We will now move back into our discussion on the archetypes of the numbers 1 to 10 through a five part series on the Hexad, or Sixness.
The Hexad – 6 – The Symbol of Structure-Function-Order
The Hexad is intimately related to the Monad (circle) and Triad (triangle).
It uses the principles of wholeness and balanced structure in advanced ways.
The Hexad is called: “The Form of Form”, “The Unwearied Anvil”, and “The Perfection of Parts”.
The hexad gives form (or structure) to all forms that is why it is called the ‘form of form’.
There are Six directions: above, below, left, right, front, back or up, down, north, west, east, south.
“To the ancients, six represented the parents (1 & 2) of all numbers with their firstborn (3), thus making a completed whole.”1
Six is the natural division of a circle, as the radius of the circle measures around the circumference creating six equal divisions making the hexagon the easiest polygon to construct.
Recall that Hexad/Cubic/Octat geometry is known for its role in structure, function and order. It is seen throughout the natural world in holding and making membranes, or creating barriers.
This is commonly seen in the structure of molecules, heat convection cells, cellular structure in plants, insects, animals and humans, as well as the shape of crystals. We will take a look at many examples at the end of this article.
Atomic Number 6 = Carbon
Carbon is an extremely important chemical element. It is the fourth most abundant element in the universe. About 20% of the weight of living organisms is carbon. More compounds are known which contain carbon than don’t. Soot, charcoal, graphite, diamonds, graphene and fullerenes are all different forms of carbon.
Diamond is the hardest substance used by humankind. It also has the highest thermal conductivity.
Graphene is the thinnest, strongest material ever known. It was discovered in 2004. It consists of a single layer of carbon atoms arranged in hexagons.
Graphite is used in pencils. The typical mechanical pencil diameter is 0.7 mm. This equals 2 million layers of graphene.
Fullerenes are carbon atoms arranged in soccer-ball shapes (truncated icosahedra). The best known fullerene is carbon-60. NASA’s Infrared Space Telescope Spitzer has identified buckminsterfullerenes (buckyballs) equal in mass to 15 of our moons in the Small Magellanic Cloud dwarf galaxy.
Ionsdaleite is actually the hardest substance known to humankind, not diamond. Ionsdaleite is an extremely rare allotrope of carbon. In its pure form it would be 58% stronger than diamond. “Ionsdaleite is a diamond-like carbon network which has graphite’s hexagonal structure. It is made when meteorites containing graphite hit another body, such as Earth. The high temperatures and pressures of the impact transform the graphite into Ionsdaleite.”2
Eight allotropes of carbon. Notice the hexagonal structure. a) Diamond; b) graphite; c) lonsdaleite; d) C60 Buckminsterfullerene; e) C540 Fullerene; f) C70 Fullerene; g) Amporphous carbon; h) single-walled carbon nanotube
Atomic Number 60 = Neodymium
Neodymium is a lanthanide rare earth metal that is soft, bright and silvery white. It is used with iron and boron to create powerful magnets called NIB magnets. These are used in many forms of technology.
Neodymium is also used as a crystal in lasers. These are used to treat skin cancers, laser hair removal, and also used to cut and weld steel and make specialized goggles for glass blowers.
It has two allotropic forms: hexagonal that transforms to body-centered cubic above 800K.
Atomic Number 66 = Dysprosium
Dysprosium is a bright, soft, silvery-white rare earth metal. Dysprosium and holmium have the highest magnetic strengths of any elements. It is good at absorbing neutrons so it is used in control rods in nuclear reactors. It is also used in sonar systems, sensors and transducers, data storage applications and rare-earth lamps to produce an intense white light used in the film industry.
Arithmetic Properties of the Hexad
Six is the first of only 2 terms within the Decad composed by the multiplication of 2 different factors (other than Unity):
1 = 1 x 1 6 = 2 x 3
2 = 1 x 2 7 = 1 x 7
3 = 1 x 3 8 = 2 x 2 x 2
4 = 2 x 2 9 = 3 x 3
5 = 1 x 5 10 = 2 x 5
Six is a doubling of three. It partakes of the Triad’s principle of balanced structure, but to a higher level.
Six is the third of four triangular numbers within the Dekad (1, 3, 6, 10).
Six is the factorial of 3: (3!) = 1 x 2 x 3
The area and semi-perimeter of the 3-4-5 triangle = 6.
6 is the first ‘Perfect Number’.
6 is the sum and product of its divisors (1, 2, 3). That is 1 x 2 x 3 = 1 + 2 + 3 = 6.
