In this article we will cover the octagon, octagrams and octagonal art.

Medieval Octagonal Korean Design

# Octagon – 1080º

The octagon is built upon the ‘root-2’ ratio, like the square.

The ratio of the side of a square to its diagonal = 1:√2.  That is, if the side of a square = 1, then the diagonal = √2.

The octagon is constructed of two squares at 45º to each other.  These are seen in red and blue below.

Its exterior angle = 135 √.

Its interior angle = 45º.

The sum of its exterior angles = 1080º.

1080 hertz = C#

1080 = radius of the Moon in miles.

The traditional metal associated with the moon is silver (as gold is to the Sun).  Incidentally, the atomic weight of silver is extremely close to 108.

Furthermore, 108 Moons fits between the Earth and Moon.

There are other 108s as well:

# 108 in the Solar System

• Sun diameter = approximately 108 Earth diameters
• 108 Sun diameters fit between the Sun and Earth
• radius of the Moon = 1080 miles
• diameter of the Moon = 1080 x 2 miles (2160)
• 10080 miles = diameter of the Earth + Diameter of the Moon
• Distance from Sun to Venus = 108 million km
• Diameter of Mars = 1080 x 4 miles (4320)
• Diameter of Jupiter = 10800 x 8 miles (86400)
• Orbit of Jupiter = 1080 x 4 days (4320)
• Diameter of Saturn = 108000 km
• Saturn’s orbit = 10800 days
• Saturn’s hexagon edge length = 1080 x 8 miles (8640)

Note these synchronicities apply to both the standard measurement system and the metric system.  Both of these systems are based upon the Earth measurements.

# Tessellations and Octagons

Recall that equilateral triangles, squares and hexagons all tessellate – they fit together on a flat plane with no gaps.

Octagons do not tessellate; though they do tessellate with small squares as the gaps.

These can be oriented different ways for different looks.

# The Area of an Octagon

A = S2 – a2                  S = span of the octagon (medium diagonal)          a = length of side

“When one comes face-to-face with the eightfold figure that is the octagon,” writes Freddy Silva, “one is essentially looking at the completion and rejuvenation of the Universal cycle, as exemplified by the horizontal figure 8, the infinity symbol.”

# The Diagonals of the Octagon

The Octagon has three diagonals as seen above:

• Short diagonal
• Medium diagonal (the span or height), which is twice the length of the inradius
• Long diagonal, which is twice the length of the circumradius

The short diagonal = a√(2+√2)

The medium diagonal = (1 + √2)a

The long diagonal = a √(4 + 2√2)

Reference Construction Lesson #74:  Construction of the Octagon and Octagram.

# The Silver Ratio & The Octagon

The silver ratio is found in the octagon as seen here.

This means the central portion of the octagon is a silver rectangle.

# The Silver Ratio

#### The Silver Ratio = 1 + √2.

The Silver ratio is any two quantities in which:

The sum of the smaller and twice the larger of the quantities is to the larger quantity as the larger is to the smaller.

(smaller + 2 * larger) : larger :: larger : smaller

This ratio, like the golden ratio and square roots of 2, 3, and 5 is an irrational or supra-rational number.

It is: 2.4142135623730950488… or as we said 1 + √2.

Like the Golden Ratio and Square Roots, the Silver Ratio can be expressed as a continued fraction:

Like the Fibonacci sequence converges on the Golden Ratio, the Pell sequence converges on the Silver Ratio.

Pell numbers are an infinite sequence of integers that comprise the denominators of the closest rational approximations to the √2.

This sequence begins: 1/1; 3/2; 7/5; 17/12; 41/29…etc.

The Pell number sequence is therefore: 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860…

# The Octagram

There are two stellations of the octagon – or two 8-pointed stars.

The first is the interlaced or double squares.  These are two squares at 45 degree angles to one another and is constructed by connecting every 2nd corner of the octagon (skipping 1 corner).

These double squares are a solar symbol recognized all over the world.  This is often called the ‘Star of Lakshmi’ in Hinduism.

It is also called the Roub el Hizb in Islam.

“Eightness is sometimes called the auspicious number:  it is a form of perfection of the earthly order, giving the four cardinal directions and their intermediates.”1

The second octagram is constructed by connecting every 3rd corner of the octagon (skipping 2 corners).  This can be called the Octalpha.

These octagrams can be presented in many ways.

# The Octalpha & Unicursal Octalpha

The Octalpha is the 2nd octagram we saw above.

