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Masonic Symbolism of the Arithmetical Number Five and Its Plane Geometric Construct the Pentagram and Solid Geometric Construct the Dodecahedron
by Stanley J. Bransgrove
For presentation to the California College of the Societas Rosicruciana in Civitatibus Foederatis on 20 September 1998
The number five  and its byproduct, the pentagram, have symbolic prominence and have been attributed extraordinary qualities. Possibly before ancient Sumerian astronomers observed the curious natural phenomena that of the seven visible planets, only Venus traced a regular geometric pattern, a pentagram in eight-year cycles, this number and figure may have already achieved symbolic import. We can confirm at least that what was observed above, was repeated by man here below. In examining remaining monuments of this distant past we find evidence that in fact the symmetry of the pentagram was utilized in mans most important architectural edifices. We may also interpret the stone work of this initiatic tradition in light of literary references to such western traditions such as those of the Pythagoreans, Jews, Christians, Druids, as well as the Lataif of Sufi tradition, and Hindu Tattwas. (See Appendix.) The pentagram served the Pythagoreans as a salutation and a symbol of health. Medieval masons considered it a symbol of deep wisdom and integrated its proportions in ecclesiastical edifices. In recent times, professor Matila Gyrka called the pentagram the "gateway to absolute independence;" "symbol of deep wisdom;" and an "ideal archetype of dynamic growth." It is not surprising that we find such symbolism reoccurring in speculative Masonry and Masonic Rosicrusianism.
As A Masonic Symbol
The number five and the pentagram are prominent symbols in speculative Masonry, and its concordant bodies. They appear in both patent and latent forms. This trestleboard will explore where the number five occurs in the three degrees, and will attempt to explain why they have been and remain such potent symbols.
In Masonry, the pentagram first appears in the monitorial section of the first degree. It appears as the blazing star on the checkered pavement. In the second degree the symbolism of five becomes most prevalent. We find that a lodge of fellowcrafts is composed of five officers (two masters and three fellowcrafts).  In due-form, the candidateís arms and legs form four right angles in different planes inside a fifth right angle. Our attention is drawn to the five senses, and the five orders of architecture. Our attention is particularly drawn to the fifth liberal art, geometry. In some countries rather than the letter "G" the candidate is introduced to a blazing star suspended in the East. Another symbol that has become obsolete in American Masonry is the broached thurnel also known as the pointed cubical stone.  A fellowcraft is said to have attained the age of five years. In the third degree, the speculative mason is introduced to the five points of fellowship, which on some Masonic charts is represented as a star inscribed with clasped hands.  We might also see in the three interlaced triangles that make up a pentagram a demonstration of the intimate connection between the three ancient grand masters.  This tri-part relationship becomes even more apparent in the symbolism of certain supernumerary degrees.
Addition And Multiplication Of Numbers
Geometry provides a clue to what these symbols seek to reawakening in the mind of the initiate. In St. Paulís Epistle to the Ephesians, Paul questions how men, "May be able to comprehend with all saints what is breadth, and length, and depth and height." Ephesians, 3:18. We have learned in Lodge that geometry treats of the powers and properties of magnitudes in general, where length, breadth, and thickness are considered. From a point to a line, to a superfice, to a solid: Or it is the transition from the first, to the second, to the third dimension.
The beginning of all geometrical matter is the point.  A point has no length, breadth, or thickness. Hildegard  described the creator as a sphere with the center everywhere and circumference nowhere (i.e., a point within a circle). The point might be described as unity, the monad,  the source, the unrealized beginning of all matter. In symbolic mathematics one is omnipotence since one raised to any power (squared, cubed, etc.) is still one. 
The first extension into the first dimension is a line.  The line has length, but no breadth or thickness. It is the dyad, polarity, positive-negative (but without connection, still just a potentiality), yin-yang, hot and cold, wet and dry. It is the male-female relationship unmanifest because no child manifests their relationship. It is the artist, brush in hand before touching the canvas. In Masonry consider Boas and Jachin as terminal points in a line. Symbolically it is the will of God to manifest yet remaining unmanifested. 
