it is a single fourness of four. The right angle is one fourth of a complete rotation; 360/90 = 4. There are four of them in the square, adding back up to a complete rotation. The square, then, is a multiple symbol of rightness. People who strictly follow social norms are sometimes referred to as ‘squares’, and those who adhere to virtue are known as ‘upright’.
The square, like its key component, the cross, is also associated with the realm of the material. Lawlor says of this [SG, p 24], “By definition, the square is four equal straight lines joined at right angles. But a more important definition of the square is the fact that any number, when multiplied by itself, is a square. Multiplication is symbolized by a cross. When we cross a vertical with a horizontal, giving these line movements equal units of length… we see that this crossing generates a square surface: a tangible, measurable entity comes into existence as a result of crossing. The principal can be transferred symbolically to the crossing of any contraries such as the crossing of male and female which gives birth to an individual being, or the crossing of warp and weft which gives birth to a cloth surface, or the crossing of darkness and light which gives birth to tangible, visible form, or the crossing of matter and spirit which gives birth to life itself. So the crossing is an action principle which the square perfectly represents. The word nature means, ‘that which is born’ and all birth into nature requires this crossing of opposites. So the square came to represent the earth and as such symbolized the conscious experience of finite existence, of that which is born into nature.”
Fiveness The Pentagon. The five sided regular polygon may be generated as shown in figure 3c. The Vesica starts with a circle, center at A, which divides to become two circles. The center of the second circle moves to point B, where its boundary touches the center of the first circle. If circle A divides again, and the movement of this new third circle continues until its center is at F, and its boundary is touching both of the original centers, A and B, a figure is created which contains three vesica shapes.
The pentagon starts with the same baseline, AB, used in the previous constructions. Again, the baseline will be rotated from both of its ends A and B, which are the centers of the Vesica circles, to form two more sides of the polygon. The question, as with the square, is where to stop the rotation. This is found by projecting lines from the apexes of the two new vesicas, points G and H, through the point K, where the boundary of circle F crosses the central axis of the original vesica. Extend these lines until they cross the boundaries of the original vesica circles at points L and M. These are the points at which to stop the rotation of the sides. The two remaining sides of the pentagon are positioned by setting the compass to the length of the side AB. With the point of the compass at L and then M, mark the point N, where the radii intersect the center axis of the Vesica. Connecting this point to the top of each side completes the pentagon.
This process of utilizing the Dyadic nature of the original Vesica plus the Triadic nature of the expanded triple Vesica yields the fivefold figure of the Pentagon, as in 3 + 2 = 5. According to Critchlow, this construction was first made public by Albrecht Durer in the fifteenth century, though it is probably much older.
The Composite Polygons. The triangle, square and pentagon may be said to arise directly from the Vesica, as has been shown. The hexagon, octagon, decagon [10 sides] and the dodecagon [12 sides], figure 4, may be thought of as ‘composite’ in the sense that they can be generated through combinations of the properties of the primary polygons.
The Hexagon is constructed from the doubling or reflection of the triangle, as in 3 x 2 = 6. It can also be seen as created by six movements of the triangle, each of which is 1/6 of a rotation, or 60 degrees x 6 = 360.
The Octagon is constructed using the diagonals of the square and its ‘shadow’ or ‘anti-square’. Here, the properties of two squares are utilized to generate the eight sided figure, similar to the arithmetic 4 x 2 = 8.
The Decagon or ten sided polygon begins with the Vesica and pentagon, as constructed in figure 3c. The decagon, then, is created by a doubling of the pentagon, as in the arithmetic 5 x 2 = 10.
The Dodecagon. The twelve sided polygon, the dodecagon, begins with the Vesica, the triangle and the square. The initial vertices are A and B. The remaining vertices may be found by various projections of sides or diagonals of the square and triangle. Here, the properties of the three and the four are utilized to generate the twelve, as in the arithmetic 3 x 4 = 12.
