The Cuboctahedron

Sacred Geometry is defined as drawing without measurement. It involves the tools of drawing compass and ruler, but unlike traditional geometry, all shapes and dimensions are interrelated and arise from a pattern of overlapping circles. The ‘sacredness’ of geometry comes from the intuitive side, containing meaning – the basis of sacred geometry. Here are a few key-points:
It is related to spiritual symbolism and architecture.
It explains that all of nature’s patterns can be related back to shape.
It suggests that the Universe is designed by a blueprint that expresses itself on all scales, from the atom to the galaxy.
In this Ultimate Guide to Sacred geometry, you will find an in-depth overview of all there is to know about the most intriguing patterns of the history of humankind!

Before starting to understand cuboctahedron, let’s first revise some basics.

Polygon

As you might remember, a polygon is a two-dimensional (2D) shape that is entirely closed and all its sides are straight. It is worth noting that it is made up of lines and not curves (which means a circle would not be a polygon). Some examples of polygons are triangle, hexagon, pentagon, etc. At least 3-line segments have to meet to make a polygon, as two straight lines cannot make a closed shape. You can observe the same in figure 1.1. as well.

Types of Polygon

Although there are several types of polygon, according to the context, the only two we would need are:

  1. Regular Polygon and
  2. Irregular Polygon

A regular polygon is a polygon with all its sides and angles equal. The best example of a regular polygon is a square.

Whereas, an irregular polygon does not have equal sides and angles. A good example could be that of a rectangle, in which the length and breadth measure differently.

Now let’s move on to understand polyhedron.

Polyhedron

A polyhedron is similar to a polygon. The only striking difference here is that it is a three-dimensional shape (3D) while polygon in 2D. The concept of straight lines applies here as well. Hence, a cylinder or cone cannot categorize as a polyhedron. There are two types of the polyhedron as well, regular and irregular. A regular polyhedron has equal faces (not sides, as it is a 3D shape) while the faces of an irregular polyhedron are not equal. Mainly there are five polyhedrons are tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

In figure 1.2., A good example of a regular polyhedron would be a cube, which has 6 faces and all are square.

There are also polyhedrons whose faces are made of more than 1 type of polygon. Some of these are categorized under the Archimedean solids. In case you have never heard this term, look at traditional football. You will notice on its faces that it is made of hexagons and pentagons. This is called a truncated icosahedron. The cuboctahedron is also one of the Archimedean solids, made of more than one polygon.

Cuboctahedron

A cuboctahedron is a polyhedron with the faces of 8 triangles and 6 squares. It is also categorized as a quasiregular polyhedron.

General

In total, a cuboctahedron has 14 faces, out of which 8 are equilateral triangles and the remaining 6 are squares. It has 12 vertices and 24 edges. Figure 1.3. is a graphic representation of cuboctahedron.

Area

The area of the edge length of cuboctahedron is calculated by the following formula:

Area = (6 + 23) a² is almost equal to 9.4641016a²

While the surface area is calculated by:

2a² (3 + 3)

Volume

To calculate the volume of a cuboctahedron, use the following formula:

V = 5a³32 

Division

Being a quasiregular polyhedron and belonging to Archimedean solids, it can be divided equally into two parts. Due to the faces of squares and triangles, the divided section is of the shape of a hexagon, it can also be observed the figure 1.3., notice how the different sides of the cuboctahedron, make a hexagon around the middle. Even if you see a 2D prospective cuboctahedron, it would resemble a hexagon. Another reason is that all Archimedean solids are modes of regular but different polygons.

Faces, Vertex, and Edge

Following are the visual representations of cuboctahedron’s faces, vertex, and edge. It is worth noting that from the square face, it appears like a square in a two-dimensional figure while in its triangular face, it appears like a hexagon.

Square Face

Triangle Face

Vertex

Edge

Coordinates

Following are the cartesian coordinates on a graph based on the vertices of a usual cuboctahedron.

(±1,±1,0)

(±1,0,±1)

(0,±1,±1)

Symmetry

In a cuboctahedron, the distance between center to vertex (long radius) is equal to the edge length. This makes it one of the most unique polyhedrons. Its basic symmetry is that of an octahedral, about which you will learn more in the coming section.