Only in the second dimension does manifestation occur.  A triad is a superfice with length and breadth. It is the visible dimension. It is the mechanism to channel hot and cold or the wire completing the circuit. It is manifestation of a creative potential: A male-female relationship creating a child, an artist's application of paint to canvas. Boas and Jachin together with Israel. (See, Corinthians at 3:17 and Kings at 7:21.) In Masonry, the twin columns of Boas and Jachin balanced with the candidate (strength, wisdom and beauty together).  It is God's will to manifest striking out to create.  It is God given form. It is a trinity: Father, Son, and Holy Ghost; Brahma, Vishnu, and Siva; Bouddha, Dharma, and Sanga; Osiris, Isis, and Horas.
The number four may be described geometrically as the first solid: It has length, breadth, and thickness.  It takes us into the third dimension. It has been described as the first descent into matter. Hence we find the occurrence of such descriptions as the four winds of heaven, foursquare firmness of Earth, four corners of the Earth, four points of the compass, and the ancients four elements.  In the Old Testament we see that Israel was divided into four principal tribes, Judah, Rueben, Ephraim, and Dan. These tribes were represented respectively by the lion, ox, man, and eagle, which are in turn related to the astrological signs of Taurus, Leo, Aquarius, and Scorpio. In the New Testament we see the four evangelists Matthew, Mark, Luke, and John. Symbolically these might be said to represent the fixed, eternal signs of One reality. 
Finally, the number five brings us to the expansion beyond three dimensions. It is raising the potential of matter.  It is breathing life, and giving intelligence to matter. It mirrors unity. Symbolically, it is unity added to matter.  "And the Lord God formed man of the dust of the ground, and breathed  into his nostrils the breath of life; and man became a living soul." Genesis 2:7. When man is expressed as a pentagram, man symbolically joins heaven and earth.
From geometry, we might deduce that only God adds.  God had to breathe life into the proto-man formed from the clay of the earth. Man cannot similarly animate a clay figure: Man cannot add the breath of life. Man can reproduce by multiplication and division. Man cannot infuse life where none exists.  Five times, Genesis enjoins, "Be fruitful and multiply." Genesis 1:22 (fish/animals); 1:28 (Adam & Eve); 8:17/9:7 (Noah); 35:11 (Jacob and the new nation of Israel).
We see from this that the elements are accessible to man because they can be reached through multiplication.  We see too that man's basic duty or responsibility to nature is fulfilled by reproduction of the species. Beyond this, seeking a conscious communion with the creator requires a conscious exercise of human will - i.e., the candidate must seek and ask for initiation and even then, the route is not direct but circuitous like a spiral stair case where the end is hidden and never in view during the ascent.
We are given another clue in another monitorial section of our ritual. The Master extols the candidate to consider symmetry, order, and proportion in nature. "The candidate is told that a survey of nature, and the observation of her beautiful proportions, first determined man to imitate the Divine plan, and to study symmetry and order," which "gave rise to societies and birth to every useful art." In this regard, five and in particular the pentagram or "blazing star," helps to rationally explain our relationship to the Creator. It is a blueprint of the figurative winding staircase.
The Divine Proportion And The Way Of The Initiate
Heraclitus  insisted that "nature loves to hide" and that this hidden reality is beyond the reach of men who trust implicitly in their sense perceptions - the mystery is discoverable through reason and analogy.  St. Paul wrote, "And to make all men see what is the fellowship of the mystery, which from the beginning of the world hath been hid in God, who created all things by Jesus Christ." Ephesians, 3:18.
On observing nature the Mason or Rosicrucian may discern curious facts which can be summarized in the relationships that occur in the geometric figure of the pentagram, and has been variously described as the "Golden Proportion," "Golden Section," and by the twenty-first letter of the Greek alphabet, F (phi). The proportion is defined as the division of a line in the most economical way possible so that the lesser part is to the greater as the greater is to the whole.
In symbolic language, it is: Three things working through two things to yield one thing.  As a symbol it becomes a means of reawakening, forming a symbolic bridge between heaven (circle) and earth (square). It is a means through which we can examine the relationship between nature-man-creator. It is the only value in the universe that relates to One by creating four elements that pass through four, three, and two. 
