Sacred Geometry

Note:  This is a fragment of our upcoming e-book “Ancient Design Principles of the Giza Pyramids”.

“Philosophy is written in this grand book – I mean the universe — which stands continually open to our gaze. But it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it.”    
— Galileo Galilei, Il Saggiatore (1623)

Sacred Geometry – Introduction

Sacred geometry is the geometry used in the planning and construction of religious structures such as temples, churches, mosques, religious monuments, altars, tabernacles; as well as for sacred spaces such as temenoi, sacred groves, village greens and holy wells, and the creation of religious art.

In sacred geometry, symbolic and sacred meanings are ascribed to certain geometric shapes and certain geometric proportions. According to Paul Calter:

In the ancient world certain numbers had symbolic meaning, aside from their ordinary use for counting or calculating … plane figures, the polygons, triangles, squares, hexagons, and so forth, were related to the numbers (three and the triangle, for example), were thought of in a similar way…

The golden ratio, geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra.

Many of the sacred geometry principles of the human body and of ancient architecture have been compiled into the Vitruvian Man drawing by Leonardo Da Vinci, itself based on the much older writings of the roman architect Vitruvius.  — Source: Wikipedia, the free online encyclopedia

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratios, that is , the Golden Mean. The first way may be compared to a measure of gold, the second to a precious jewel.”  — Johannes Kepler, an astronomer (1571-1630)


The relationship between number, proportion, astronomy and music must have been well known to the ancient people well before Greeks and Romans.

Herodotus, the father of history and a native Greek, stated in 500 BCE:

 Now, let me talk more of Egypt for it has a lot of admirable things and what one sees there is superior to any other country.

The ancient Egyptians considered certain numbers as “sacred”, for example numbers 1, 2, 3, 4, 7, and their multiples and sums.

The Ancient Egyptian works, large or small, are admired by all, because they are proportionally harmonious. This harmonic design concept is popularly known as sacred geometry and  used in the design of sacred architecture and sacred art.

The basic belief is that geometry and mathematical ratios, harmonics and proportion are also found in music, light, and cosmology.

Another aspect of sacred geometry is  that all figures could be drawn or created using a straight line (not even necessarily a ruler) and compass, i.e. without measurement (dependent on proportion only).

The principles of sacred geometry are of Ancient Egyptian origin, which constituted the basis of harmonic proportions, as evident in their temples, buildings, theology, …etc. The Ancient Egyptian design followed these principles in well-detailed canons. Plato himself attested to the longevity of the Egyptian harmonic canon of proportion (sacred geometry), when he stated:

That the pictures and statues made ten thousand years ago, are in no one particular better or worse than what they now make.

The key to divine harmonic proportion (sacred geometry) is the relationship between progression of growth and proportion. Harmonic proportion and progression are the essence of the created universe. It is consistent with nature around us. Nature around us follows this harmonious relationship.


The System of Proportions in Ancient Architecture

Three main systems of proportion in architecture.

  1.  A system based on the musical ratios
  2. A system based on the golden ratio
  3. A system based on the square, Sacred Cut Square and Octagon

Roman architect, engineer and writer, Vitruvius* (1st century B.C. ) noticed that ancient architects always used a system of proportions, rather that picking each dimension with no regards to the others in a structure. It was an easy, pragmatic way to insure harmonious design without having to do complicated calculations. Also, this way almost all the constructions (an on-site layout) could be done with straightedge, compass, and stretched cord (just drive pegs and swing arcs.)

“Symmetry is a proper agreement between the members of the work itself, and relation between the different parts and the whole general scheme, in accordance with a certain part selected as standard.”

 “Therefore since nature has proportioned the human body so that its members are duly proportioned to the frame as a whole, in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme”.    — Vitruvius* 

* Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC) was a Roman writer, architect and engineer, active in the 1st century BC. He is best known as the author of the multi-volume work De Architectura (“On Architecture”).

A 1684 depiction of Vitruvius (right) presenting De Architectura to Augustus.