The Heptagon. The 7 and 9 sided polygons also can be shown to arise from the Vesica. Their constructions start from the same baseline AB, the line connecting the centers of the Vesica circles. This will provide the location of the first two vertices. But in these constructions another element will be added; a circumscribing or ‘controlling’ circle, on the boundary of which all the vertices of the polygon will fall. The key to the constructions will be to locate the center of this circle. The seven sided polygon, or heptagon, figure 5, is constructed starting with the Vesica, the triangle, and the square, similar to figure 3b. To find the center of the controlling circle draw the diagonals of the square, then bisect the distance from the center of the square C, to the apex of the triangle D. Then, using the compass, lay off a distance equal to this, above the apex of the triangle E. This is the center point of the heptagon’s controlling circle. The first two vertices are known from the baseline AB. The second two occur where the controlling circle crosses the Vesica circles F and G. The fifth is where the Vesica axis crosses the controlling circle H. The remaining two, points J and K, can be found by marking off with the compass set to the length of side, AB for example. Please note that this is an approximate construction. The Heptagon, then, is realized through utilizing the properties of the three and the four, similar to the arithmetic 4 + 3 = 7.
The Ennagon or nine sided polygon, figure 6, is constructed starting with the Vesica and the triangle. The center of the controlling circle point E, is determined by setting the compass to half the length of the baseline, AC, and then marking off this distance along the axis of the Vesica starting from the apex of the triangle, point D. The first two vertices of the ennagon are A and B. The second two fall where the controlling circle crosses the Vesica circles, points F and G. The fifth vertex is where the axis of the Vesica crosses the controlling circle, point H. The sixth and seventh fall where the projections of the sides of the triangle cross the controlling circle, points J and K. The last two can be found with the compass or, more interestingly, by drawing a circle centered at E, of radius ED which equals AC, half the side of the original triangle. Point E is also the center of the ennagon’s controlling circle. Where the projections of the sides of the original triangle ABD, cross this circle at points L and M, draw a line, which will be parallel to the baseline AB, and extent it until it crosses the controlling circle at points N and O. These are the last two vertices of the ennagon. This last line also creates two triangles NJL and MKO, similar to the original triangle ABD, who’s bases coincide with every third side of the ennagon. This is an approximate construction. As we saw in the Arithmetic chapter, the nine is internally structured by the three.
The eleven sided polygon and those with more than twelve sides are omitted here both for the sake of brevity, and due to their absence from the classical corpus of the Quadrivium. An approximate construction of the eleven sided polygon is given by Critchlow [IP p 169]. Others of higher sidedness can be found in various contemporary textbooks or can be worked out from the principles of the Vesica and the primary polygons, the triangle, square, and pentagon.
We may see in this series of constructions the suggestion of a sort of geometric creation myth: beginning at the mysterious point of emergence; the expansion of this seed into the Monadic circle; the division of the Monad from within to form the Dyad; the Dyad’s ‘co-creation’, so to speak, of the Triad in the form of the Vesica. From this triplicity the indefinite multitude of polygons is generated.
The Root Systems
The Square and the Square Root of Two
The root is the part of a plant that provides a structural foundation by its penetration of the earth and is the basis of nourishment through its ability to break down or divide the earth into parts. Both of these functions are related to growth through transformation. In geometry transformations of size and shape are also governed by functions known as roots.
Start with a square who’s side is set equal to 1, figure 7a. The area of the square is 1. The length of the diagonal is equal to the square root of 2. This number, like Pi, is not resolvable into a whole number. It is known in modern mathematics as an ‘irrational’ number, or a ‘transcendental’, because it transcends or goes beyond our rational ability to know.
If we construct a second square on the diagonal of the first, who’s side is equal to the square root of 2, shown shaded in figure 7a, the area of this new square is equal to 2. So, the diagonal of a square who’s area is 1 is the ‘root’, that is, the structural and generative principle, of a larger square who’s area is 2, double that of the first. This is yet another way of looking at that paradoxical moment when the Monadic One becomes the Dyadic Two by division from within itself. In this case the square of dark earth is divided from within itself by a root which transforms the earth into a visible appearance or flowering.