In nature it may be observed in what is sometimes termed the Spiral of Life. From the imperceptible level of the double helix of the DNA  to the Milky-Way Galaxy, our nature unfolds her mystery as a golden proportion spiral: As below, so above. More readily we can observe in looking at life forms around us. Such living things as the scales of a pine cone, the seeds in a sun-flower, the spikes on a pine apple, or the arrangement of leaves on a rose stem and distribution of its petals  manifest this spiral of life, since no matter at what stage of life the plant is examined, its growth radiates in a logarithmic and equiangular spiral consistent with the golden proportion.  Likewise the golden proportion is apparent in a nautilus shell, the arrangement of the scales on a fish, and in the proportions of limbs of man.  Even in observing the sides of our own hand uncurling from a clinched fist we can readily see the spiraling effect of the Golden Proportion. 
A mathematical expression of this spiraling proportion is found in the Fibonacci series.  The Fibonacci series demonstrates the peculiar proportion that results from a sequence of integers in which each integer after the second is the sum of the two preceding integers, i.e., 1, 2, 3, 5, 8, 13, 21, 34, etc.  By dividing any number in the series by its preceding number, the remainder is approximately 1.61803Ö, which is the approximate equivalent of F (÷ 5+1/2 or F ) which metaphysically might be termed the mathematical embodiment of perfection or literally translated as the addition of the monad to the initiate who together dominate duality.  Any number in this series, when divided by the following number approximates 0.61803Ö, the decimal equivalent of 1/F also represented as F - 1, which metaphysically represents the monad dominating the perfected individual or in other words, the difference between the monad and perfected individual.  The slight divergence of the Fibonacci series from precise mathematical F might metaphysically be explained as the disparity that exists between material perfection and divine perfection. In other words, while the mathematical relationship mirrors the divine, is not divinity itself. However, we might see in its reflection an existent ability in man to attain an eventual complete realization of his moral possibilities. 
Symbolically the F relationship coordinates and balances the five senses of human nature, which are analogous to the four elements (Sight - Fire - zeal; Hearing - Air - thought; Taste - Water - emotion; Feeling - Earth - body/physical; Quintessence - Unity - Love - respiration).  Symbolically, a candidate who successfully passes through the trials of Earth, Water, Air, and Fire "sets himself free from material life, philosophy, and religion to reach Absolute initiation at last.  In a practical sense, when an individual is awake to and conscious of F , he knows what faculty needs to be used and in what measure. In other words, he has subdued his base passions. Such mastery of the passions is essential to avoiding the distractions that cloud higher thoughts.
It might be perceived that a lack of balance in coordinating these senses, "elements" or mental states creates a state of disequilibrium or "disease." In such a state of un-risen potentiality the person cannot be awake. Their consciousness has through involution become wholly subsumed in matter. Initiation is a means of awakening the latent state, beginning the person on the path of return and of perfection as an initiate. The perfected initiate, that is, one who has been awakened and not merely been the recipient of titles, hence becomes conscious of God.
The process of involution, evolution along the path of initiation and perfection, can be expressed mathematically in F relationships. Phi, as previously discussed, is mathematically expressed in terms of the square root of five. The root-five family of relationships, as demonstrated geometrically, consists of just seven members: 1, 2, ÷ 5/2, F /2, ÷ 5, F ≤, and ÷ 5+1/2 (or F ). 
In symbolic terms, each member of the root-five family carries a different meta meaning expressive of the relationship to the creator. One, as discussed more fully in the examples above, is representative of the monad, or unrealized beginning of all things. Two, represents the dyad, or the will of God to manifest that is yet unmanifested.
The square root of five divided by two (÷ 5/2) represents involution.  Involution might variously be described as the regressive alteration of a body, or a decline marked by a decrease of bodily vigor. It might also be seen in metaphor as the fallen state of man as described in the story of Adam and Eve in Genesis.
The opposite of involution is evolution, represented in the relationship of Phi divided by 2 (F /2).  Evolution may be considered as a process of continuous change from a lower, simpler, or worse to a higher, more complex or better state. "Verily, verily I say unto you, if any one be not born again he cannot see the realm of God." John, 11:3. In the reconstruction of Solomonís Temple, the dimensions of the portico containing the pillars of Boas and Jachin yield the proportion of evolution.
The way of evolution is the path, represented by phi squared (F≤).  As expressed in theological terms, "I am the resurrection and the life [or path]: he that believeth in me, though he were dead, yet shall he live." John 11:25. In the reconstruction of Solomonís Temple, the dimensions of the interior of the Temple yield the proportion of the path. Visually, it is apparent that the aspirant must tread the path to reach the Sanctum Sanctorum or Holy of Holies represented by the perfect cube.