Vitruvius uses “symmetrical relationships” to mean the same proportions, rather than some kind of mirror symmetry. Such a system would use the repetition of a few key ratios, to insure harmony and unity. It would have additive properties, so the whole could equal the sum of its parts, in different combinations. This would give a pleasing design, and maintain flexibility. Finally, since builders are most comfortable with integers, it would be based on whole numbers.

“One might conjecture that the Egyptians hold in high honor the most beautiful of the triangles, since they liken the nature of the Universe most closely to it, as Plato in the Republic seems to have made use of it in formulating his figure of marriage.  This triangle has its upright or three units, its base of four, and its hypetenuse of five.  The upright may be likened to the male, the base it the female, and the hypotenuse to the child of both…”       –-  Plutarch**, Isis and Osiris  [ From the English translation of Plutarch’s work by Frank Cole Babbitt as printed in pp1.191 of Vol. V of the Loeb Classical Library edition of the Moralia, published in 1936.]

** Plutarch (c. 46 – 120 AD), later named, on his becoming a Roman citizen, Lucius Mestrius Plutarchus, was a Greek historian, biographer, essayist, and Middle Platonist known primarily for his Parallel Lives and Moralia. He was born to a prominent family in Chaeronea, Boeotia, a town about twenty miles east of Delphi.


The Architecture of Marcus Vitruvius Pollio

Here (below) are few quotes from The Architecture of Marcus Vitruvius Pollio, translated by Joseph Gwilt, London: Priestley and Weale, 1826.

Book I, Chapter 2

Architecture depends on fitness (ordinatio) and arrangement (dispositio); it also depends on proportion, uniformity, consistency, and economy. Fitness is the adjustment of size of the several parts to their several uses, and required due regard to the general proportions of the fabric: it arises out of dimension (quantitas). Dimension regulated the general scale of the work, so that the parts may all tell and be effective. Arrangement is the disposition in their just and proper places of all the parts of the building, and the pleasing effect of the same; keeping in view its appropriate character. It is divisible into three heads, which, considered together, constitute design: they are called ichnography, orthography, and scenography. The first is the representation on a plane of the ground-plan of the work, drawn by rule and compasses. The second is the elevation of the front, slightly shadowed, and shewing the forms of the intended building. The last exhibits the front and a receding side properly shadowed, the lines being drawn to their proper vanishing points. These three are the result of thought and invention. Thought is an effort of the mind, ever incited by the pleasure attendant on success in compassing an object. Invention is the effect of this effort; which throws a new light on things the most recondite, and produces them to answer the intended purpose. These are the ends of arrangement.

Proportion is that agreeable harmony between the several parts of a building, which is the result of a just and regular agreement of them with each other; the height to the width, this to the length, and each of these to the whole.

Uniformity is the parity of the parts to one another; each corresponding with its opposite, as in the human figure. The arms, feet, hands, fingers, are similar to, and symmetrical with, one another; so should the respective parts of a building correspond. In the balista, by the size of the hole; in ships, by the space between the thowls, we have a measure, by the knowledge of which the whole of the construction of a vessel may be developed.

Consistency is found in that work whose whole and detail are suitable to the occasion. It arises from circumstance, custom, and nature. From circumstance,  when temples are built, hypæthral and uninclosed, to Jupiter, Thunderer, Coelus, the Sun and Moon; because these divinities are continually known to us by their presence night and day, and throughout all space. For a similar reason, temples of the Doric order are erected to Minerva, Mars, and Hercules; on account of whose valour, their temples should be of masculine proportions, and without delicate ornament. The character of the Corinthian order seems more appropriate to Venus, Flora, Proserpine, and Nymphs of Fountains; because its slenderness, elegance and richness, and its ornamental leaves surmounted by volutes, seem to bear an analogy to their dispositions. A medium between these two is chosen for temples to Juno, Diana, Bacchus, and other similar deities, which should be of the Ionic order, tempered between the severity of the Doric and the slenderness and delicacy of the Corinthian order.

