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t was long ago that I first noticed that the builders of edifices were discontinuing the use of natural materials, and I was disquieted by the diminution of activity in the stone quarries that our ancestors traditionally exploited so fruitfully for construction. Questioning these conditions that troubled the fortunes of numerous enterprises, I sought to discover their cause, and set off to travel throughout the world, guided by the “seven spirits who stand before the throne of God.”
Having thus collected and recorded many observations, I was on the point of terminating my travels when, overcome by fatigue, I fell asleep by the roadside and dreamt that I saw a marvelous garden in the middle of the desert, surrounded by three octagonal enclosures and planted in quincunxes, in which trees bore golden apples. I went to cross the entrance in order to take some of the fruit, but a coiling green dragon blocked my access, spewing flames at me. A hundred times I tried to surprise the vigilance of the dragon with the help of the clear and distinct ideas that my masters had taught me as being of obvious efficacy; a hundred times surrounded by thick flames and striving in vain, I was forced to remain outside the enchanted enclosure. Discouraged, I was about to abandon the struggle, and was filled with doubts and confusion amidst the darkness; but then, having invoked the Archangel Saint Michael, I saw the word ABRACADABRA written in the sky, and immediately, the dragon having disappeared, an invisible force drew me into the garden of Eden. At this same moment I awoke and, well rested, reached the end of my route. I immediately began my quest for the magical power of ABRACADABRA.
I already knew that this word symbolized Father, Spirit, and Word[1] or Triad; writing them out, according to their meaning, in the following triangular form:
I formed all the combinations of letters, writing the words A, AB, ABR, ABRA, etc., starting at the left angle of the base, without skipping any lines and without descending.[2] I then substituted the last letter of the word with the number of combinations found, thus obtaining this arrangement:
I immediately recognized Pascal’s arithmetical triangle.
This arrangement of numbers indicated many hidden things relating to human nature, and I located them in circumnavigating the famous ancient triangle in the direction of the earth’s rotation (Fig. 2).
According to “forma” or discontinuity, the sum of the numbers, for each column, enumerates all the possible options between things defined by their number, combining these in all possible ways, and thus it signifies the liberty or human will which, being one, can at the same time be all, in that it participates in Unity but is also limited by the multiplicity of things.
According to “materia” or continuity, the numbers on each line represent the most simple geometric elements that can be conceived in each dimension of space: line, triangle, tetrahedron, hyper-pyramid, etc.
According to “complex-um,” or the product of continuity and discontinuity, the sum of the numbers constitutes the series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, which Fibonacci first described in the Liber Abaci, counting the number of offspring produced each month by a pair of rabbits, fertile at the end of a month and producing two offspring each month; thus it composes a continuous series in which each term equals the sum of the two terms preceding it.
Pursuing my studies, and considering the ratios of two successive terms of this series, I saw that it was possible to represent them with a simple continuous fraction:
It has been known for a long time that the limit “g” toward which this continuous division tends can be expressed by the following equations:
g = (1 + √5) / 2 = 1 / (2 cos [2π/5]) ;
whence, if g = a / b and a > b,
(a + b) / a = a / b.
This ratio g = a / b, thus defined, divides a given length in such a way that the total (a + b) has the same ratio to the larger part (a) as the larger has to the smaller (b), and thus the whole is divided in parts so that the largest is the geometric mean of the smallest and the whole.
This ratio, which the ancients called “the rule of mean and extreme proportion” or “divine proportion” because it indicates the universal analogy of being in the most simple way possible, has discontinuous and continuous properties at the same time, that is to say, natural properties that are illustrated by the following series of additions:
1 g ^{n} – 8 | |||||
1 g ^{n – 6} | 4 g ^{n – 7} | ||||
1 g ^{n – 4} | 3 g ^{n – 5} | 6 g ^{n – 6} | |||
1 g ^{n – 2} | 2 g ^{n – 3} | 3 g ^{n – 4} | 4 g ^{n – 5} | ||
1 g ^{n} | 1 g ^{n – 1} | 1 g ^{n – 2} | 1 g ^{n – 3} | 1 g ^{n – 4} | |
————— | ————— | ————— | ————— | ————— | etc. |
g ^{n} | g ^{n} | g ^{n} | g ^{n} | g ^{n} |
and in this way, through the coefficients, the arrangement of numbers of PASCAL’s arithmetical triangle appears, from which the ratio “g” emerges.
