In this article we will go into detail about the cuboctahedron, an Archimedean solid, and its dual, the rhombic dodecahedron, a Catalan solid.
Cuboctahedron (a.k.a. Vector Equilibrium) – 3600º
The cuboctahedron, an Archimedean solid, is also known as the Vector Equilibrium (VE). We will use these terms interchangeably.
It was Buckmister Fuller who coined the term ‘vector equilibrium’.
According to Bucky, “The vector equilibrium is the zero starting point for happenings or nonhappenings: it is the empty theater and empty circus and empty Universe ready to accommodate any act and any audience.”
In other words it contains infinite potential, at rest, waiting for action.
It is “an omnidirectional equilibrium of forces in which the magnitude of its explosive potentials is exactly matched by the strength of its external cohering bonds.”1
This means the distance between the center and any vertex matches the distance of each edge. All its forces are equal.
It is a quasiregular polyhedra.
This means it is made of 2 types of regular polygons (squares and equilateral triangles), each being surrounded by polygons of the other type.
The radial projections consist entirely of complete great circles.
The cuboctahedron is a perfect 12-around-1 system.
12 spheres pack around an equal 13th to produce a cuboctahedron (hexagonal close-packing).
“Given that the diameter is the same for all of the spheres, the centers of each sphere will be equidistant from all of their adjacent neighbors, including the center one. The lines connecting their centers are the vectors of the VE.”2
There are many important symbolic 12-around-1 systems including:
The 12-tone chromatic musical scale
The Astrological Zodiac
The 2D Fruit of Life
The Cuboctahedron is the Archimedean Solid that forms halfway between the transformation from a cube to an octahedron. It is the degenerate truncation (rectification) of both the cube and octahedron.
It has:
- 14 faces (8 triangular and 6 square)
- 24 edges
- 12 vertices
It combines 6 squares of the cube with 8 triangles of the octahedron.
The Projections of the Cuboctahedron
Below are some commonly encountered views.
All internal angles are 60°.
The Sum of its Angles = 3600º.
It has octahedral symmetry and is commonly found in the shapes of crystals. (fluorite & diamond pictured below)
It is composed of 8 tetrahedra pointing inward to one central point, alternating with 6 octahedra cut in half (3 total octahedra). In the image below you can see the tetrahedra. The empty space is where the half octahedra fit in.
The 11 solids (8 tetrahedra and 3 octahedra) make a total of 10,080°.
The combined diameters of earth and moon in miles = 10,080. There are also 10,080 minutes/week.
The Vector Equilibrium’s “twelve radii form eight symmetrically arrayed regular tetrahedra – corresponding to the VE’s eight triangular faces. The tetrahedra, which radiate outward edge to edge, create six cavities in the shape of square-based pyramids. Because of the uniform edge lengths everywhere, these cavities are actually perfect ½ octahedra, corresponding to the six square faces of the VE, which in turn correspond to the six faces of the cube.”3
It is an extremely stable shape.
It is the only geometry in perfect equilibrium in all vectoral possibilities.
All lengths are equal.
- The triangles and squares share the same vector (side) length.
These are equal to:
- The vectors from its center point to its circumferential sides, and
- The vectors connecting all of those points.
In other words, “All the edges of the figure are of equal length, and this length is always the same as the distance of its vertexes from the center of the figure.”4
Bucky Fuller writes, “If the force of a vector is its length and two vectors are driven against each other in opposite directions you will get equilibrium. But it will be unstable since any other force in any other direction could break the balance. You needed a vector equilibrium so all forces would cancel each other out and appear as empty space.
With all vectors being exactly the same length and angular relationship, from an energetic perspective, the VE represents the ultimate and perfect condition wherein the movement of energy comes to a state of absolute equilibrium, and therefore absolute stillness and nothingness.”
Hexagonal Planes
In a way, the VE consists exclusively of hexagons. It has 4 Hexagonal Planes.
These are seen below in red, yellow, black and white.
The regular hexagon is the only polygon that has edges equal in length to the distance between the polygon’s center and its vertices.
The edges define four regular hexagons, seen above. They are all 60° from each other.
The angles they define are exactly the same as those of the faces of a tetrahedron.
Genesa Crystal
The Genesa Crystal uses circles instead of hexagons for the 4 hexagonal planes.