In respect of custom, consistency is preserved when the vestibules of magnificent edifices are conveniently contrived and richly finished: for those buildings cannot be said to be consistent, to whose splendid interiors you pass through poor and mean entrances. So also, if dentilled cornices are used in the Doric order, or triglyphs applied above the voluted Ionic, thus transferring parts to one order which properly belong to another, the eye will be offended, because custom otherwise applies these peculiarities.

Natural consistency arises from the choice of such situations for temples as possess the advantages of salubrious air and water; more especially in the case of temples erected to Æsculapius, to the Goddess of Health, and such other divinities as possess the power of curing diseases. For thus the sick, changing the unwholesome air and water to which they have been accustomed for those that are healthy, sooner convalesce; and a reliance upon the divinity will be therefore increased by proper choice of situation. Natural consistency also requires that chambers should be lighted from the east; baths and winter apartments from the south-west; picture and other galleries which require a steady light, from the north, because from that quarter the light is not sometimes brilliant and at other times obscured, but is nearly the same throughout the day.

Economy consists in a due and proper application of the means afforded according to the ability of the employer and the situation chosen; care being taken that the expenditure is prudently conducted. In this respect the architect is to avoid the use of materials which are not easily procured and prepared on the spot. For it cannot be expected that good pit-sand, stone, fir of either sort, or marble, can be procured every where in plenty, but they must, in some instances, be brought from a distance, with much trouble and at great expense. In such cases, river or sea-sand may be substituted for pit-sand; cypress, poplar, elm, and pine, for the different sorts of fir; and the like of the rest, according to circumstances.

The other branch of economy consists in suiting the building to the use which is to be made of it, the money to be expended, and the elegance appropriate thereto; because, as one or other of these circumstances prevails, the design should be varied. That which would answer very well as a town house, would ill suit as a country house, in which store-rooms must be provided for the produce of the farm. So the houses of men of business must be differently designed from those which are built for men of taste. Mansions for men of consequence in the government must be adapted to their particular habits. In short, economy must ever depend on the circumstances of the case.


Sacred Geometry – Examples

Golden Ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Expressed algebraically:

The golden ratio is often denoted by the Greek letter “phi”. The figure of a golden section illustrates the geometric relationship that defines this constant.

 The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Golden Ratio in Architecture

Many  architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.


Parthenon, Acropolis, Athens. 
This ancient temple fits almost precisely into a golden rectangle.


Squaring the Circle

The square, with its four corners like the corners of a house, represents earthly things, while the circle, perfect, endless, infinite, has often been taken to represent the divine or godly. So squaring the circle is a universal symbol of bringing the earthly and mundane into a proper relationship with the divine.


Squaring the circle:
Image on the left: Circumference of the circle equals the perimeter of the square.
Image on the right:  the areas of this square and this circle are equal.

A square whose surface is equal to the surface of the circle.This construction is sacred because it contains both square and circle, uniting the earthly and the divine as in the Vitruvian man. Furthermore, it squares the circle. The length of the four arcs equal the four diagonals of the half-square.


The “Pi” is embedded in the design of the Great Pyramid.
 Arcs AC and CB are equal half of the Base (AC and CB).

Vitruvian Man

The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci around the year 1487. It is accompanied by notes based on the work of Vitruvius.

A passage from Roman architect Vitruvius (Marcus Vitruvius Pollio), describing the perfect human form in geometrical terms, was the source of inspiration for numerous renaissance artists. Only one of these, the incomparable Leonardo da Vinci, was successful in correctly illustrating the proportions outlined in Vitruvius’ work De Architectura, and the result went on to become the most recognized drawings in the world, and came to represent the standard of human physical beauty. It was the version produced by Leonardo da Vinci, whose vast knowledge of both anatomy and geometry made him uniquely suited to the task.

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Other artists had attempted to depict the concept, with less success. The drawing is traditionally named in honour of the architect.