Here we see that six is the only number that is the sum and product of the same 3 integers.
5 & 6 represent ‘marriage’.
2 + 3 = 5
2 x 3 = 6. This is a multiplication (cross-fertilization) of even & odd.
5 & 6 are the only numbers (except Unity) that generate progeny like themselves:
The self-multiplying powers of 5 always end in a 5.
5 x 5 = 25; 5 x 5 x 5 = 125…etc.
This expresses self-replication in living forms.
The powers of 6 always end in 6.
6 x 6 = 36; 6 x 6 x 6 = 216…etc.
This expresses self-similarity in self-reinforcing structure-function-order.
- 6 faces of a cube
- 6 vertices of an octahedron
- 6 edges of a tetrahedron
- 6 x 2 faces of a dodecahedron
- 6 x 2 vertices of an icosahedron
- 6 regular 4D polytopes:
- Simplex, tesseract, 16-cell, 24-cell, 120 cell, 600-cell
Base-6 is the measure for structure: (inches; feet; etc.)
It is also the measure for order (time): (seconds, minutes, hours, etc.)
Powers of Six
62 = 36
- Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- 36 is the number of years required for the equinoctial sun to complete a precessional shift of half a degree along the ecliptic. (36 is half of 72).
- There are 36 degrees in the interior angle of a regular pentagram.
- There are 36 edges in the truncated cube and truncated octahedron. (Archimedian solids)
- There are 36 edges in the tetrakis hexahedron and triakis octahedron. (Catalan solids).
- The sum of the integers from 1 to 36 = 666.
- 36 is a square, triangular and circular number.
- A circular number is a square number that ends with the same integer by itself (6×6=36)
- 36 has significance in Jewish tradition, Maori legend, and the 36 tattvas in Shaivism.
- Atomic Number 36 = Krypton
- 3600 = The Sum of degrees in the Icosahedron, Cuboctahedron, and Truncated Tetrahedron.
63 = 216
- Divisors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
- 216 = 33 + 43 + 53 + 63
- 216 = 107 +109 (sum of a twin prime).
- 216 = 6 x 6 x 6
- 216 has a significance in the Kabbalistic tree of life.
- (216 = numerical value of Gevurah, the 5th sephirot; and the sum of the letters in God’s 72 hidden names.)
- 216 x 10 = 2160
- 2160 = 72 x 30 = the number of years required for the sun to complete a passage of 30 degrees along the ecliptic (through one of the 12 zodiacal constellations).
- That is, there are 2160 years in One Great Month of the Precession of the Equinoxes.
- 2160 = the sum of interior angles of the cube, triakis tetrahedron, and rhombic dodecahedron.
The number of the cube = 2160. Recall that the cube and hexagon are intimately related. The 2D version of a cube is a hexagon. Also, cross-sections of a cube produce hexagons when the cube is intersected by a plane perpendicular to its diagonal and is cut in half.
2160 (related to 432)
- cube = 2160 (sum of interior angles)
- 432/2 = 216
- 2160/2 = 1080 = octagon
- 2160/3 = 720 = hexagon
- 2160/4 = 540 = pentagon
- 2160/5 = 432
- 2160/6 = 360 = square and circle
- 2160 = diameter of the Moon in miles (half 432: 4320/2 = 2160)
- The diameter of the Sun in miles = 864,000 (twice 432: 432000 x 2)
- 21600 = The sum of the angles of the trapezoidal hexecontahedron and the disydakis triacontahedron (Catalan solids).
The Theology of Arithmetic – Iamblichus
As we have been doing, we will now take a look at the meaning of the Hexad as prepared by Neo-Platonist philosopher, biographer of Pythagoras, Greek mystic and mathematician Iamblichus (245-325 AD).
The hexad is the first perfect number, for it is counted by its own parts, as containing a sixth, a third and a half.
When squared it includes itself (6 x 6 = 36).
When cubed, it does not maintain itself as a square. It includes 6, but not 36. (6 x 6 x 6 = 216)
The hexad arises out of the first even and first odd numbers, male and female, as a product by multiplication – hence it is called ‘androgynous’.
It is also called ‘marriage’ – it arises not by addition, as the pentad does, but by multiplication.
It is also called ‘marriage’ because it is equal to its own parts, and it is the function of marriage to make offspring similar to parents.
The hexad’s parts (1, 2, 3) have a certain arithmetical proportion.
The harmonic mean is first formed by the hexad: 6, 8, 12.