As Lawlor puts it [SG, p 31], “..let us philosophically raise square 1 to represent the principal of Unity, or that quality of absolute Unity which is represented by the finiteness of the square as a unit, an individual, a wholeness, or a system. Square 2 can likewise be extended to represent Duality, and the power of proliferation that is multiplicity. When one becomes two, we have automatically the potential of endless multiplicity through progression … Thus, the extreme essential polarity of the universe, Unity and Multiplicity, is perfectly represented and observable in the simple drawing of the square and its diagonal.” Elsewhere he writes [SG p 29], “Like the vegetal root, the root of 2 contains the power of nature which destroys in order to progress (it severs the initial square) and it also contains the power which instantaneously transforms 1 into 2. The plant grows progressively out of a previous breaking down, but there is no rational theory which can explain how a flower or a squash can spring forth from a tender, narrow stem, like the explosion of one square out of another. This is a transformative power existing a priori in the causal root.”
Another aspect of this diagram is the natural appearance of the Geometric proportion or progression. As noted in the previous session on Arithmetic, this is written a:b :: b:c, or as stated by Lawlor [LL10 p 82], “each successive term unfolds by being multiplied by the first term.”, and [SG p 27], “… each term, or ratio, is multiplied by a constant value in order to achieve the next term in the progression.” As Wittkower puts it [APH p 109], ” the first term is to the second as the second is to the third.” ; that is, a:b as b:c. Or, to put it yet another way, the mean equals the square root of the product of the extremes, b = √ac. This is the process described in Timaeus [36] by which the Demiurge proportioned the world soul and it is also the source of both legs of the Lambda sequence. If the construction of figure 7a is continued several more steps, figure 7b, it illustrates the principles of progression through continued generation of the double sequence.
The relationship of the square and its diagonal also illustrates the concept that the apparent, visible or changeable world, the flower of the plant, so to speak, may be controlled by an unseen, fixed constant, the root. Lawlor is [SG p 27], “reminded of the alchemical axiom that everything in creation is formed from a fixed, immutable component (proportion) as well as a volatile, mutable component (progression). The relationship between the fixed and the volatile (between proportion and progression) is a key to Sacred Geometry: everything which is manifest, be it in the physical world or in the world of mental images and conceptions, belongs to the ever flowing progressions of constant change; it is only the non-manifest realm of Principals which is immutable.”
The Vesica, the Triangle and the Square Root of Three
We have seen in figure 2c-d that the Vesica gives birth to the right angle cross and to the equilateral triangle. If we look at the Vesica again, figure 8a, in the light of the above discussion of the root principle it can be shown that the length relationship of the two axes AB and CD, within the Vesica is 1:√3. Setting the baseline AB, of the triangle equal to 1, we extend the sides of the triangle until they reach the boundaries of the Vesica circles at points E and F. This creates triangle ECF who’s side length is 2. It can also be seen as two right triangles ECD and DCF, each with base, a, =1, altitude, b, and hypotenuse, c, = 2. Using the ‘Pythagorean’ theorem, a2+b2=c2, we get b2 = 4-1 or b = √3.
Thus the Vesica, considered as the generator of form, gives rise naturally to the Triadic aspect of material structure, in an outward aspect as the triangle, and in an inward aspect as the √3. In conformation of the role of √3 as a ‘formative’ agent, it might be cited that, just as the diagonal of a square who’s side is equal to 1 is the √2, the diagonal of a cube who’s side is equal to 1 is the √3. This is established, as per Lawlor [SG, p 31], figure 8b, by cutting a plane through the cube along the diagonal of a face. This gives a right triangle with base a = 1, altitude b = √2, and hypotenuse c = diagonal. (√2)2 + 12 = c2; 2+1=c2; c = √3.
The Root Rectangles.