The Vitruvian Man image exemplifies the blend of art and science during the Renaissance and provides the perfect example of Leonardo’s keen interest in proportion.

Vitruvian Man composition is based on a square, which is duplicated and rotated 45º to form an octagram.  The distance between the base line of the first square and the apex of the rotated one simply represents the diameter of the circle.


The Second Pyramid (Khafre) shows “Vitruvian Man” proportions


Vitruvius, De Architectura: THE PLANNING OF TEMPLES, Book 3, Chapter I

1. The planning of temples depends upon symmetry: and the method of this architects must diligently apprehend. It arises from proportion (which in Greek is called analogia). Proportion consists in taking a fixed module, in each case, both for the parts of a building and for the whole, by which the method of symmetry is put to practice. For without symmetry and proportion no temple can have a regular plan; that is, it must have an exact proportion worked out after the fashion of the members of a finely-shaped human body.
2. For Nature has so planned the human body that the face from the chin to the top of the forehead and the roots of the hair is a tenth part; also the palm of the hand from the wrist to the top of the middle finger is as much; the head from the chin to the crown, an eighth part; from the top of the breast with the bottom of the neck to the roots of the hair, a sixth part; from the middle of the breast to the crown, a fourth part; a third part of the height of the face is from the bottom of the chin to the bottom of the nostrils; the nose from the bottom of the nostrils to the line between the brows, as much; from that line to the roots of the hair, the forehead is given as the third part. The foot is a sixth of the height of the body; the cubit a quarter, the breast also a quarter. The other limbs also have their own proportionate measurements. And by using these, ancient painters and famous sculptors have attained great and unbounded distinction.
3. In like fashion the members of temples ought to have dimensions of their several parts answering suitably to the general sum of their whole magnitude. Now the navel is naturally the exact centre of the body. For if a man lies on his back with hands and feet outspread, and the centre of a circle is placed on his navel, his figure and toes will be touched by the circumference. Also a square will be found described within the figure, in the same way as a round figure is produced. For if we measure from the sole of the foot to the top of the head, and apply the measure to the outstretched hands, the breadth will be found equal to the height, just like sites which are squared by rule.
4. Therefore if Nature has planned the human body so that the members correspond in their proportions to its complete configuration, the ancients seem to have had reason in determining that in the execution of their works they should observe an exact adjustment of the several members to the general pattern of the plan. Therefore, since in all their works they handed down orders, they did so especially in building temples, the excellences and the faults of which usually endure for ages. [Source:]

Arithmetic Rope

The rope-stretcher’s triangle is also called the 3-4-5 right triangle, the Rope-Knotter’s triangle, and the Pythagorean triangle.

A rope knotted into 12 sections stretched out to form a 3-4-5 triangle. It also makes right angle. The rope stretchers triangle (3-4-5) when opened out gives a zodiac circle, with the number of knots the most important of the astrological numbers (note: 2 drawings below are not to the same scale).

. __ . __ . __ . __ . __ . __ . __ . __ .  __ . __ . __ . __ .



The Great Pyramid and the Tibetan Sand Mandala

Tibetan Sand Mandala seems to have very similar modular design…

Note: This image is explained in our upcoming e-book:
Ancient Design Principles of the Giza Pyramids


Sacred Numbers

 The Pythagoreans adored numbers. Aristotle, in his Metaphysica, sums up the Pythagorean’s attitude towards numbers.

 “The (Pythagoreans were) … the first to take up mathematics … (and) thought its principles were the principles of all things. Since, of these principles, numbers … are the first, … in numbers they seemed to see many resemblances to things that exist … more than [just] air, fire and earth and water, (but things such as) justice, soul, reason, opportunity …”

One of fascinating ancient discoveries is Tetractys. It is a symbol composed of ten dots in an upward-pointing triangular formation. It was a sacred pattern for the school of philosophers who followed the teachings of the Greek sage Pythagoras.