Spherical Close-Packing
The Cuboctahedron – 13 spheres – The closest packing of spheres around a nucleus of equal size gives the cuboctahedron (vector equilibrium). Each sphere touches the nucleus and 4 others.
The Spherical Cuboctahedron:
The Stereographic Projections of the Cuboctahedron:
Square-centered on left; triangle-centered in middle; vertex-centered on right
The Cuboctahedron Net:
Area and Volume of the Cuboctahedron
Area = (6 + 2√3)s2 s = side length
Volume = 5/3 √2s3 s = side length
The Dual of the Cuboctahedron is the Rhombic Dodecahedron, a Catalan solid.
We will discuss the rhombic dodecahedron in greater detail later in this article.
Buckminster Fuller’s Cosmic Hierarchy
It starts with the tetrahedron and its dual (star tetrahedron).
This forms a cube and its dual the octahedron.
Then comes the icosahedron and its dual the dodecahedron.
Then comes the vector equilibrium and its dual, the rhombic dodecahedron.
Jitterbugging
Buckminster Fuller’s explanation of ‘jitterbugging’ once again relates to the nesting properties of Platonic solids. The jitterbugging motion is a result of the vector equilibrium’s ability to transform into each and every Platonic solid, remembering that the vector equilibrium is the ground state geometry of the Aether.
The jitterbugging exhibits a pulsating dynamic that arises when it moves out of equilibrium and returns back again. During its non-equilibrium phase it manifests the primary structural forms of the Platonic solids.
The vector equilibrium (VE) has square and triangular faces.
The square faces are unstable structurally – this allows for the VE to collapse in a spiraling motion (jitterbugging).
It can rapidly collapse and expand in both left and right spirals, pulsating and oscillating like a dancer.
As it does, it transforms through phases that include symmetrical articulation of the icosahedron, dodecahedron, octahedron, cube and tetrahedron.
Each stick represents a vector of energy that strengthens by doubling and then tripling during the folding.
“As the VE collapses inward and the square faces contract across one of their diagonals, the length of that diagonal distance becomes the same length as the VE’s edges. At this moment the symmetry of the icosahedron arises.”5
The dodecahedron is energetically implied at this phase as it is the dual of the icosahedron.
“Continuing on its inward journey, the square faces of the VE continue to contract across the diagonal until the gap is completely closed. At this moment the symmetry of the octahedron arises. This octahedral phase now displays a doubling of the vectors of the VE, creating an extremely strong bonding tension as is found in atomic elements that have octahedral symmetry.”6
The cube is implied at this point, as it is the dual of the octahedron.
From an octahedron, the motion continues, folding in on itself to form the tetrahedron.
Here is the tetrahedron expanded into a cuboctahedron then expanded into the dual tetrahedron (tetrahedron pointing in opposite direction).
Jitterbugging as Toroidal Flow
“As the vector equilibrim (VE) jitterbug spins inward it sets up a differential of energy density (i.e. pressure, electromagnetic charge) that sets in motion a dual vortex flow that creates the form of a torus. The pumping of the jitterbug sustains this toroidal form in a balanced rhythmic exchange of energy that flows through the manifest system. From a fractal-holographic perspective, it is this fundamental dynamic that takes place at every scale, first expressed as photons, then sub-atomic particles, which then aggregate into the geometric arrays of atoms, which aggregate into compounds that form crystals, minerals, cells and organs, and then whole organisms such as trees, animals, us, and then ecosystems, atmospheres, planets, stars and galaxies.”7
The Spinning Vector Equilibrium as a Dual Torus Feedback Loop
Instead of a spherical smooth surface, the VE spins to form a torus. A dual torus to be exact – one donut spinning clockwise; the other spinning counter-clockwise.
This represents a spherical flow process that meets at the Equator, goes back to the poles then meets again at the Equator.
This system shows how a feedback structure is formed: in atoms, planets, the sun’s magnetic field and galaxies. It is all based on geometry and fluid dynamics.
The center of the torus is the wormhole that acts as a gateway between time/space and space/time. There is continual oscillation between these two realms as the flow falls through the center of the torus and back out again.
This dual torus feedback loop also expresses the consciousness feedback loop.