The square root of five (÷ 5) is representative of the initiate.  Man in his fallen state, evolving toward his potential awakening as he treads on the path. In the reconstruction of Solomonís Temple , the dimensions of the courtyard or parnaos outside the Temple yield the proportion of the initiate.
That awakening is expressed by the addition of the monad to the initiate, which results in the domination of the dyad, expressed as ÷ 5+1/2, which is phi (F ) itself. This is the state of perfection, which in theological terms, might variously be described as Christ consciousness, Krishna consciousness, transcendence, nirvana, opening the eye of Horus,  etc. It is being awake to the breath of Divinity animating us. It is true initiation. We read in the Egyptian Book of the Dead at Chapter 80: "I am the woman, an orb of light in the darkness. I have brought my orb to darkness, it is changed to light." Or, as stated in John at 1:5, "Öand the light shineth in the darkness and the darkness comprehendeth is not." In the reconstruction of Solomonís Temple, the outer dimensions of the Temple, i.e. the whole completed, yield the proportion of perfection.
Returning to the Masonic charge imparted by the Master, we are told that from his study of symmetry and order, man imitated the Divine plan, and this imitation gave rise to societies and birth to the useful arts. The truth of this statement, can be found in an examination of the proportions used in the monuments of early society, e.g., Egyptian Temples and Pharaonic tombs, Myan pyramids, or Gothic Cathedrals such as Chartres.  An examination of sacred architecture demonstrates that the foregoing F relationships have been rigidly adhered to in constructing temples for man to commune with God. 
This fundamental proportional of Phi (together with the porportionals of the square root of two, and the square root of three),  which Plato showed in his Timaeus through the superimposition of the five regular solids  - cube (earth), icosahedron (water), octahedron (air), and tetrahedron (fire) - "hold together and differentiate the basis of our experience of the world, the cosmic structure, this holy shrine we inhabit. Thus it should come as no surprise to find just these exact proportionals present in the sacred and traditional Temple, Mosque, and Cathedral structures throughout human history, for it is precisely through these proportionals that the Sacred Buildings, as the cosmos, could become shrines, the same proportionals maintaining the greater cosmos as maintain the microcosmoses of our bodies." 
A Secret Lost
Coming full circle, we see the foregoing proportionals in Masonry. From the construction of the temple, modeled on the proportions of Solomonís Temple.  The continuing theme of the star (first subsumed in the matter of the carpet, second elevated as a beacon, and finally met as a symbolic embrace) and emphasis placed on "five" affords a direction for our investigation of proportion that can be confirmed in the work of our operative brethren. Returning to another speculative symbol,  the broached thurnel  may be interpreted so as to demonstrate the foregoing principles. It is a model based on pentagonal symmetry by which one can study nature's relationship to man, man's to the creator, and the creator.
Viewed from the exterior, the pointed cubical stone symbolizes the structure of reality itself. The cubical portion is suggestive of the quaternary of the four traditional elements - air, water, fire, and earth.  These material elements are surmounted by the divine ternary. The axe broaching the pointed cubical stone suggests going inside to seek unity through proportion. Unfolding the broached cubical stone, we find a figurative representation of the quintessence, regarded by the alchemists as the fifth element, created by a synthesis of the other four.  However, only four parts of the pentagon are physically present.  The invisible fifth part, when again folded into a point, "dissolves" into and gives form to the other four parts, just as the creator gives life by being within all living things.
In symbolic language, five and its natural geometric constructs (pentagram, pentagon, dodecahedron) have been used to demonstrate the fundamental relationship between man, nature, and the creator for over 2,600 years. From clues supplied in Masonic ritual and symbolism, we are led to rediscovery of the symbolic significance of Five. In this regard, it could be said that the three degrees of masonry represent different aspects of F relationships creating a ÷ 5 or the initiate: 1į - Nature - Creator; 2į - Man - Creator; 3į - Creator.
* Bro. Bransgrove is a member of: Mill Valley Lodge No. 356, F & AM, Mill Valley, CA; Marin Chapter No. 102, R.A.M., San Rafael, CA; Oakland Valley A... A... S... R, Oakland, CA; California College, SRICF, IVį.
 "Five" has been called providence, because it makes unequals equal since any odd number added to five becomes even. "Five" has been identified with nature because if multiplied by itself it returns into itself (a peculiarity shared only with the number 6).