Reference Construction Lesson #75: Sacred Cut, Octalpha & Unicursal Octalpha.

# The Octad in Art

Octagonal art shows up in many places over the ages in art and architecture.

See page 299 in A Beginner’s Guide to Constructing the Universe for some examples.

Examples include:

The earliest example found is the Star of Inanna – a Sumerian goddess

Detail of a kudurru (stele) of King Melishipak I (1186–1172 BC), showing a version of the ancient Mesopotamian eight-pointed star symbol of the goddess Ishtar (Inana/Inanna).

Bronze mirror from the Tung Dynasty (China) – represents the “hare in the moon”

Aztec calendar stone

Star of Lakshmi – Hindu (various versions)

Kali Yantra – Hindu

Taoist I-Ching symbol

Japanese Kimono designs

Egyptian Ogdoad w/ cosmic egg in center

The sun rises from the mound of creation at the beginning of time.  The central circle represents the mound, and the three orange circles are the usn in different stages of its rising.  AT the top is the “horizon” hieroglyph with the sun appearing atop it.  At either side are the goddesses of the north and south, pouring out the waters that surround the mound.  The eight stick figures are the gods of the Ogdoad, hoeing the soil.  From the 21st dynasty (c. 1075-945 BC)

Dendera zodiac – Egyptian

Celtic pagan calendar

Slavic religious symbols

The kolovrat (“spoked wheel”) is a symbol of the supreme God in Rodnovery (Slavic Native Faith), Rod and its manifestations. Kolo means “wheel”, and its vrat (“spokes”) are spinning. It is a symbol of spiritual and therefore secular power. The kolovrat represents the endless cycle of birth and deaths, time, the sun and fire, strength and dignity. Each turn of the wheel is a cycle of life in our world.

Star of Rus symbol

Valentinian Gnosticism

Buddhist wheel of dharma

Pharos at Abuquir, Egypt – likely modeled after the Lighthouse of Alexandria

The Vatican – St. Peter’s Square

The Dome of the Rock octagonal floor plan

Roman Pantheon floor plan

The Tower of Winds in Athens, Greece (Horological Monument of Andronicus Cyrrhestes)

Jabalieh (Rock Dome) in Kermin, Iran

Chinese Pagodas

Lingxio pagoda (860) & Liaodi Pagoda (1000s)

St. George’s Cathedral in Addis Ababa, Ethiopia

Basilica of San Vitale in Ravenna, Italy

Castel del Monte in Apulia, Italy – octagonal towers and floorplan

Ely’s Cathedral, Ely England

Labyrinth of Reims Cathedral – quasi-octagonal

Aachen Cathedral central space in the Carolingian Palatine Chapel

The Dome at 333 Collins Street, Melbourne, Australia

Clearly, throughout the world the octagon and symbols of eightness have been used to symbolize the divine aspects of humanity and the higher aspects of religion.

More octagonal architecture can be found here: http://www.wikiwand.com/en/List_of_octagonal_buildings_and_structures

It is also found in “many other traditions of finding one’s inner light”.

One of these is Islamic Design.

# Islamic Design

Daud Sutton tells us, “The role of sacred art is to support the spiritual life of those whom it surrounds, to instill a way of perceiving the world and the subtle realities behind it.  The challenge thus facing the traditional artisan is how to build with matter so as to best embody spirit.”2

Freddy Silva adds, “Sacred geometry enables humanity to see the archetypal world of God.  At its heart, the Arabic faith still contains an unadulterated snapshot of this primordial truth, in the geometric figures adorning its mosques and art forms.  Consequently, Islam has served as curator and preserver, maintaining the purity of the philosophy of geometry, ‘akin to the Pythagorean-Platonic tradition of antiquity but in a totally sacred universe free of the nationalism and rationalism which finally stifled and destroyed the esoteric traditions of Greek intellectuality” to quote the eminent Arabic historian S.H. Nasr.”

Islamic design forbids depiction of animals or humans.

There are two key aspects:

• calligraphy using Arabic script and
• abstract ornamentation using a varied but remarkably integrated visual language including:
• geometric pattern
• idealized plant forms embodying organic life and rhythm

There are many examples of octagonal art in Islamic design.

One beautiful design is the Breath of the Compassionate – the two interlaced squares.