Consciousness demands feedback.
The Perennial Philosophy teaches that all in the universe is composed of consciousness.
We are constantly disappearing and reappearing at the speed of light.
The structure of the double torus allows feedback from the outside to go inside, then back outside again…
When our consciousness fluctuates into the invisible realm we pick up information there as we simultaneously inform it with our thoughts, feelings, expectations, and beliefs.
This information is shared by all other consciousness when they in turn fluctuate into the invisible realm.
It is an exchange and sharing of internal understanding and perspectives of everyone and everything.
Physical reality is constantly created and adjusted by new input from all consciousnesses.
We input, input creates reality, we learn from reality, we input new understanding, new input creates an altered reality, we learn from the altered reality…and so on…
While we are in the metaphysical reality what are we saying? How are we informing it when we are present there? What is your general state of mind? What are your beliefs, thoughts, emotions, and expectations?
This consciousness feedback occurs on an individual level and on a mass level.
We are co-creating our reality with everybody and everything at all times.
We are in a relationship with the universe at all times and exchanging information on all scales.
The Torus
“A torus consists of a central axis with a vortex at both ends and a surrounding coherent field. Energy flows in one vortex, through the central axis, out the other vortex, and then wraps around itself to return to the first incoming vortex.
The torus is the fundamental form of balanced energy flow found in sustainable systems at all scales.
It is the primary component that enables a seamless fractal embedding of energy flow from micro-atomic to macro-galactic wherein each individual entity has its unique identity while also being connected with all else.
In this way we can see that there is a seamlessly dynamic exchange of energy and information (aka consciousness) occurring throughout the entire cosmic experience.
Even the most fundamental energy event – a photon of light – can be seen as a toroidal fluctuation emanating from the underlying Unified Field [Aether].”8
The torus represents a process, not just a particular form.
The Universe = Processes.
Every dynamic manifestation (photon, electron, atom, flower, person, hurricane, planet, galaxy…etc.) is an energy event, an energy process.
Buckminster Fuller & the Isotropic Vector Matrix (IVM)
Buckminster Fuller concluded the fundamental blueprint of the universe was a “4-frequency isotropic vector matrix”.
Nassim Haramein and those at cosmometry.net agree that the IVM is the fundamental structure of the Aether. In essence it is the ‘ground state’.
The Isotropic Vector Matrix: The 64 Tetrahedral Grid
The Isotropic Vector Matrix (IVM) is composed of octahedra and tetrahedra. The vector equilibrium sits in the center.
You can extend the Vector Equilibrium (VE) outwards (due to equal vectors and equal 60° angles) from the center to form the IVM.
Isotropic – all the same
Vector – line of energy
Matrix – pattern of lines of energy
The IVM consists of an arrangement of alternating tetrahedral and octahedral geometries.
This is the same matrix as our cosmic gravitational cells (tetrahedra & octahedra that tessellate). This is from the work of Conrad Ranzan and the Dynamic Steady State Universe model.
The IVM is related to the Flower of Life. It is the straight line active version of the passive Flower of Life.
It is composed of 20 tetrahedra: ten on bottom; six in middle; three on the 3rd floor and 1 on top.
The negative space naturally creates octahedra – base to base pyramids.
There is another set in the middle reversed and rotated 180 degrees.
They are there because one IVM could not be alone. It needed an opposite one pointing downwards. This is the principle of polarity at work that is an intrinsic part of physical reality.
They lock into each other to form a star tetrahedron.
The geometry in the middle is the extremely stable cuboctahedron (vector equilibrium).
The edges of the matrix still had open spaces. 24 more tetrahedrons were added. This created a total of 64 tetrahedra.
The IVM: A 64 Tetrahedral Grid
There are:
- 32 positive tetrahedra (upward pointing)
- 32 negative tetrahedra (downward pointing)
It is a true 3D fractal structure that grows in perfect octaves.
The octave structure is a part of the geometric laws that govern the universe. Growth occurs in octaves. This includes light, sound, color, the chemical periodic table of elements, and so on.
It can be built from 8 star tetrahedra to produce the 64 tetrahedral grid.
The 8 star tetrahedra have points radiating out.
Put together it creates 8 vector equilibriums with the points reaching inward.