 Two, three, and five are in the Fibonacci series, and as a proportion approximate F , infra.
 In comparison, in the First Degree, the candidateís body forms three right angles in different planes inside a fourth right angle. In the Third Degree, the candidateís body forms two right angles in different planes inside a third right angle. A progression may thus be deduced from the foursquare matter, through the aspirated - "living" number of man, to the number of manifested deity.
 The Pythagorean Triangle, at xi-xii.
 The Pythagorean Triangle, at 125.
 The nonagon or nine-sided figure representing the camp of the Scottish Rite degrees might be considered as a variant of this. Compare also the nine-sided figure identified as an enneagram in the Gurdjieff-Ouspensky school that in general terms also speaks symbolically to balance and equilibrium. However, it should be noted that the enneagram geometrically represents the law of seven (if you divide any number under 7 by 7, the same series of digits results and repeats into infinity) and the law of three (dividing numbers up to 3 by 3 will result in recurring infinite decimals whose digits do not appear in the series of digits created by division by 7) rather than the square root of five relationships discussed below. The symbol of the enneagram, inter alia, expresses the use of awakened will to straighten the spiral of natural existence, or in other words making the spiral staircase a ladder such as that envisioned by Jacob. Consider in this regard Godís injunction to the tribes of Israel not to turn either to the left or right as they sought to escape from darkness (Egypt) in Numbers at 20:17, to be strong and courageous observing the law and not turning left or right in Joshaua at 1:7, and to listen for a voice designating the way from which they should not deviate in Isiah at 30:21.
 See generally, A Beginnerís Guide to Constructing the Universe, Chapter 1 at 1et seq.
 St. Hildegard (1098-1179) was a German abbess and mystic whose revealed visions influenced St. Bernard of Clairvaux. Encyclopedia Britannica, 11th Ed. (Cambridge, England, 1911). Bernard of Clairvaux favored division of monastic orders along lines of function - spiritual, ministerial, and militant. As a Cistercian, Bernard was devoted to primitive, ascetic monasticism. The Cistercians relied heavily on recruitment of peasantry as lay brothers to perform useful trades that the monks could not carry out owing to their choir and religious duties. As the Cistercian Order grew under the tremendous influence of Bernard, the Cistercians became noted builders whose abbeys are especially noteworthy because of their strict adherence to proportions. Bernardís influence on dividing monastic orders functions is also evident in his successful efforts in helping the Knights Templars gain recognition as a new monastic order whose rule (said to be written by Bernard) permitted Termplar monks to wage war. Encyclopedia Britiannica, "Bernard of Clairvaux," and "Cistercians," 11th Ed. (Cambridge, England, 1911) and also Priv. Diag.
 The Pythagorean Triangle, at 28-29.
 Priv. Diag.
 See generally, A Beginnerís Guide to Constructing the Universe, Chapter 2 at 21et seq.
 Priv. Diag.
 See generally, A Beginnerís Guide to Constructing the Universe, Chapter 3 at 38 et seq.
 Consider the interpretation of a lodge room as representative of a physical body, and the candidate being born into the lodge from between the "legs" of Boas and Jachin as quoted critically in The Lodge and the Craft, Roland Blackmer (McCoy).
 Priv. Diag.
 See generally, A Beginnerís Guide to Constructing the Universe, Chapter 4 at 60 et seq.
 In Hebrew, Iammim (sea), Nour (fire), Ruach (air), and Iebeschah (dry earth).
 Penguin Dictionary of Symbols, "Four," at 403-404. And see, Metaphysical Bible Dictionary, "Judah" the fourth son attributed there to the spiritual faculty, "Ruben" the twelfth son attributed to faith, "Ephraim" the second son attributed to will, "Dan" the fifth son attributed to judgment.
 See generally, A Beginnerís Guide to Constructing the Universe, Chapter 5 at 96 et seq.
 Priv. Diag.