Tetractys itself can be interpreted as the symbolic blueprint of creation. Its image is an equilateral triangle based on the essential numbers 1 (top), 2, 3 and 4 (base), whose sum is the “perfect” number 10  ( 1 + 2 + 3 + 4 = 10).

 These numbers were considered by the Pythagoreans to be holy and at the origins of the universe. They believed that a four-fold pattern permeated the natural world, examples of which are the  point, line, surface and solid and the four elements Earth, Water, Air and Fire.

Musically they represent the perfect consonants: the unison, the octave, the fifth and the fourth.

 The importance of the tetractys to the Pythagoreans is illustrated by their oath of fellowship:

 I swear by the discoverer of the Tetractys,
Which is the spring of all our wisdom,
The perennial fount and root of Nature.


Sacred Tetractys

One particular triangular number that they especially liked was the number ten. It was called a Tetractys, meaning a set of four things, a word attributed to the Greek Mathematician and astronomer Theon (c. 100 CE). The Pythagoreans identified ten such sets.







In Plato’s Timaeus, we find that God created the Cosmic Soul using two mathematical strips of 1, 2, 4, 8 and 1, 3, 9, 27. These two strips have the shape of an inverted “V” or the “Platonic Lambda” since it resembles the shape of the 11th letter of the Greek alphabet “Lambda” (?).

The Platonic Lambda interestingly describes 3 dimensional space:

  • Top – 1 — single point
  • 1st  row: 2 – 3 — linear dimension
  • 2nd  row: 4 – 9  — surface (2-D: 2×2 and 3×3)
  • 3rd  row: 8 – 27 — cubic volume (3-D: 2x2x2  and 3x3x3)

The Summation Series

The natural progression follows a “summation series” that is known today as the Fibonacci Series [ of course this Series was in existence before Fibonacci (born in 1179 CE) – he simply “re-discovered” it.] The Summation Series is a progressive series, where you start with the first two numbers, then you add their total to generate the next number, and so on. By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. The first summation progression:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

It is worth to note that the first summation (Fibonacci) progression generates  approximation of the golden ratio (PHI):

3/2 =1.5000
5/3 =1.666
8/5 =1.6000
13/8 =1.6250

The second of Fibonacci progression (provides approximate geometry of the pentagram and also lead to golden ratio ‘phi’ just like in the above example:  322/199=1.618… ):

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …

The Summation Series is reflected throughout nature. The number of seeds in a sunflower, the petals of any flower, the arrangement of pine cones, the growth of a nautilus shell, etc…all follow the same pattern of these series.

The overwhelming evidence indicates that the Summation Series was known to the Ancient Egyptians. Throughout the history of Ancient Egypt, temples and tombs (including pyramids) show in their design  expression of the Summation Series: 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .

Masculine and Feminine Numbers

The ancient philosophers considered odd numbers masculine. Even numbers were thought of as feminine because they are weaker than the odd. When divided they have, unlike the odd, nothing in the center. Further, the odds are the master, because odd + even always give odd. And two evens can never produce an odd, while two odds produce an even.

Since the birth of a son was considered more fortunate than birth of a daughter, odd numbers became associated with good luck. “The gods delight in odd numbers,” wrote Virgil.

1   Monad. Point. The source of all numbers. Good, desirable, essential, indivisible.

2   Dyad. Line. Diversity, a loss of unity, the number of excess and defect. The first feminine number. Duality.

3   Triad. Plane. By virtue of the triad, unity and diversity of which it is composed are restored to harmony. The first odd, masculine number.

4   Tetrad. Solid. The first feminine square. Justice, steadfast and square. The number of the square, the elements, the seasons, ages of man, lunar phases, virtues.

5   Pentad. The masculine marriage number, uniting the first female number and the first male number by addition.

  • The number of fingers or toes on each limb.
  • The number of regular solids or polyhedra.

Incorruptible: Multiples of 5 end in 5.