We have seen that the diagonal of the square produces the √2 and the Vesica produces the √3. These √2 and √3 systems are related in another remarkable way shown in figure 9a. Starting with a square of side = 1, swing the diagonal down to the base and extend the top and bottom of the square to meet it. This new figure is called a √2 or ‘root 2’ rectangle. The diagonal of this rectangle is equal to √3. Continuing the process, the diagonal of the root 3 rectangle is the √4, which is 2, so a root 4 rectangle is a double square. The diagonal of the root 4 rectangle is the √5. The process, as Hambridge tells us [EDS p 19], can be continued indefinitely. He then points out that a square of side length equal to root 2 has an area of 2, which is twice the area of a square of side length 1, as we have demonstrated above (figure 7a), and that a square of side length equal to √3 has an area 3 times that of a square of side length 1, etc. The root rectangles, then, are related commensurably by area, that is by the square of their sides, figure 9b.
Figure 10 is a key diagram relating the Vesica and the three root systems. It is based on one provided by Lawlor who notes [SG p 37], “The two principal elements of sacred geometry, circle and square, in the act of self division give rise to the three sacred roots. The roots are considered as generative powers, or dynamic principles through which forms appear and change into other forms.”
The Root 5, the Pentagon and Phi.
The diagonal of the double square, which is equal to the √5, geometrically gives rise to the half diagonal of the square as shown in figure 10. Using this length, √5/2, as a radius, a root 5 rectangle can be generated that is made up of a square and two equal, smaller rectangles. Each of these have sides that are in the ratio of 1 : √5/2 -1 or 1 : .618…
This is the famous ‘Golden Section’ usually represented by the Greek letter phi, Ø. Ghyka tells us [GAL p 4], “This proportion , called in the textbooks ‘division into mean and extreme ratio,’ has got … the most remarkable algebraical properties. The Greek geometers of the Platonic school called it h tomh, ‘the section’ par excellence, as reported by Proclus (On Euclid); Luca Pacioli, Leonardo’s friend, called it ‘The Divine Proportion.”
Huntley adds [DP p 25], “Kepler called it ‘the divine proportion.’ … Phidias, a famous Greek sculptor, made use of it. … It was suggested in the early days of the present century that the Greek letter f ]Ø] – the initial letter in Phidias’s name – should be adopted to designate the golden ratio.”
The properties of this proportion answer to a special need in arithmetic and geometry. It may be remembered that for Pythagoreans, a goal of studying the Quadrivium is to accomplish the Socratic motto, “Know Thyself”. At the risk of exposing his own level of ignorance, your author may dare to suggest that this may be paraphrased as ‘experience the One within yourself’, or ‘experience that aspect of yourself that connects you to the One’. Whatever points the way from the many to the One, is considered valuable. Just as the golden sun and the metal gold have been symbolically related to Oneness, the phi proportion embodies an approach to Oneness in terms of geometric construction.
To follow this pathway we may start with the concept of the reciprocal. Hambridge writes [EDS p 30], “… the reciprocal of a rectangle is a figure similar in shape to the major rectangle but smaller in size. The [short] end of the major rectangle becomes the [long]side of the reciprocal. Greek art shows us several simple methods for determining reciprocals, but they all depend upon the fact that the diagonal of a reciprocal cuts the diagonal of the major shape at right angles.”
Figure 11a demonstrates that the reciprocal of a root rectangle divides it into as many equal parts as the root of its side. A √2 rectangle is divided into two equal reciprocals, a √3 rectangle into three equal reciprocals, etc. A square, being representative of unity, is its own reciprocal.
Pursuing the search for Unity, we may ask, ‘Is there any rectangle who’s reciprocal leaves a remainder of unity [a square]?’; or ‘Is there a figure that can be continuously, proportionally divided, and produce unity through division?’ Such a rectangle, shown in figure 11b, has its sides in a geometric progression such that a + a2 = 1. Paraphrasing Lawlor’s discussion [SG p 46]: a ratio is a comparison, a:b. A proportion is “a relationship of equality between ratios.” , and may be written, a:b :: c:d, for example, 2:4 :: 3:6. This is called a four term, discontinuous proportion. Pursuing the path to Unity, we find that the relationship between some ratios can be expressed as a three term, continuous proportion, a:b :: b:c, for example 2:4 :: 4:8.