 Air is a palpable symbol of invisible life, universal driving force and purifier, universally regarded as principle of life. The Spirit of God that brooded upon the face of the primeval waters in the Genesis was ruach, "breath." In addition to the story of Adamís creation discussed in the text above we find in Job, 33:4: "The spirit [ruach] of God hath made me, and the breath of the Almighty hath given me life." (4 + 1 = 5) Theology holds that the breath of life given by God to mortals is imperishable: "Then shall the dust return to earth as it was: and the spirit shall return unto God who gave it" (Ecclesiastes 12:7) This may be considered as 5 - 1 = 4. Psalm 104:29-30: "Thou hidest thy face, they are troubled; thou takest away their breath, they die, and return to dust. Thou sendest forth thy spirit [or breath], they are created; and thou renewest the face of the Earth." This accords with the Hindu belief in the domain of Vayu, the wind and breath of life joins the two worlds of heaven and earth. Believing there is an identity between the structure of the microcosm and macrocosm, Yoga tradition holds that the individual is held together by breaths, of which there are five different types that regulate the vital functions.
 Priv. Diag. Consider, 1+1 = 2 + 1 = 3 + 1 = 4 + 1 = 5 Ö. In contrast, One causes multiplication and remainder but does not admit of either itself: 1 x n = n, and n/1 = n, but 1 x 1 = 1 and 1/1 = 1.
 Consider the moral implications of man seeking to usurp this power. For instance, in the Kabalistic story of the Golem, a sorcerer imitates Genesis using red clay to mold a statue of a man. By writing the Hebrew letters (which have numerical equivalence) on the forehead, the statue comes to life to do the magicianís bidding. However, the statue is speechless, and continues to grow to gigantic proportions. Without the magicianís guidance it does only evil acts. It is in effect a slave to the magicianís passions. The magician is forced to erase one letter, changing the word "life/truth" (Emeth = 441) to "death" (Meth = 440) and thereby returns the Golem to inanimate clay. The inability of the magician to impart ethical qualities while aping God hints that good is linked to divine inspiration. Dictionary of Symbols, 444-445. "And God saw every thing that He had made, and, behold, it was very good. Genesis, 1:31.
 Priv. Diag.
 Heraclitus was a Greek philosopher sometimes called the "Dark Philosopher" or "Weeping Philosopher from the profundity of his philosophy and in a real sense the founder of metaphysics. In his philosophy, real knowledge consists in comprehending the all-pervading harmony of which constitutes the law of the universe. Real virtue consists in the subordination of the individual to the laws of this harmony as the universal reason wherein alone true freedom is to be found. The doctrine of immortality comes prominently forward in his ethics. Encyclopedia Britannica, 11th Ed. (Cambridge, England, 1911).
 Homage to Pythagoras, at 139.
 Priv. Diag.
 Mathematically, this can be expressed as a/b = c/d (or x2 = x + 1, x = F ).
 Deoxyribonucleic acid: The basic material in the chromosomes of the cell nucleus containing the genetic code and transmitting the hereditary pattern.
 Fittingly, with respect to Rosicrucianism, Paul Foster Case identifies the five petals of a wild rose as a symbol of secrecy, the word made flesh, the absolute natural philosophy and natural science. The True and Invisible Rosicrucian Order, at 138-140, 165.
 No inorganic matter, i.e., lifeless matter which is neither animal or vegetable, is ever structured on this pattern. Crystals are a visible example of inorganic structure.
 The Power of Limits, 4.
 Illustration 3. Leonardo de Vinciís drawing of a man inscribed within a squared circle is reflective not only of the fact that the members of the body have golden proportion symmetry, but symbolically that we unite within ourselves the diversities of heaven and earth, an idea shared by many mythologies and religions. Power of Limits, at 93. It should be noted that Leonardo, as well as Albrecht DŁrer were initiated by Franciscan Fra. Luca Pacioli, mathematician and author of the Divina Proportione, which was illustrated by Leonardo. The Geometry of Art and Life, at 87. Luca Paciolli was threatened with excommunication because of his attitude as a free-thinker. The Dictionary of Art, "Pacioli," ed. Jane Turner in 34 volumns (Macmillian, 1996). "He was not so much an original thinker as a gatherer and expounder of previously held knowledge. He did Ö arrange and extend his material by bringing to it philosophical speculation and the intellectual aspirations of the circles in which he moved." Id.
 Discovery of this series is credited to 13th Tuscan mathematician Leonardo Fibonacci Pisano, also known as Leonardo of Pisa.
 The Power of Limits, at 5, 53-59, 82, 93-102.