The arithmetic mean also falls under six: 6, 9, 12.
Six also forms a geometric mean: 3, 6, 12.
There are six extensions of solid bodies: forward, backward, up, down, left, right.
The universe is ensouled and harmonized by the hexad, and thanks to it, comes by both wholeness and permanence, and perfect health.
Number itself is found to have formed its progression to infinity by means of the hexad, in perfect additions, for primary perfection is having beginning, middle and end (1, 2, 3); and secondary perfection is being equal to one’s own parts (1 + 2 + 3), without excess or deficiency.
The primary type is found in the triad.
The secondary type is found in the hexad.
But the triad’s perfection is also found in the hexad (2 + 2 + 2 = 6).
Quantities occur in triads and that by adding these quantities you get a hexadic identity:
1 + 2 + 3 = 6
4 + 5 + 6 = 15 = 1 + 5 = 6
7 + 8 + 9 = 24 = 2 + 4 = 6
10 + 11 + 12 = 33 = 3 + 3 = 6…etc.
All number is formed by the dependence of triad on hexad.
The hexad is considered to be the form of forms.
The hexad is found stably to be maker of soul and causer of the condition of life – All soul is harmonic and the most elementary concordant intervals are the sesquitertian (4:3) and sesquialter (3:2).
The hexad is constituted to bring together and into unison things which are altogether different.
Solidity turns out to fall under six and to be not single, but triple.
36 (6 x 6) encompasses harmony as well: it is the summation of 6, 8, 9 and 12 (and the Monad, their common source) and these numbers in which the musical intervals which most properly constitute harmony in general are said by musicians to reside.
The hexad has been called ‘wholeness of limbs’ because it alone of the numbers within the decad is a whole equal to its parts or limbs.
It is called ‘reconciliation’ for it weaves together male and female by blending, and not by juxtaposition as the pentad does.
It is called ‘peace’ based on the fact that it organizes things.
It is also called ‘universe’, for the universe, like 6, is often seen as composed of opposites in harmony, and the summation of the word universe is 600.
It is also called ‘health’ and ‘anvil’ because it is reasonable to think that the most fundamental triangles of the elements of the universe partake in it, since each triangle is six, if it divided by three perpendiculars it would be divided into six parts.
There are 6 edges to a tetrahedron.
There are 6 faces of a cube.
There are 6 vertices in an octahedron.
There are 6 bases of a dodecahedron.
Nothing pertaining to their faces or angles or edges is altogether free from the hexad.
There are also six signs of the zodiac over the Earth and six under the Earth.
The hexad is called ‘presider of crossroads’ because the hexad is the first to acquire three movements of the dimensions.
It is called ‘measurer of time in twos’ because the distribution of all time, which is accomplished by a hexad of zodiacal signs over the Earth and another under the Earth, or because time has three parts (past, present, future).
The hexad is a very close neighbor of the pentad, therefore it is called ‘dweller by justice’.
The sequence from the monad to the hexad is continuous; music starts with the hexad and proceeds by doubles and triples.
The hexad is a clear likeness, more than any other number of the even-odd monad because it is the very first to contain parts with opposite names and opposite denominations (for its third is 2, its half 3, its sixth 1, and the whole is 6).
It alone of all numbers within the decad is half even, half odd (2 x 3) and is therefore patently a mixture of indivisible being and divisible being.
The Hexad in Chemistry
See Articles 166-167.
There are Six Reactions in Chemistry:
- Single & Double Displacement
- Acid-Base Reactions
Hexagonal bonding is one of the most common in molecular chemistry. Seen below is the molecule of THC, protein, kevlar and the benzene ring.
The benzene Ring (C6H6) is the basic structure of organic chemistry. It came to German chemist Friedrich Kekule after he dreamed of the ouraboros.
Heat convection cells are hexagonal in structure. This can be seen in liquids and weather patterns
Heat convection cells seen in the cloud structure over the west coast of Africa
Soap bubbles eventually settle into 6-around-1 arrangements. The molecules strive to balance forces with minimum surface tension.
Crystals often show a hexagonal structure. This includes snow crystals.
See Articles 173-177.
Six water molecules form the core of each snowflake. More molecules build upon the seed pattern to become beautiful snowflakes of 6-fold symmetry.
The Hexad in Nature
See Articles 173-175.
We see hexagonal formations in geology, land masses & soil.
See Articles 178-197.
There are Six Kingdoms:
- prokaryotes (Archaebacteria & bacteria)
- Animalia (eukaryotes)
As we have seen hexagonal geometry is also seen in many life forms from plants to insects, to animals to the human body.