6   The first feminine marriage number, uniting 2 and 3 by multiplication.
The first perfect number (One equal to the sum of its aliquot parts, IE, exact divisors or factors, except itself. Thus, (1 + 2 + 3 = 6).
The area of a 3-4-5 triangle

7   Heptad. The maiden goddess Athene, the virgin number, because 7 alone has neither factors or product. Also, a circle cannot be divided into seven parts by any known construction).

8   The first cube.

9   The first masculine square. 
Incorruptible – however often multiplied, reproduces itself.

10   Decad. Number of fingers or toes.
Contains all the numbers, because after 10 the numbers merely repeat themselves.
The sum of the archetypal numbers (1 + 2 + 3 + 4 = 10)

27   The first masculine cube.

28   Astrologically significant as the lunar cycle. 
It’s the second perfect number (1 + 2 + 4 + 7 + 14 = 28). 
It’s also the sum of the first 7 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 = 28)!

35   Sum of the first feminine and masculine cubes (8+27)

36   Product of the first square numbers (4 x 9) 
Sum of the first three cubes (1 + 8 + 27) 
Sum of the first 8 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 )

[ Note: Masculine and Feminine numbers source: ]

“Special” Numbers

This article will reveal deeper meanings of the Pythagorean doctrine concerning the tetractys, which link it to current research into the theories of superstrings and bosonic strings:

Ten Sets of Four Things





















Living Things


growth in length

in breadth

in thickness
















Ages of a Person




old age

Parts of living things



three parts of the soul



11 + 21 + 31 + 41 = 10

  1!+2!+3!+4! =33

The 33rd “prime number” is 137 (regular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. )

  • 10 + 20 + 30 + 40 = 4
  • 11 + 21 + 31 + 41 = 10
  • 12 + 22 + 32 + 42 = 30
  • 13 + 23 + 33 + 43 = 100
  • 14 + 24 + 34 + 44 = 300
  • 15 + 25 + 35 + 45 = 1300
  • 10 + 20 + 30 = 3
  • 11 + 21 + 31 = 6
  • 12 + 22 + 32 = 14
  • 13 + 23 + 33 = 36
  • 14 + 24 + 34 = 98
  • 15 + 25 + 35 = 267
People that were so nuts about numbers that they made up little stories about them and arranged pebbles to make little pictures of them. Then they discovered that all the musical intervals they felt was beautiful, these five sets of ratios, were all contained in the simple numbers:  1, 2, 3, 4 and that these were the very numbers in their beloved sacred tetractys that added up to the number of fingers. They must have felt they had discovered some basic laws of the universe.

Quoting Aristotle again … “[the Pythagoreans] saw that the … ratios of musical scales were expressible in numbers [and that] .. all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number.”

In Plato’s Timaeus, we find that God created the Cosmic Soul using two mathematical strips of 1, 2, 4, 8 and 1, 3, 9, 27. These two strips have the shape of an inverted “V” or the “Platonic Lambda” since it resembles the shape of the 11th letter of the Greek alphabet “Lambda”. Here’s Benjamin Jowett’s translation 34 of Plato’s Timaeus 35b:

“Now God did not make the soul after the body, although we are speaking of them in this order; for having brought them together he would never have allowed that the elder should be ruled by the younger… First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [i.e. between 1, 2, 4, 8] and the triple [i.e. between 1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals”

 “This entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two-and three, three of each, and bade the orbits proceed in a direction opposite to one another; and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion.”

Cyclic number 142857

 The Great Pyramid has embedded in its design “pi”. An ancient approximation to ? is  22/7 = 3.142857

 142857 is the six repeating digits of 1/7, 0.142857, and is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7, respectively.

 142857 = 33 x 11 x 13 x 37

  • 1 × 142,857 = 142,857
  • 2 × 142,857 = 285,714
  • 3 × 142,857 = 428,571
  • 4 × 142,857 = 571,428
  • 5 × 142,857 = 714,285
  • 6 × 142,857 = 857,142
  • 7 × 142,857 = 999,999

 Multiplying by a multiple of 7 will result in 999999 through this process

     142857 × 74 = 342999657
    342 + 999657 = 999999

If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.