 Priv. Diag. NB. Duality as used herein is not a dualism of good (spiritual) and an evil spirit (material) such as in Manichaeanism, nor is it the co-existent evil principal (Ahriman) opposing God (Ohrmazd) as in Zoroastrianism. See, Teachings of the Magi, at 52-66; Compare, The Pythagorean Triangle, at 67. Here, duality is considered as a linear projection of the creative force of the Deity.
 Priv. Diag.
 Rosecrucians might consider the Fama Fraternis. Bro. C.R.C. waits five years to found the Order after returning to Germany from foreign countries. He assembles eight brethren. Five are dispatched to several countries. The walls of Bro. C.R.C.ís vault measure five by eight.
 Penguin Dictionary of Symbols, at 344-347, and Priv. Diag.
 Penguin dictionary of Symbols, at 346, citing Joules Boucher, La symbolique maconnique.
 The Elements of Dynamic Symmetry, at 25-27; The Geometry of Art and Life, at 30; and Priv. Diag.
 Priv. Diag.
 Priv. Diag.
 Priv. Diag.
 Priv. Diag.
 See, Symbolism of the Gods of the Egyptians: The Light They Throw on Freemasonry, Dr. Thomas Milton Stewart, at 17-18, 80-81 and 85-86 (Lewis, Eng., 1927, 1977 ed.).
 After sacred architecture passed from being an initiatic tradition celebrating the Divine to a technical skill celebrating innovation and novelty (i.e., ego) on the part of the draftsman, ridged adherence to root-five proportionality was lost. The human was emphasized over the Divine. Priv. Diag.
 As demonstrated in the accompanying slide presentation from Egyptian Temple structures to the Gothic Cathedral at Chartres, masons have adhered to a strict usage these proportions. The proportion applied to each portion of the structures and iconography related specifically to their function in building "that house not made with stone." For instance, chambers for storing food for the Pharaoh's after life were built only in ÷ 5/2, or involution. In Christian Cathedrals, representations of the Garden of Eden were similarly in ÷ 5/2 proportion. The aisles of Egyptian Temples and naves of Cathedrals might be expected to demonstrate F ≤ proportionality. Side aisles in Cathedrals demonstrate ÷ 5 proportionality. Similarly, iconography (including stained glass) provides a clue: E.g., the five precursors of Christ adorning the east side of the center bay of the North Transept (a.k.a., Virgin Portal) to Chartres are each in ÷ 5 proportion and the five together form another rectangle that is also in ÷ 5 proportion: In other words, the operative Masons were expressing that Malchizadek, Abraham, Moses, Samuel, and David (mirrored on the west side by Isaih, Jeremiah, Simeon, St. John the Baptist, and St. Peter who opposite Malchizedek marks the transition in dispensations) were each initiates in the mystery. Priv. Diag. For photographs of the iconography discussed, see Chartres Cathedral, ed. Robert Branner, at 195-199 and plates 64-67 (W.W. Norton & Co., N.Y. 1969, 1996 ed.) And see, The Geometry of Art and Life, at 59-69.
 The square root of 2 family has three members: 1, 2, ÷ 2. The Elements of Dynamic Symmetry, at 38-42. It is used architecturally in gates, fortresses, and gallows. It is noteworthy that operative stone masons were free not to build fortresses or prisons since such buildings restrict freedom. Priv. Diag. The square root of 3 family consists of five members: 1, 2, ÷ 3, ÷ 3/2. The Elements of Dynamic Symmetry, at 38, 48-50. This proportion represents the deity or divine perfection (as opposed to human perfection - ÷ 5 +1/2) as is evident in its architectural usage. In Christian cathedrals it is associated with the manifestations of the GodHead, Christ. It is constructed out of the vesica pisces. E.g., at Chartres, the south and central tympanum of the West Portal (Second Coming and Last Judgment), Mary is pictured before the birth of Christ on a couch: The blessed virgin is in F symmetry. In a different plane, she is within a rectangle representing the Christos which is in ÷ 3 symmetry and the couch on which she rests is detailed to suggest its division into three rectangles, each of which has ÷ 3 symmetry, i.e., Father, Son, and Holy Ghost. Pictured in Chartres Cathedral, Id. at plates 23, 24 and 25. Even certain pagan architecture, to wit, the Parthenon, is built in root three, evidencing as it is named that it was a structure for the gods in which man is separated and cannot partake of communion with the gods. Priv. Diag.
 The Pythagorean Triangle, at xx-xxi.
