The main thing to know about hexagons is that they impose structure onto the free-flowing fractal structures we see in the archetype of 5-ness. While five-ness sets up the property of regeneration, six-ness handles that offspring. It is like a disciplinary parent, imposing restrictions on a child who, while may be having a good time, is bound to leer into chaos.
The Pythagoreans referred to the archetype of six-ness as “form of form” and the “unwearied anvil”. I interpret this is “having the properties of inherent structuring” and “resistance to degradation”, due to the stability in the forms of six-ness, as we’ll see.
Here we will learn how the pure number properties of six give hexagonal forms its archetype of functionally structuring into order.
Archetype of Six-ness: Structure, Function, and Order
In the book of Exodus, it says “Six days shalt thou labor, and do all thy work.” The seventh day was for rest. The first six days establishes all the order necessary to have things “up and running.” The number six contains interesting relationships in its additive identities, and, for the first time up to this number, a bit more complexity in the multiplicative identities.
Other than four, six is the first composite (non-prime) number. Though four is the first composite number, it only has one factor besides one and itself (2×2=4). The factor of two is in duplicate, so four has just three factors.
Six, on the other hand, with four unique factors, is the product of two different prime numbers. Let’s explore all the numerical relationships here.
Birth of the number 6
Six in a way is like four.
Six is the “fourth” number that introduces entirely new properties. In some ancient traditions, one and two give birth to three. In this way, the numbers one and two (representing Monad and Duad), are the “parents” of the first number three. So then the first four numbers are 3, 4, 5, and 6. The way the numerical archetype of four represents stability and order, on the Earthly plane, six borrows a bit from this.
–>Threeness
–>Fourness
–>Fiveness
–>Sixness (you are here!)
The most highly divisible numbers have six as a factor. Six’s two prime factors, two and three, are the most common prime factors in general, since they are the first non-unity primes.
Math of the Six-ness and Hexagons
For example, 12 and 60 are commonly used in accounting and recording systems, like clock faces (12 divisions), seconds in an hour (60), inches in a foot (12).
The half equilateral has angles 30-60-90, and an irrational side proportion (√3).
The triangle having side lengths of the first Pythagorean triple (3, 4, and 5) has an area of exactly 6 units squared.
60° is also the measure of all three angles in any equilateral triangle. Equilateral triangles are frequently used in design and construction for the stability and three-fold symmetry it possesses. Read more about the stability inherent in the archetype of three-ness. Also ubiquitous in math and engineering is the half-equilateral triangle, also known as the 30-60-90 triangle.The 30-60-90 triangle side ratios have the root of six-ness in the irrational side, described below. This triangle is also inside the hexagon.
Another interesting property of six is rather unexpected. Every single power of 6 actually ends in a six. For examples, 62 is 36, 63 is 216, and so on.
| 63 | 216 |
| 64 | 1296 |
| 65 | 7776 |
| 66 | 46656 |
| 67 | 279936 |
| 68 | 1679616 |
| 69 | 10077696 |
To become even more acquainted with the archetype of six-ness, one can learn to construct the hexagon on paper. This can be done through the vesica piscis, equally dividing a circle into 6 parts, or surrounding a circle with 6 circles.
The hexagon has sixfold symmetry, meaning that within a full rotation, there are six equivalent sections. Since a full circle is 360°, each of these six equivalent sections is 60° each. Remember when I said multiples of six are often chosen for systemization? The choice of 360° for a full circle is one such choice.
Ancient Pattern, Modern Appearances
Patterns with the archetype of six are found in many ancient artifacts. We see the common motifs of hexagonal symmetry in pottery, tile, wallpaper, fabric, and more.
As we gaze upon such patterns for a while, various patterns can begin to appear and change. Our own nervous systems fluctuate as they pick up on these.
Pennsylvania Dutch Hex Signs
No one is quite at a consensus about the exact origin of Pennsylvania Dutch symbolism, or what it really means or was potentially used for. But the Protestant settlers of the Pennsylvania area began putting these nice signs up on their barns. This became a bit of a trend in the late 1800’s early 1900’s when paint was more available.
Tulips are also related to three-fold and six-fold symmetry. The tulip flower has three or six petals and three or six stamen inside. Daffodils also follow the three-fold and six-fold style. Read about other flowers and plants with three-fold symmetry.
Some say the different symbols could represent luck, fertility, protection, or other magical benefits. There has been a bit of literature about this, such as a 1924 book by Nutting in which he claimed the signed were used to ward off evil. These were often on gravestones, pottery, as well as certificates of baptism and marriage on Fraktur.