The shape reflects balanced polarity (upward and downward pointing tetrahedra).
The star tetrahedra with points radiating outwards represent the radiating and expanding qualities present in the Aether.
The vector equilibriums, with points radiating inwards, represents the absorbing and contracting qualities of the Aether.
It defines the most balanced array of energy structures (tetrahedra) where the positive and negative polarities are equal and without gaps in the symmetry.
The outer surface of the IVM is composed of 144 triangles.
There are 180 degrees in each triangle.
180 x 144 = 25920 = the amount of years in the Precession of the Equinoxes.
“Beyond the VE’s primary zero-phase symmetry, the 64 Tetrahedron Grid, as it is known, represents the first conceptual fractal of structural wholeness in balanced integrity. It is noteworthy that the quantity of 64 is found in numerous systems in the cosmos, including the 64 codons in our DNA, the 64 hexagrams of the I Ching (Chinese Book of Changes), the 64 tantric arts of the Kama Sutra, as well as in the Mayan Calendar’s underlying structure.”9
The Isometric Vector Matrix and the Sri Yantra
If you look at a 3D IVM from the North Pole downwards you will see the nine interlocking triangles of the Sri Yantra.
The Rhombic Dodecahedron – 2160º – Catalan Solid, Dual of the Cuboctahedron
Now we will discuss the dual of the cuboctahedron, the rhombic dodecahedron.
It has:
- 12 Faces (rhombi)
- 24 Edges
- 14 Vertices
The Sum of its Angles = 2160º, the same as the cube.
The long diagonal of each face is exactly √2 times the length of the short diagonal.
The long diagonal on this image would be from the right point to the left point. The short diagonal here would be from the bottom to top.
The acute angles of the rhombi of the faces = 70.53º.
The cross-section of a rhombic dodecahedron is a hexagon.
The Projections of the Rhombic Dodecahedron
The Spherical Rhombic Dodecahedron:
The Net of the Rhombic Dodecahedron:
The rhombic dodecahedron tessellates – it fills space without any gaps.
The Cuboctahedron & Rhombic Dodecahedron in Science
In chemistry, the cuboctahedron is associated with the noble elements.
As the structure of the atoms become more complex they periodically reach stability with the inert or noble elements. These are neon, argon, krypton, xenon, radon. They all have 8 electrons in their outer shell.
These 8 vortices match the 8 outer triangles of the extremely stable ground state Vector Equilibrium. That is why they exhibit equilibrium on their own.
The rhombic dodecahedron is found in the crystal world, in the garnet and boracite for example.
It is also found in the insect world. “Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.”11
The rhombic dodecahedron is also found in the large-scale structure of the universe according to Conrad Ranzan’s work on the Dyanamic Steady State Universe (DSSU).
Cosmic cells are negative pressure cells. Negative pressure is the manifestation of the process of Aether expansion.
Ranzan writes, “A hypothetical slice through a pair of dodecahedral cosmic bubbles reveals the main features of galaxy distribution; rich clusters, voids, walls of galaxies and right-angled walls.”
The Rhombic Dodecahedron and the Cube
“If a cube is divided by the six diametral planes which pass through pairs of opposite edges, it breaks up into six square pyramids. If these pyramids are assembled outwards on the faces of another cube, the result is a rhombic dodecahedron.”10
See below, the pyramids facing inward form the cube. Below that the faces pointing outward form the rhombic dodecahedron.
Here is an animation of this transformation:
The Stellated Rhombic Dodecahedron
The rhombic dodecahedron has 4 known stellations. The first is the most well-known. It is simply called the stellated rhombic dodecahedron.
It is created by attaching a rhombic-based pyramid to each face of the rhombic dodecahedron.
The stellated rhombic dodecahedron also tessellates.
We will now discuss several polyhedra that are related to the cuboctahedron. These are the stellated cuboctahedron, the truncated cuboctahedron and the rhombicuboctahedron.
The Stellated Cuboctahedron
The first stellation of the cuboctahedron is the compound of a cube and the octahedron. The vertices of the cuboctahedron are located at the midpoints of the edges.
Here is the second stellation of the cuboctahedron:
Truncated Cuboctahedron – 16560° & Dual Disdyakis Dodecahedron – 8640°
The Truncated Cuboctahedron (Great Rhombicuboctahedron)
The truncated cuboctahedron is also known as the Great Rhombicuboctahedron.