This proportion has the property that the product of the extremes equals the square of the mean, ac = b2 or b = √ac. It is thus known as a Geometric proportion. We may reduce the number of terms even further, to two, by setting c = a + b. We then have b = √(a2 + ab). If we now set b = 1, we have a2 + a =1. That is, root plus square equals unity. Phi is the only number that satisfies this condition. Lawlor points out [SG, p 46], “This completes the mathematical metaphor for the Trinity: ‘Three that are Two that are One’. It is the ultimate reduction of proportional thought to the causal singularity.”
Numerically, we may substitute (√5-1)/2 for a, we have (2.236-1)/2 + [(2.236-1)/2]2 = .6180 + .3819 = .9999…. Phi, as a transcendental number, does not actually reach 1 but makes a ‘near approach’. Lawlor further reminds us [SG p 46], “The Golden Proportion is a constant ratio derived from a ‘geometric’ relationship … . …it is first and foremost a proportion, not a number… .”
Geometrically, the relationship of the Pentad and Ø is shown in figure 11c. Just as the diagonal of the square is related to the √2 and the altitude of the equilateral triangle is related to the √3, the diagonal of the pentagon is related to Ø. If the side of the pentagon is equal to 1, the diagonal is 1 + Ø = 1.618. The arms and center of the star pentagon in figure 11c further divide the diagonals at the phi points.
Character of the Root Systems
For Lawlor, √2 is characterized as Generative because it [SG p 28], “… contains the power which instantaneously transforms 1 into 2.” , and it establishes the principle of Alternation; the √3 is said to be Formative [SG, p 34]“The √3 contained within the Vesica Piscis is the formative power giving rise to the polygonal ‘world’.” ; and the √5 can be considered as Regenerative, because through its relation to phi, it may be said to ‘return to unity through division’, [SG p 37], “The √5 is the proportion which opens the way for the family of relationships called the Golden Proportion. The Golden Proportion generates a set of symbols which were used by the Platonic philosophers as a support for the ideal of divine or universal love. It is through the Golden Division that we can contemplate the fact that the Creator planted a regenerative seed which will lift the mortal realms of duality and confusion back toward the image of God.”
Aspects of Phi
The Fibonacci Series
Any series of numbers in which each term is derived from the sum of the previous two terms is called a summation series. The particular summation series named for the early Italian Renaissance mathematician Fibonacci, is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … Such number series, however, were known in the ancient world. , [DP p 47] shows the series beginning with 0, 1, 1, 2, … Although this may account for the seeming anomaly of the initial two 1’s, this could not have been the reasoning of the ancients. As noted in the Arithmetic session, zero was not known before the seventh century AD; and in Pythagorean thinking, zero was not a number, but rather the absence of number. Another basis for starting with two 1’s, shown in figure 12a, is that the Fibonacci series can also be considered as a series of squares who’s side length is the sum of the two previous squares. A third, and perhaps deeper understanding is presented by Lawlor [SG p 38], in his discussion of Theon’s exposition of lateral and diagonal numbers, in which Theon begins with the assumption of a square who’s side and diagonal are both equal to 1. This may be seen as a rounding off of the ‘irrational’ √2 but it is also the beginning of a process in which rational, or whole numbers are used to lead us closer and closer to the transcendental.
If we take the ratio of each term in the Fibonacci series to the next term we find that they form more and more precise approximations of Ø. Even more interestingly the values of
the ratios alternate above and below phi. 1/1 = 1, 1/2 = .5, 2/3 = .667, 3/5 = .6, 5/8 = .625, 8/13 = .615, etc. ; further illustrating the concept of the ‘near approach’.
Huntley notes that any two terms can begin the series [DP, p 46], “We may take as a random example 5 and 2 as the initial terms … giving the sequence 5, 2, 7, 9, 16, 25, …, 280, 453, 733, …, 13153, 21282, … from which we may determine approximations to Phi: 16/9 = 1.777, 453/280 = 1.6178, 733/453 = 1.6181, 21282/13153 = 1.61803.” Lawlor illustrates this principle in figure 12b [SG p 42], “Each coefficient oscillates first above then below, coming closer and closer to the perfect irrational state. This is a basic element in what is called Diaphantine mathematics, which sets up numerical progressions which can be seen as representations of vibrational systems in that a vibrating string also moves above and below an abstract node or inexpressible still point. We may more poetically conceive of this as a model of the pulsation of Cosmic Life.”