Considering the preceding triangular tables, it was easy for me to see that the sums of the numbers, according to “forma,” in relation to the five vowels A, were: 1, 8, 32, 64, 1. These numbers, not counting the 1, represent the number of squares in a row of a chessboard, the number of pieces, and the total number of squares. Also, in order to define the arithmetical triangle, I divided the chessboard with a diagonal line stretching from the bottom left to the top right, then wrote the number 1 in the squares along the bottom, and in the other squares, the sum of the number in the adjacent square to the left on the same row plus the number of the square to the left and one row down (Fig. 3). Thus, having numbered the squares in a triangular form, I confined the FIBONACCI sequence to seven numbers.[4]
Lagrange demonstrated that every continuous periodic simple fraction can be represented by a construction with a ruler and compass. Such a construction of the continuous fraction revealed by ABRACADABRA, where only the unity appears, is the simplest possible, all multiplicity being thus excluded. To reveal this construction, I first recognized that the vowel A was represented by an isosceles triangle, all of whose angles were acute. Then, in the interest of simplicity, I naturally drew the angle that is unequal to the two others as half of each of the equal angles, such that a circle divided by 5 measures this angle.
Having placed the three vertices of this triangle upon a circle, I proceeded to inscribe four other triangles similarly and regularly on the same circle, thus forming a regular pentagonal star or pentalpha (Fig. 4).
In this figure I effectively rediscovered the ratio “g” between the long and short sides of the isosceles triangle, because:
cos (2π / 5) = 1 / (1 + √5) = 1 / 2g.
Furthermore, it was easy to demonstrate that between the longest side of the triangle and the radius of the circumscribed circle, there is a ratio “h” with the value √(1 + g2), and that THE RATIO h/g IS THE RATIO OF THE SIDE OF A REGULAR PENTAGON TO THE SIDE OF A REGULAR HEXAGON WHEN BOTH ARE INSCRIBED IN THE SAME CIRCLE.[6]
Finally, the pentalpha forms a similar pentalpha at its center, inverse and consonant with the exterior pentalpha according to the divine proportion.
Thanks to ABRACADABRA I thus discovered, on the chessboard (discontinuous) and on the pentalpha (continuous), the divine proportion, symbolizing the universal analogy that rules the multiple states of being in the opposite direction, and according to which the “forma” and “materia” of the created beings are unified while still remaining distinct. I then sought to use this proportion to discover the key to the construction of edifices, conceived in a way that uses the natural stones with the maximum natural commodity and harmony.
To this end, inspired to pursue my studies, I represented Unity with a very small square, of side 1, and studied the manner by which this little square—or pole, by its own proliferation—could generate a surface of a certain area. For this reason I traced a second square, identical and adjacent to the first, and thus obtained a rectangle of module[7] 2, called, according to the traditional language, “silver rectangle” (Fig. 5a).[8] But I did not continue this construction, which had given me an indefinite line and not a defined surface, and I adjoined a square of side 2 to two opposite squares A and B, thus assuring their development (Fig. 5b). I then obtained a rectangle of module 3/2.
In the same way, one after another, I constructed a series of rectangles, almost identical to each other, whose modules were equal to the ratio of two successive numbers in the FIBONACCI sequence; and since this ratio has the number “g” (the golden number) as its limit, the small square (pole), or principle of evolution, tends in its proper proliferation to manifest itself in a rectangle of module “g” known as the “golden rectangle.” I thus saw the series of rectangles develop in such a way that the corners of the squares were situated on BERNOUILLI’s logarithmic spiral, the pole of the spiral being placed at the meeting point of the orthogonal diagonals of two contiguous figures of the series, this pole being quadrupled by the symmetry of the figure along two axes, and finally merging with the position of the small square whose proliferation generated the golden rectangle (Fig. 6).
Working in this way, bit by bit, I forged the key to open the door to the temple of Natural Architecture. First, I wanted to make the house symmetrical according to the positioning of the parts of the human body, that is to say by breadth alone and not by depth or height.[9] To this end, tracing each symmetrical façade around a vertical axis, I drew the sides of the edifice as identical, the façades as differing, and the roof as sloping.
It has been said with good reason that the concrete being is an existing one, realization of an essence, its immediate principle, enveloped by the realizing substance; therefore, the plans of houses constructed in the image of nature should be divided into squares signifying the principle of the existing, then into golden rectangles signifying the existing itself. They should then be surrounded by gardens whose ground plan is a silver rectangle symbolizing the realizing substance (see p. [*]21).