It is an Archimedean solid.
It has:
- 26 Faces (12 square; 8 regular hexagonal; 6 regular octagonal)
- 48 Vertices
- 72 Edges
The Sum of its Angles = 16,560º.
Note, “the name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.”12
Projections of the Truncated Cuboctahedron
The Spherical Truncated Cuboctahedron:
Net of the Truncated Cuboctahedron:
Disdyakis Dodecahedron – 8640°
The Disdyakis Dodecahedron is the dual of the truncated cuboctahedron. It is a Catalan solid.
It has:
- 48 Faces (scalene triangle)
- 72 Edges
- 26 Vertices
Its faces look as follows:
The Sum of its Angles = 8640º.
The disdyakis dodecahedron is a Kleetope of the rhombic dodecahedron. That means, if you replace each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion then you will end up with a disdyakis dodecahedron.
Projections of the Disdyakis Dodecahedron:
The Spherical Disdyakis Dodecahedron:
The Net of the Disdyakis Dodecahedron:
Rhombicuboctahedron – 7920° & Dual Deltoidal Icositetrahedron – 8640°
The Rhombicuboctahedron (Small Rhombicuboctahedron)
We will now look at the rhombicuboctahedron, an Archimedean solid. It is also called the small rhombicuboctahedron.
It has:
- 26 Faces (8 triangular; 18 square)
- 24 Vertices
- 48 Edges
Six of the 18 square faces share vertices with the triangles. The other 12 share an edge.
The Sum of its Angles = 7920º.
The rhombicuboctahedron is a rectified rhombic dodecahedron.
Recall that rectification means extreme truncation. It creates new vertices mid-edge to the rhombic dodecahedron, creating rectangular faces inside the original rhombic faces, and new square and triangle faces at the original vertices.
It is also an expanded cube or expanded octahedron. This is seen in the animation below.
It is also a cantellated cube or cantellated octahedron.
Recall that cantellation means truncating the edges. Here the edges are truncated and the corners trimmed to create the rhombicuboctahedron.
Projections of the Rhombicuboctahedron
It is interesting to note that, “the lines along which a Rubik’s cube can be turned are, projected onto a sphere, similar, topologically identical, to a rhombicuboctahedron’s edges.”13
Spherical Rhombicuboctahedron:
Net of the Rhombicuboctahedron:
Deltoidal Icositetrahedron – 8640º
Now we will discuss the dual of the rhombicuboctahedron, the Catalan solid the Deltoidal Icositetrahedron. It is also called a Trapezoidal Icositetrahedron.
It has:
- 24 Faces (kites)
- 48 Edges
- 26 Vertices
The face shape is as follows:
The ratio of the short edges to the long edges of each kite are 1: (2 – 1/√2).
“The deltoidal icositetrahedron is topologically equilvalent to a cube whose faces are divided in quadrants.”14
Projections of the Deltoidal Icositetrahedron
The Spherical Deltoidal Icositetrahedron:
Net of the Deltoidal Icositetrahedron:
The deltoidal icositetrahedron shows up in the mineral analcime and occasionally in the garnet crystal.
Deltoidal Icositetrahedron garnet crystal. Credit: Fine Mineral Photography by Laszlo Kupi
- Edmondson, Amy, A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Burkhauser Boston, 1987
- http://cosmometry.net/vector-equilibrium-&-isotropic-vector-matrix
- Edmondson, Amy, A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Burkhauser Boston, 1987
- ibid.
- Lefferts, Marshall, Vector Equilibrium & Isotropic Vector Matrix, http://cosmometry.net/vector-equilibrium-&-isotropic-vector-matrix
- ibid.
- ibid.
- http://www.cosmometry.net/home
- ibid.
- http://www.matematicasvisuales.com/english/html/geometry/rhombicdodecahedron/rhombicdodecahedron.html
- https://en.wikipedia.org/wiki/Rhombic_dodecahedron
- https://en.wikipedia.org/wiki/Truncated_cuboctahedron
- https://en.wikipedia.org/wiki/Rhombicuboctahedron
- https://en.wikipedia.org/wiki/Deltoidal_icositetrahedron
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