The top shape represents the out-breath and the bottom shape represents the in-breath.  Together they represent the Cosmic process of the inward & outward oscillations of the wave-structures of all matter & life.  These inward/outward oscillations create geometric interference patterns upon which photons, subatomic particles, atoms & molecules coalesce, to form standing waves, or matter as we know it.

It is no coincidence that Cymatics patterns and Islamic tiling patterns are similar.

Cymatics Patterns – Sound (Vibration) creates geometry.

It is also no coincidence that Islamic tiling patterns resemble molecular lattices upon which matter is built.

A molecular lattice of proteins.

# The Breath of the Compassionate

The Breath of the Compassionate is called the Roub el Hizb in Islam.

Remember, the atomic number of 8 is oxygen, and oxygen is essential for breath.

There are 100 names of God in Islamic tradition.  Only 99 are pronounceable.

The highest pronounceable name is “The Compassionate”.

“From its infinite goodness the Compassionate exhales and inhales.  Through the polar cycle of divine breath the universe is periodically created, maintained, dissolved and renewed.”3

According to Mohammed (570-632) in the Hadith, “God created the universe through the Breath of the Compassionate.”

The name refers to the teachings of the Great Master Ibn al’Arabi which expound the Divine Breath as the basis of creation, liberating the possibilities of the four Elements: Fire, Air, Water and Earth.

Ibn al’Arabi stressed the ‘unity of being.’  He said all beings are in reality manifestations of God’s attributes.

He said, “Do not praise your own faith so exclusively that you disbelieve all the rest.  If you do this, you will miss much good.  Nay, you’ll fail to realize the real truth of the matter; God, the omnipresent and omniscient cannot be confined to any one creed, for he says in the Qu’ran, ‘For wheresoever you turn, there is the face of Allah.’ Everybody praises what he knows.  His God is his own creature, and in praising it, he praises himself.  Consequently, he blames the beliefs of others, which he would not do if he were just – but his dislike is based on ignorance.”

These concepts are the core of The Perennial Philosophy – that each of the world’s religious traditions shares a single, metaphysical truth or origin from which all esoteric and exoteric knowledge has grown.

Ibn al’Arabi spoke much on the Oneness of the Universe.  At times he uses a metaphor of the mirror.

“In this metaphor al-Arabi compares an object being reflected in countless mirrors to the relationship between God and his creatures. God’s essence is seen in the existent human being, as God is the object and humans being the mirrors. Meaning two things, that since humans are mere reflections of God there can be no distinction or separation between the two and without God the creatures would be non- existent.

When an individual understands that there is no separation between human and God they begin on the path of ultimate oneness. The one who decides to walk in this oneness pursues the true reality and responds to God’s longing to be known.  The search within for this Reality of oneness causes one to be reunited with God, as well as, improve self-consciousness.

Al- Arabi details that the perfect human is of the cosmos to the divine and conveys the divine spirit to the cosmos.”4

The Breath of the Compassionate symbolizes the interplay of polarities that manifest form.

“This motif tells us that the process by which the archetypal patterns crystallize into material configurations is a polar process as simple and ephemeral as breathing.  And so it is in the basic positive and negative charges of the atoms that configure the universal patterns.”5

It commonly appears throughout Islamic art and architecture including:  wall tiles, doors, screens, rugs, manuscript illumination, ornamentation…etc.

It is a tessellating design, leaving no gaps “just as Divine Compassion fills the universe without gaps creating a geometric motif seen throughout Islamic design as a reminder of divine presence.”6

This basic design is the starting point of a vast family of patterns.

Repeating them in each square makes the fundamental pattern of stars and crosses.

# Constructing the Breath of the Compassionate

Reference Construction Lesson #76: The Breath of the Compassionate.

These can be used as patterns or stencils to create larger designs.

# Other Eight-Fold Patterns in Islamic Art

Eight-Fold Rosettes are prevalent in Islamic patterns.

The petals are arranged around a central star like an archetypal crystalline flower.

See pages 70-71 in Designa.

## The Eight-fold Khatam – Pieces of Eight

See pages 86-87 in Designa.

## Zillij Design – Eight-Fold Extravaganzas

See pages 88-89 in Designa.

Read Islamic Design: A Genuis for Geometry by Daud Sutton for more information on the exquisite art forms of Islam.

1. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
2. Sutton Daud, Islamic Design: A Genius for Geometry, Bloomsbury USA, 2007
3. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
4. https://en.wikipedia.org/wiki/Ibn_Arabi
5. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
6. ibid.
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