I saw also that in plans, private houses had the ratio of golden rectangles generating one another and formed by a number of small elementary squares defined by the FIBONACCI sequence, and that they were divided into three orders, as follows: order B with five small squares along the façade and three along the sides, order D with eight small squares along the façade and five along the sides, and order G with thirteen small squares along the façade and eight along the sides, this order having as its complement order F, whose plan, in a horseshoe shape like that of order G, is inscribed in a square of thirteen small squares per side.
Each plan is divided by the partition walls into three consonant parts according to the divine proportion: two parts, on the sides, identical and opposite, with the same ratio as the two whorls of the key (Fig. 7), are reunited by a third part, central but different, like the strophe, antistrophe, and epode in ritual chants. In the extremities, I assigned the area reserved for men to the left, the area reserved for women to the right, and the doors, passageways, staircases, and closets to the central part. The size of the small square defined the dimension of the edifice.
Façades and sides correspond to each plan, included either in the squares or in the golden rectangles, in such a way that all the faces of the parallelepiped that encloses the house are either squares or golden rectangles, and the height of the roof is determined in this manner.
With open windows: in order B, only the main façade; in order D, the two façades; in orders F and G, all the façades and sides. The placement of the windows was such that each façade or side was divided by the openings into squares and into golden rectangles.
Tradition tells us that every state of a being finds its explanation in its principle, which rests immobile, governing evolution, and that the various principles of the multiple states of the same being are joined by a straight line perpendicular to the planes of manifestation, this straight line thus representing the continuity of the unique principle which rules all the states of this being. Naturally, I traced the two columns (dotted lines) of the roof on the vertical planes that contain the “poles” of the planes of the different stories of the edifice, and I determined the extent of the roof itself in such a way that the ratio of its height to that of the walls was the same as that of mean and extreme proportion (Fig. 7).
The rules of natural architecture being thus defined, I sought to synthesize the essential proportions of an edifice in a pantacle. Also, knowing that multiplicity, or evolution, was contained in the Unity or principle, so that the rectangle was contained within the square, I set about constructing four silver rectangles and four golden rectangles inside a given square.
For this, I divided one square, with equal sides of 1, into four squares, with the lines εε′ and φφ′ (Fig. 8). In each silver rectangle thus formed (αβφ′φ, βγε′ε, γδφφ′, δαεε′), I drew a diagonal. The four diagonals (αφ′, βε′, γφ, δε) bisect each other in pairs, symmetrically, at the corners of square α′β′γ′δ′.
Then, on diagonal αφ′, I used the length α″φ′, (= βφ′), and on the side αβ of the square, the length αμ (= αα″), and thus obtained the golden rectangle αμμ′δ, because:
αδ/ αμ = αβ/αμ = 2 / (√5 – 1) = (1 + √5) / 2 = g.
Furthermore, by joining corner β to point α″ with a line whose extension intersected side αδ at point ω, I obtained another golden rectangle αβω′ω, because αω = αμ.
[1] In Hebrew: Father, Ab (אב); Spirit, Rvach (רוה); Word, Dabar (דבר).
[2] For more clarity, I used arrows to represent the three different paths to take so that the word ABRA will end with the first A of the second line.
[3] The 9 (= 5 + 4) divinities that dominate the garden recall the 5 consonants and 4 vowels of the word ARSENICVM, which is the name of the Philosopher’s Stone.
[4] This limitation served to define the rules of natural architecture (see p. [*]6).
[5] The sum of the numbers composing the lines on one side of the diagonal, and that of the numbers on the other side, is in each case 33 (= 12 + 21 = 1 + 10 + 15 + 7).
[6] The value approached by “g” is 1.6180..., that of “h” is 1.9021..., and that of h/g is 1.1756....
[7] The module is the ratio of the long of a rectangle to its short side.
[8] The silver rectangle defines the ratio of mean and extreme proportion through the ratio of the sum of its diagonal and its short side to its long side: g = (√5 + 1) / 2.
[9] “Symmetry, in what one sees at a glance, is based upon that which cannot be reasonably done otherwise, and also based upon the human figure, whence it comes about that one only sees symmetry in breadth, not in height or depth.” (Pascal, Pensées, 28).
[10] The symbolism of this figure, where the Mother and Child are placed at the center of the monstrance, between the Grail and the Athanor, is highly representative of the thoughts that inspired the Author to write this “Report.”
[11] The double spiral can be considered a symbol of the winding and unwinding of the “Shakti” (see pp. [*]22-23), as well as of the waxing and waning of the moon.