Lawlor illustrates this principle in figure 12b [SG p 42], “Each coefficient oscillates first above then below, coming closer and closer to the perfect irrational state. This is a basic element in what is called Diaphantine mathematics, which sets up numerical progressions which can be seen as representations of vibrational systems in that a vibrating string also moves above and below an abstract node or inexpressible still point. We may more poetically conceive of this as a model of the pulsation of Cosmic Life.”
The Equiangular Spiral
The geometric progression, A:B :: B:C, forms the structure of the spiral, as shown in figure 13a. In this figure the Phi rectangle is used but as Hambridge notes [EDS p 59], “The first geometrical discovery made by the Greeks, in fact the first general law discovered by man, of which there is a record, was that the angle in a semi-circle is a right angle. … Another great discovery, made later, was that a line dropped from the juncture of the two legs of a right angled triangle to meet the hypotenuse, was a mean proportional between the two segments of the hypotenuse as cut by this dropped line.” Later on, he adds [EDS, p 61], “In geometry a mean proportional line is equivalent to a square root. By definition a mean proportional line is the side of a square equal in area to the rectangle contained by the two extreme lines.” Therefore, any right triangle can generate a spiral.
It is debatable whether this is the earliest rule on record, especially as Hambridge offers no reference, but it almost certainly predates the Greeks. As to the history and properties of the spiral, Huntley writes [DP p 168], “The spiral has been known by a variety of names corresponding to one or another of its features. By Descartes, who discussed it in 1638, it was designated the equiangular spiral, because the angle at which a radius vector cuts the curve at any point is constant. Because its radius increases in geometric progression as its polar angle increases in arithmetical progression, it has been called the geometrical spiral. Halley, noting that the lengths of the segments cut off from a fixed radius by successive turns of the curve were in continued proportion, named it the proportional spiral. Jakob Bernoulli (1654-1705), who was so fascinated by the mathematical beauty of the curve that he asked that it might be engraved on his tomb- stone, called it ( for reasons that will be stated later) the logarithmic spiral. … Sir Christopher Wren, considering its architecture, perceived that the spiral was a cone coiled about an axis.”
Huntley continues, paraphrasing D’arcy Thompson [DP p 168-169], “The fundamental mathematical property of the equiangular spiral corresponds precisely to the biological principle that governs the growth of the mollusk’s shell [figure 13b]. This principle is the simplest possible: the size increases but the shape is unaltered. … The only mathematical curve to follow this pattern of growth is the logarithmic spiral. Because of this, Bernoulli described it as ‘spira mirabilis’. … This interested not only the Greeks of 500 BC but also the Egyptians a thousand years earlier.”
D’arcy Thompson adds [OGF p 181], “This singular property of continued similarity, which we see in the cone and recognize as characteristic of the logarithmic spiral, would seem, under a more general aspect, to have engaged the particular attention of ancient mathematicians even from the days of Pythagoras and so, with little doubt, from the still more ancient days of that Egyptian school whence he derived the foundations of his learning. (footnote, I am well aware that the debt of Greek science to Egypt and the east is vigorously denied by many scholars, some of whom go so far as to believe that the Egyptians never had any science, ‘save only some rough rules of thumb for measuring fields and pyramids’ [Burnet’s Greek Philosophy, 1914, p 5].”
The Gnomon
The equiangular spiral grows by the principle of change in size without change in form; known as ‘gnomonic expansion’. This has already been discussed in the Arithmetic session, but to restate, as per Hambridge [EDS p 103], the gnomon “is that figure which, when added to a figure, increases the size of the latter but maintains its original shape.”; and from Huntley [DP p 169], “A gnomon is a portion of a figure which has been added to another figure so that the whole is of the same shape as the smaller figure.” ; as in figure 13c.