     8572 = 734449
    1422 = 20164
    734449 ? 20164 = 714285

It is the repeating part in the decimal expansion of the rational number 1/7 = 0.142857. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:

     1 ÷ 7 = 0.142857
    2 ÷ 7 = 0.285714
    3 ÷ 7 = 0.428571
    4 ÷ 7 = 0.571428
    5 ÷ 7 = 0.714285
    6 ÷ 7 = 0.857142
    7 ÷ 7 = 0.999999
    8 ÷ 7 = 1.142857
    9 ÷ 7 = 1.285714    …

Number 137

137 is the 33rd “prime number” (regular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. 33=1!+2!+3!+4! )

It is also a “primeval number” [The number of primes that can be obtained from the primeval numbers is: 0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, …] , the fifth harmonic number 137/60 [ The first five harmonic numbers are H1 = 1, H2 = 3/2, H3 = 11/6, H4 = 25/12, H5 = 137/60 ] , fine structure constant number 1/137:

1/137 defines “fine structure constant”

In physics, the fine-structure constant (usually denoted ?, the Greek letter alpha) is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction.
Anthropic explanation

The anthropic principle is a controversial argument of why the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were much different. For instance, were ? to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If ? were > 0.1, stellar fusion would be impossible and no place in the universe would be warm enough for life.

Numerological explanations

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists. Richard Feynman, one of the originators and early developers of the theory of quantum electrodynamics (QED), referred to the fine-structure constant in these terms:

“ There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! ” —Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0691083886

Arthur Eddington argued that the value could be “obtained by pure deduction” and he related it to the Eddington number, his estimate of the number of protons in the Universe. This led him in 1929 to conjecture that its reciprocal was precisely the integer 137.

1/137.036 is the value of the fine structure constant at zero energy.

 – – –

Ancient Design Principles of the Giza Pyramids

The Great Pyramid of Egypt has fascinated historians, travelers, scientists and mystics for thousands of years. How were pyramids built? Did ancient architects embedded in their design of the pyramids secret knowledge and messages to the future generations?  Several proponents of ancient astronauts claim that the Great Pyramid of Giza was influenced by extraterrestrial beings…The height of it multiplied by one billion equals the distance from Earth to the sun. The side of its base, divided by its height, equals the approximate value of  ½ pi.  What accounts for these coincidences?

In our upcoming e-book: “Ancient Design Principles of the Giza Pyramids” we re-discover the “lost” principles of design applied to the three pyramids at Giza and their layout. 




Sacred Geometry – Art

Rippleink and graphite on paper – 10.5” x 8” 2011

“What are we but the ripple a sound makes …
self-created from within the space of our own desire.”

Katarzyna Vedah is a Canadian artist based in the greater Vancouver region of British Columbia. Artist’s website:


  1. says

    Hello! Relative to Metatron’s Cube (and all 5 Platonic Solids as well as the phi ratio contained within it), I have crafted a new way to tune musical instruments to the mythologized A=432hz standard. It involves making the chromatic scale sexagesimal based, tuning with a different unison note than usual. Metre also coincides perfectly when done this way with the series of divisors used to derive the note intervals in the scale as well! It’s perfect. I wrote an article, soon available in Barnes and Noble, iBookstore, and NOOK stores, called “A = 432hz: On the Proper Concert Pitch and a New Standardization of Tempo,” and I have a FB page dedicated to this.

  2. says

    This one could get good Alex. It represents the very core of our being and becoming. You are on the threshold of a new forum of knowledge and Wisdom and I am waiting for much more insight from this new book and future posts. I would think that you could now offer up a special site where folks like you and many others would pay for the deeper intellect that could follow this line of thought. From the Line comes the Gestalt and multidimensional aspects. I have high hopes for this new direction I see you burgeoning into…anon.


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