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We will now continue our discussion of the Pentad by covering the quintessential geometric representation of the Pentad – the Pentagon.



Pentad Born through the Vesica Piscis as:

  • Pentagon – 2D

  • Pentagram

  • Dodecahedron – 3D

  • Golden/Fibonacci Spirals


The Golden Ratio is embedded within each of these forms.


This entails: self-replication, self-accumulation, self-recurrence, and self-similarity.

The Golden Ratio is the precursor, or seed, of fractalization and holography.

“The pentagonal star was the sacred emblem of the Pythagorean fellowship, which consisted of men and women living in communities and abstaining from all luxuries, dedicated to a life of moderation and the practice of healing.  The Pythagoreans applied the letters of the name “Hygeia” the goddess of healing, to the tips of the sacred emblem.  The five-pointed star has remained a universal symbol of good portent, appearing in the national flags of many countries.

The pentagon and pentagram, like all patterns, are defined by their limits.  Incorporated in the harmonious patterns of fruits and flowers, they exemplify an epigram attributed to Pythagoras, that limit gives form to the limitless.  This is the power of limits.” ~ Gyorgy Docizi, The Power of Limits


Pentagon – 540º

The pentagon is a 5-sided regular polygon.

Each interior angle = 108º.

Sum of interior angles = 540º = (108 x 5)

Each exterior angle = 72º.


540 hertz = C#

This is the harmonic fifth of the triangle (180 Hz), and the square/circle (360 Hz).


12 pentagons make up the faces of a dodecahedron.

Here is a pentagon decomposed into six smaller pentagons (half a dodecahedral net) with gaps.

The net of the dodecahedron.  Fold it up like a flower and it produces the dodecahedron.


Here we see that if the side length of a pentagon is 1, then the diagonal is phi.

The pentagon divided into ten equal parts produces 10 3-4-5 Pythagorean triangles.

Credit: Gyorgy Doczi – The Power of Limits



The Area of a Pentagon

A = ¼ √5 (5 +2√5)s2                where s = side length



Recall the importance of the Numbers 72 and 108:


  • 72 = number of years required for the equinoctial sun to complete a precessional shift of one degree along the ecliptic, in the 25,920 year Precession of the Equinoxes


  • 72º F is considered ‘room temperature’.
  • 72 beats/minute is considered the average human resting heartbeat.
  • 72 years is the current average lifespan of a human.


  • 72 = The exterior angles of a regular pentagon

  • 720 = sum of the angles of a tetrahedron

  • 720 = sum of angles of a hexagon (120 x 6)

  • 720 x 9 = 6480 = sum of the angles of a dodecahedron
  • 720 x 11 = 7920 = Diameter of the Earth in miles

  • 720 x 12 = 8640; 8640 x 10 = 86400 = number of seconds in a day
  • 720 x 12 = 8640 x 100 = 864000 = diameter of the Sun in miles

  • The sum of the angles of all the Platonic solids, Archimedean solids and Catalan solids are factors of 72.




  • 11 x 22 x 33 = 108
  • 108º F is the temperature at which organ failure begins in the human body.


  • In the sacred number canon the pre-eminent number in the code is 72. To this is frequently added 36, making 108, and it is permissible to multiply 108 by 100 to get 10,800.


  • 108 = degrees in interior angles of a regular pentagon (shown above)
  • 1080 = sum of angles of an octagon (135 x 8)

  • 1080 = radius of the Moon in miles

  • Approximately 108 Suns fit between the Earth and Sun.
  • 108 Moons fit between the Earth and Moon.
  • 108 Earths fit in the Sun. (This means the diameter of the Sun = 108 Earth diameters.)



Pentagonal Symmetry – The Supreme Symbol of Life

Every molecule of water is a corner of a pentagon.

A single tetrahedral water molecule.


Water Pentamer: Water clustering forms pentagons which then combine to form a complex icosahedral structure.