The cellular structure of plants is most often hexagonal.
Hexagonal cellular packing can be seen in the structures of many plants including fruits, seeds, and pollen.
Hexagonal structure is also seen in fungi.
Insects have 6 legs. They lift alternate legs in a stable tetrahedral fashion as they walk.
Honey bees build the most well-known example of geometry in nature, the hexagonal honeycomb.
The use of 120 degree joints allows for the least wax to hold the most honey. A mere 1 ½ oz. of wax holds 4 pounds of honey!
According to Randall Carlson, “Bees and the beehive [in Freemasonry & Ancient Egypt] symbolize the existence and identity of the Ones who, from time to time, reveal themselves to Mankind through the medium of celestial apparitions and visitations.”
Wasp & hornet nests are also hexagonal and wasps, bees and flies have close-packed hexagonal facets that make up their eyes.
The eye of a wasp, fly, ant & bee
Many insect wings also have a hexagonal structure. These, of course, are not perfect hexagons but are based upon Delauney triangulation and Voronoi tessellations.
Hexagonal structures are seen in sea creatures, animals and humans as well.
Some of these include:
- water-net algae (hydrodietyon)
- star coral – hexagonal skeleton
- tube coral – cylindrical skeletons packed in hexagonal groups
- hexagonal fish scales
- hexagonal scales of reptile skin
- hexagonal plates of a tortoise shell
- striped muscle in humans is hexagonal. This type of muscle is available for voluntary movement.
- human lung alveoli is in a fish net pattern. This maximizes the passageways for oxygen-carbon dioxide transfer.
The Hexad in Space
See Articles 251-262 and 98-110.
There is a hexagonal vortex at the pole of Saturn.
Many galaxies have a hexagonal bending of their arms.
Galactic clustering is based upon rhombic dodecahedral cells. The cross-section of the rhombic dodecahedron is the hexagon.
Credit: Conrad Ranzan
The Hexad in Music
The pentatonic scale consists of five notes per octave.
Six is therefore the return, or pentatonic octave.
Pentatonic scales are used all over the world including, but not limited to: many European folk music styles, West African music, Gospel, Bluegrass, Jazz, Blues, Rock, Andean music, Afro-Caribbean music, and the music of ancient Greece.
The Hexad in Religion
- Six Days of Creation – Abrahamic traditions:
- Light, Firmament, Land & Vegetation, Heavenly Bodies, Fish & Birds, Animals & Man
- Six Buddhist Perfections: Giving, Morality, Patience, Energy, Meditation, Wisdom
- Six Hindu-Buddhist Realms: Gods, Hells, Humans, Hungry Ghosts, Demons, Animals
The Hexad in Art
Hexameter Verse in Poetry – “Hexameter is a metrical line of verses consisting of six feet. It was the standard epic meter in classical Greek and Latin literature, such as in the Iliad, Odyssey and Aeneid.
In classical hexameter, the six feet follow these rules:
A foot can be made up of two long syllables (- -), a spondee; or a long and two short syllables, a dactyl (- u u).
The first four feet can contain either one of them. The fifth is almost always a dactyl, and last must be a spondee.”3
Classical hexameter is rarely used in English because English is a stress-timed language that condenses vowels and consonants between stressed syllables. Hexameter relies on the regular timing of phonetic sounds.
Here is one example by Michael Drayton from 1612 of classical hexameter. It is the Poly-Olbion and the vertical lines mark the feet:
Nor a | ny o | ther wold | like Cot | swold e | ver sped,
So rich | and fair | a vale | in for | tuning | to wed.
Hexagonal design properties are also found in the following classical art. This is but a small list of examples.
- Great Seal of the U.S – The Eagle side contains a hexagonal array of 13 stars. The entire design is laid over a hexagon/hexagram framework.
- David Trying on Saul’s Armor – from Byzantine Constantinople (610-630). This design in laid over the same hexagon/hexagram framework as the Great Seal of the U.S.
- Aegeus before the oracle of Delphi – Greek. This design fits within two interlaced hexagons.
- Bella Coola wooden sun mask – West Coast Native American. This design in laid over two interlocking hexagons and a hexagram.
- British Royal Seal – This design in based upon a more complex geometrical arrangement of circles and hexagons.
Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe
- Egyptian Art: The two ends of the Nile symbolically unite ancient Egypt with a hexagonal knot
- This design is based on a hexagonal/hexagram framework.
- Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994