 Homage to Pythagoras, Critchlow, at 165.
 As the accompanying illustration of a reconstruction of King Solomonís Temple demonstrates, the structure follows square root of five symmetry. The Parnaos (porch) represents the initiate. The initiate and deity are separate by a double cube (duality) which together they overcome as the initiate evolves along the path. Under such an analysis, the whole complex represents the perfection of the initiate.
 It seems possible that our perfect ashlar is a truncated version of the Broached Thurnel. As becomes apparent, however, the rectangle of the perfect ashlar does not completely express the mystery that can be extrapolated from the pointed cubical stone.
 Thurl, vi (Thurl, s.) Mining: To make a breach into former workings, or gate-roads (gate-way); an opening; a passage; a way of entrance; Figurative: A passage or opening; a means or way of ingress or egress. Thurl, s. [A.S. Thyrel = a hole] [Thrill] Mining: 1) a short communication between audits (audit = a house appurtenant/next to a cathedral); 2) a short audit in a coal pit. Broached: Literally - To split, transfix on any sharp instrument; to top a cask: Figuratively - To pierce; shed, as blood; allow an liquid to flow; to open; produce; to vent, make public; start a subject, publish; to commence; set on foot. Technically - Masonry: To indent the surface of a stone with a broach or puncheon, to rough hew.
 Dictionary of Symbols, "Freemasonry" at 410.
 Priv. Diag.
 In other words, it is like a five-petal rose on a cross, but with one petal hidden from our view.
Further examples of the number five and pentagram/pentagon among Jews, Christians, Druids, Sufis, and Hindus.
Private dialogues (designated herein as "Priv. Diag.") with Bro. Bernard Lietaer, professor of economics, contributed immeasurably to this trestleboard. Bro. Lietaer suggested the topic to me, introduced me to the works of Critchlow, Doczi, Gyrka, and Hambridge, allowed me to use and photograph his personal analysis of parallax adjusted photographs of sacred architecture, probed my thoughts with insightful questions that led to new discoveries, and from mouth to ear communicated how the various elements from such disparate works fit together in an initiatic tradition manifest in sacred architecture. Bro. Lietaer was initiated in Freemasonry in Belgium under the Scottish Rite.
A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science - A Voyage from 1 to 10, Michael S. Schneider (HarperCollins, 1995)
The Divine Proportion: A Study in Mathematical Beauty, H.E. Huntly (Dover Publications, N.Y., 1970)
The Elements of Dynamic Symmetry, Jay Hambridge (Dover Publications, N.Y., 1967 ed. of 1919 Yale University Press Publication.)
The Geometry of Art and Life, Matila Ghyka (Dover Publications, 1977, reprint of 1946 ed.)
Homage to Pythagoras: Rediscovering Sacred Science, Christopher Bamford anth. ed., Critchlow, Lawlor, Macaulay, Raine, Zajonc (Lindisfarne Press, Hudson, N.Y., 1994)
King Solomon and His Followers: A Valuable Aid to the Memory, California Ed. (Allen Publishing Company, N.Y., 1927 ed.) Note: Contains the discontinued long form of lectures.
Metaphysical Bible Dictionary, ed. Charlie Fillmore and Theodosia DeWitt Schobert, 2d ed. (Unity School of Christianity, Kansas City, MO, 1942)
Order in Space: A Design Source Book, Keith Critchlow (Thames and Hudson, 1979 ed.)
The Penguin Dictionary of Symbols, Jean Chevalier and Alain Gheerbrant, trans. John Buchannan-Brown (Penguin Books, Ltd., London, 1996) NB. Includes very good Masonic references based on work of leading French Masonic writers.
Teachings of the Magi, R.C. Zaehner (Oxford University Press, N.Y., 1956, 1976 ed.)
Timaeus and Critias, Plato, trans. Desmond Lee (Penguin, 1977)
The True and Invisible Rosicrucian Order: An Interpretation of the Rosicrucian Allegory and An Explanation of the Ten Rosicrucian Grades, Paul Foster Case (Samuel Weiser, Inc., 1989)
The Power of Limits: Proportional Harmonies in Nature, Art & Architecture, Gyorgy Doczi (Shambhala, Boulder & London, 1981)
The Pythagorean Triangle: The Science of Numbers, Rev. George Oliver (London, 1875)
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