Credit: Martin Chaplin: Water Structure & Science


The overall design of any leaf fits within a stretched or compressed pentagon.  Note that leaves have the same symmetry as hands.

Credit: Michael Schneider


The flower of every edible fruit has 5 petals, however not all 5-petaled flowers yield edible fruit.

Apples, pears, blueberries and starfruit (carambola) have pentagonal symmetry…among many, many others.  See the section on plants in Cosmic Core.

“When we look at any leaf, flower, or fruit we’re seeing the invisible energy web of the archetype made visible as a pattern of living cells.”1


Pentagonal symmetry is also seen in sea cucumbers, sand dollars, star fish, and microscopic radiolarian skeletons.


Pentagonal symmetry is clearly seen in the expression of the human body:

  • 5 senses – touch, sight, hearing, smell, taste
  • 5 extensions from the torso – 2 arms, 2 legs, 1 head
  • five fingers and toes


Golden ratios are found abundantly throughout the human body.  See Article 194.


Pentadactylism is clearly seen in other species of animals.  There is fivefold symmetry of hands, paws, and claws.



Constructing the Pentagon

The pentagon is one of the more complex shapes to bring through the vesica piscis.

Its construction was taught by Euclid (300 BC) in his Elements.

There were even earlier descriptions found in Babylonian clay tablets.

In 1509 Fra Luca Pacioli, teacher and collaborator with Leonardo da Vinci, published instructions on how to construct the pentagon.  This allowed the method of construction to be revealed to the public.


On a side note, a simple knot tied in a ribbon or strip of paper and pulled tight and flattened creates a perfect pentagon.

Here a ribbon which is green on one side and blue on the other is shown folded in this way.  Non-overlapped hidden edges are shown by dotted lines of the appropriate color (except that the middle of one line has been removed where it would confusingly intersect the other hidden-edge lines if shown.


Reference Construction Lesson # 57: Constructing the Pentagon to learn various ways to construct a pentagon with compass and straight-edge.



√5 & The Pentad

The √5 (and its related function of phi, the Golden Mean) represent the Regenerative and Transformative function of life.

√5 = 2.2360679…


Remember, the square roots are not numbers, but ratios – seeds, geometric functions that continue to infinity.


The √5 can be approximated as:   161/72            This equals 2.23611...


Recall from Article 25 that the diagonal of a double square rectangle is the √5.

“It seems that the dividing and transforming root powers must be seen at the same time as powers that bind and synthesize, as such principles must often demonstrate two poles of an opposition.  The square root of 5 transverses two worlds, indicated by the upper and lower squares, the world of spirit and the world of body.”2

The √5 opens the doorway for the Golden Proportion.

The √5 is the diagonal of a double square.  This shows the inherent relationship between the √5, golden ratio and square.


The formula for the Golden Proportion:        

φ = (1 + √5)/2


“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 towards the image of God.”3


The golden ratio is the arithmetic mean of 1 and √5.

The √5 is also the diagonal of a rectangle whose side lengths are 1 and 2 (same as the double square, seen above).


Reference Construction Lesson #58: √5 & the Pentad.


These geometric demonstrations reveal the relationship of √5 both with the number 5 (as the square of √5) and with the fivefold symmetry of the pentagon.

Credit: John Michell



The diagonal of the square = √2.

The diagonal of the double square = √5.  The semi-diagonal of a single square = √5/2.

The √5/2 diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.



Pentagram – 5-pointed Star

There is a single stellation of the pentagon: the pentagram.


As we have seen, the pentagram, or 5-pointed star, is full of golden ratios.


The pentagram recurs as a symbol of excellence, brilliance, power, authority, strength, and uprightness.

There are 5 star restaurants, movies, hotels, and 5-star generals.

There are ‘rock stars’, ‘movie stars’ and ‘starstruck fans’.