D’arcy Thompson adds [OGF, p 181], “There are certain things, says Aristotle, which suffer no alteration (save magnitude) when they grow. (footnote, Categ. 14, 15a, 30: Greek text). Thus, if we add to a square an L-shaped portion, shaped like a carpenters square, the resulting figure is still a square; and the portion which we have so added, with this singular result, is called in Greek a ‘gnomon’. Euclid extends the term to include the case of any parallelogram, whether rectangular or not; (footnote, Euclid, II, def. 2) and Hero of Alexandria specifically defines a gnomon (as indeed Aristotle had implicitly defined it), as any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. Included in this important definition is the case of numbers considered geometrically; that is to say, the [greek text], which can be translated into ‘form’, by means of rows of dots or other signs (cf Arist. Metaph. 1092 b 12), or in the pattern of a tiled floor: all according to the ‘mystical way of Pythagoras and the secret magic of numbers.’ “
“For instance, the triangular numbers, 1, 3, 6, 10, etc., have the natural numbers for their ‘differences’ and so the natural numbers may be called their gnomons because they keep the triangular numbers still triangular. In like manner the square numbers have the successive odd numbers for their gnomons, as follows: 0+1=12, 12+3=22, 22=5=32, 32+7=42, etc. And this gnomonic relation we may illustrate graphically … by the dots whose addition keeps the annexed figures perfect squares (footnote, cf Treutlein, Z. Math. Phys. 28 (1883), 209).”
The gnomon of the Phi rectangle is the square, a symbol of unity. The Phi spiral is symbolic of growth through time by a return to unity. The spiral curve, proceeding outward will approach, but never reach the infinite curvature of the straight line. Proceeding inward it will approach, but never reach the pole, or mysterious center point. As in philosophical study, it can make only the ‘near approach’
The spiral as a model of time is illustrated by Lawlor [SG p 71], “Gnomonic expansion in nature makes visible patterns of successive stages of growth. This relates to our notions of time in an interesting way. Ordinarily we conceive of time either as a fleeting directional movement from a dissolving past through an imperceptible present towards an imaginary future, or, mystically, as an all-containing, eternal fullness. The gnomonic principle adds a third description of time. This is time as an expanding growth upon growth, an evolution, one might say, belonging to the conscious energies which transcend their transitory forms and substances. … In such a model, past time remains present as form, and the formation grows through pulsating, rhythmic gnomonic expansion. To remove the most recently accreted layer or compartment from a nautilus shell is actually to move backwards in its lifetime. Logarithmically developed forms always carry this element of the retention of past time, and thereby symbolize the evolution not of substance but of consciousness.
“In gnomonic time, all phases exist in ever-present layers. like the light-year structure of galactic space which makes every glance into a starry night sky a view into the past of distant bodies, while the layers of light just behind the visible light are the future energy waves which will strike and influence the earth. All aspects of the material world, including our own bodies, are therefore in the past tense, existing in a residual gnomonic layer already bypassed by inflowing cosmic energies. This is a rather disturbing idea, yet not dissimilar to our ancestors’ notions of time.”
Later he adds [SG p 73], “In the philosophic approach to geometry we are attempting the contemplation of characteristics of form as carrying meaning in themselves. There is a teleological message contained, for instance, in the spiral itself; for it moves in successively opposite directions towards the ultimate expression of both the infinitely expanded and the infinitely contracted. The spiral is constantly approaching these two incomprehensible aspects of the ultimate reality, and therefore symbolizes a universe moving toward the perfect singularity from which it arose. Thus the spiral-like arms of our galaxy constitute an image of the continuity between fundamental polarities – infinite and finite, macrocosm and microcosm.”
The equiangular spiral then, symbolizes the process of growth through time by gnomonic expansion. It is based on the geometric progression, as summed up in the Lambda sequences 1, 2, 4, 8, & 1, 3, 9, 27; and the Lambda, in its turn, begins with the One.
End of Geometry Notes