Escher type tessellation
Inventions using Archetype of Six-ness
Hexagons are a good way to approximate a circle. This is a way to get close to the efficiency of a circle using straight lines.
- hex nuts
- woven baskets (stability in the bottom)
- bicycle spokes (multiples of six)
- camera lens diaphragm
- umbrellas
- parachutes seams
- ball caps seams
Human ingenuity may make various inventions, […] but it will never devise any inventions more beautiful, nor more simple, nor more to the purpose than Nature does; because in her inventions nothing is wanting and nothing is superfluous.”
-Leonardo Da Vinci
Notice how on a ball cap there are six wedges. this minimizes fabric and maximizes the sturdiness. Umbrellas and parachutes share this feature as well. Woven baskets made by indigenous tradition follow this as well.
Hexagonal Packings and Packagings
Tessellated hexagons: “”six-around-one”. Circles and spheres pack similarly – connecting the centers forms hexagons in the plane. These hexagonal tilings also leave no gaps in the plane, with no need for other shapes. Hexagonal tessellations of this kind extend monad properties across the plane, in a way tessellated circles could not. This feels like “equality in all directions.” Such constructions minimize anisotropic (asymmetric) stress and strain. These hexagons also for 120 degree joints, which has the triangular stability discussed in the three-ness archetypes, such as in bridges.
Here are some examples of how things, especially round or radial things, tend to pack hexagonally:
- produce stacks (pyramid of oranges)
- bubble wrap
- netting
- chicken wire
- central marketplaces
Natural Hexagonal Symmetry
Pentagonal symmetry we see in living things in almost every form. Hexagonal symmetry we see in some natural structures, but we also see it in many man-made things, as discussed above. (move this up). Now let’s discuss the ways hexagonal symmetry crops up in natural fields.
First we have crystals and molecules, natural bonding configuration or tendency to precipitate. In general, a hexagonally symmetry crystal’s molecules take up 2/3 to 3/4 of the crystals space. This close packing arrangement happens to be a “sweet spot” for balancing the positive and negative forces for ground energy.
“Doth not [N]ature itself teach you?”
1 Corinthians 11:14
We definitely tend to see five-ness have a more captivating and magical effect in nature. This teaches us about the “structure-function-order” aspect of the archetype of six-ness.
Hexagonal Symmetry in Nature:
- Muscle striations
- Eye muscle arrangement
- Lung aveoli net
- insects: 6 legs
- Quartz structure
- snowflake symmetry
- carrot and pepper stems
- fly’s eyes
- silkworms
- beehives
- viruses
- radiolaria
- fish scales
- Hydrodietyon
- tortoise shells
- cellular packing (i.e. human cells)
- hormones: testosterone, estrogen
- drugs: terramycin, asprin, steroids
- vitamin c, vitamin d
- UF6
- “hexagonal” water
- graphite (including other carbon structures)
- TNT
- Cyclo-octa-deca-nona-ene (aromatics
- Benzene rings, sugars
Entering the Honey Dimension
What it be like?
The beehives are really composed of hexagonal prisms/cylinders. This is similar to wax crystal, like quartz. In honey space, Pappus c. 300 A.D. , Alexandria, Egypt – 3 figures can fill up this space, the triangles, the squares, and the hexagons. (But really hexagons have both the others by connecting different points which is why you can also see this in your mind.)
These are not selected by bees but by the genetics and nature driving them, but these contain most of the angles contained therein, and thus flexibility to choose and create structure within itself. Indeed, from a volume perspective, this shape holds more honey than any of the other two, triangular or square, prisms.
The Hexagons, so Right they’re Wrong?
So hexagons give lots of options and indeed they are also related to the flower of life. Indeed the shapes that dissipate better can not exactly be this flower of life because a blending is needed. For example, in the only patented emf harmonizer that I’m aware of, But look, when the DNA is projected it actually has decagonal symmetry, but this 3D spiral projected actually has hexagons symmetry in some cross sections, as shown by this shadow.
So these have lots of different things to look at just like in the sri yantra, magic eye paintings changing in the brain and geometric secrets are revealed to you.
Also Consulted
Constructing the Universe by Micheal S. Schneider; http://www.constructingtheuniverse.com/
You can certainly see your skills in the paintings you write. The world hopes for more passionate writers such as you who are not afraid to say how they believe. All the time go after your heart.
Thank you, I am glad that you enjoyed this.