Flags of over 60 nations include a 5-pointed star.  Some of these include: Vietnam, Morocco, Somalia, DRC, Ghana, Suriname, Cameroon, Senegal, Burkina Faso, Dominica, Ethiopia, Panama, Sao Tome & Principe, Syria, North Korea, Algeria, Pakistan, Azerbaijan, China, Central African Republic, Togo, Chile, Liberia, Solomon Islands, United States, Samoa, Singapore, Myanmar, Burma, Puerto Rico, Cuba, Djibouti, Jordan, East Timor, Western Sahara, Zimbabwe, Philippines, Honduras, Venezuela, Tunisia, Turkey, Grenada, Brazil…


The pentagram was an ancient symbol of warding off evil in Babylon, Egypt, Greece, India, China, Africa, and the Americas.

The pentagram was a symbol of health and humanity to the Pythagoreans.


The symbol for the Duat (underworld; world beyond death) in ancient Egypt was a pentagram in a circle.

In ancient Greece the pentagram star was associated with Pan (All).  Note here that Pan had horns, red hair, hooves, and a tail.



Inverted pentagram – ‘Drudenfuss’ – ‘Witch’s foot’

The inverted pentagram is feared by many.

It appears in magical witchery and witchcraft and is associated with devils, demons and Satanism.

Inverted pentagram representing Baphomet, a deity allegedly worshiped by certain sects of the Knights Templar.


The negative connotations of the inverted pentagram dates back to the Egyptian story of Set rebelling against his brother Osiris and nephew Horus.

Set had horns, coarse red hair, hooves, and a tail.  He resisted truth, order, justice and righteousness.

His 42 attributes are symbols of the down-reaching elements of our own psyche that must be purified.

Similarly, in Western culture the lustful fertility and sexuality of nature that Pan had become known for “became the great temptation of the modern devil and the star was inverted to imply the reverse of, and resistance to, its positive qualities of excellence and goodness.”4


Please note that symbols are not to be feared.  It is the belief, intention and focus of the human behind the use of a symbol that gives it power, not simply the presence of a symbol.  Just looking upon an inverted pentagram will not curse a person or cause any harm.



The Vesica, 5 and 6 – Union of the Hexagon and Pentagon

Reference Construction Lesson #9: Constructing the Linked Pentagon and Hexagon via the Vesica to learn how to construct this beautiful union.

This geometric construction was originally published by Albrecht Durer in Course in the Art of Measurement with Compass and Ruler.

It was later republished in Charles Bouleau’s The Painters Secret Geometry & John Michell’s The City of Revelation.


This construction symbolizes the emergence of the Universe (6) and Life (5) from the cosmic womb of creation.

“The reconciliation of six and five and many of the other objects of sacred geometry may be achieved by means of the figure known as the Vesica piscis.”5

Master geometer Randall Carlson remarks of it, “The reconciliation of six and five refers to the potential to develop, from the Vesica as a starting point, polygons of six and five sides, as well as many of the other polygons, with all their symbolic connotations and unique properties.”


The hexagon is easier to draw and precedes the pentagon.

This represents how the cosmos had to exist before humankind could emerge from the combination of matter and energy generated through the evolution of the universe.


The pentagon has psychic and lunar qualities.

The hexagon has rational and solar qualities.


The three angles of an equilateral triangle, six of which make up a hexagon, are each of 60 degrees, reflecting the solar number 666.

The pentagon angle is 108 degrees, 1080 being the lunar number as the radius of the Moon is 1080 miles.


According to John Michell, the pentagon and hexagon combined also symbolizes “the purpose of a true scholar, to relate his own interests to those of the world at large…As a symbol of reconciliation between different orders in nature, its presence here prepares the student of life’s mysteries for the contradictions and paradoxes he/she will encounter and provides the geometric model to contain them.”6


  1. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  2. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  3. ibid.
  4. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  5. Michell, John, The Dimensions of Paradise, Inner Traditions, 2nd edition, 2008
  6. ibid.

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