A cuboctahedron is obtained between a truncated octahedron and a truncated cube. Let’s look at this entire process closely.

General

Cuboctahedron, also known as a rectified cube and a rectified octahedron, is a result of truncation, as are many non-regular polyhedrons.

Truncation is the process of cutting off or shortening the edges of something. Before we begin understanding the place of cuboctahedron in the truncation between octahedron to a cube, it is important to know a bit about the other solids as well. You can view the truncation process through a graphic here as well.

From Octahedron to the Cube

Octahedron – Truncated Octahedron – Cuboctahedron – Truncated Cube – Cube

In a nutshell, these are the five stages involved in truncation from an octahedron to a proper cube.

  1. Octahedron: As by the name one can understand, an octahedron is a polyhedron with 8 faces. This hexagon-like prism has all equilateral triangles and hence is a regular polyhedron. It has exactly 12 edges, 6 vertices and is one of the platonic solids. The journey of both cuboctahedron and a cube starts from here. In figure 2 the rightmost figure is an octahedron.

  1. Truncated Octahedron: Another of the Archimedean solids, a truncated octahedron is formed after truncation (as explained earlier) of the edges of an octahedron. It can be seen in the bottom-most figure in figure 2, right below the cuboctahedron. It has a total of 14 faces consisting of square and hexagonal shapes. It has 24 vertices and 36 edges. It is also known as a zonohedron.

  1. Cuboctahedron: With further truncation, we get a cuboctahedron, which is the centremost shape in figure 2. As mentioned earlier, it is a quasiregular polyhedron with 8 triangular faces and 6 square ones. It has 12 vertices and 24 edges.

  1. Truncated Cube: Also known as a truncated hexahedron, a truncated cube is obtained after further truncation of a cuboctahedron. It is in this stage that the shape changes closely to that of a cube. It can be found as the topmost shape in the figure. It has 14 faces, 36 edges, and 24 vertices.

  1. Cube: Finally, we get the popular cube, having 6 faces, 12 edges, and 8 vertices. It is in the shape of a perfect square (in 3D).

Let’s now move on to one of the most important topics that one needs to understand about a cuboctahedron.

Close packing of several spheres is one of the building blocks of theoretical geometry. Particularly, in hexagonal packing systems, different dimensions give different results.

One Dimensional Approach

In one dimensional approach, the spheres are laid horizontally in a line, where they are closely attached.

Two-Dimensional Approach

As you can see in figure 3.1., in the two-dimensional approach, the two components are width and length. Due to this, the second layer of spheres is arranged right above the first line of spheres. It is to be noted that these are aligned in such a manner that the spheres of the second line tend to fill the void between two spheres of the first line, this is why although not perfectly aligned, the spheres are right above each other. Then subsequent layers are added one above the other in such a manner that each alternate layer is identical. This classification is also known as ABAB layering as each group of spheres above fill the void of the one below (fitting in the depressions). In such an arrangement, each sphere has at least 6 spheres around it. With these, a hexagon can be formed.

Three-Dimensional Approach

According to the previous figure, first, imagine that the image in the two-dimensional approach is laid down horizontally, this is because, in a three-dimensional figure, there would also be depth. The alignment of spheres is similar alternatively, as could be observed from the two-dimensional approach. Here the coordination number is 12, which means the number of particles nearby concerning one sphere. This is what looks like in figure 3.2. The hexagonal packing system is thus useful in describing shapes like polyhedrons and specifically, cuboctahedron.

3.1. FCC and HCP

There are two arrangements in the close packing system that give the highest density. One is FCC (face-centered cubic) and HCP (hexagonal close-packed). The former can be observed in squares and triangles, as in a cuboctahedron. The latter is found more in orientation with triangles.

The configuration of hexagonal close-packed (HCC) is AB AB AB, while that of face-centered cubic (FCC) is ABC ABC ABC. Hence, in FCC, every third layer is identical while in the case of HCC, every second one identical.

3.2. Hexagonal Packing in Cuboctahedron

As mentioned above, FCC (face-centered cubic) is an identical model to cuboctahedron. The requirement for such a hexagonal packing system in cuboctahedron is to place a sphere at the center, its diameter should be equal to the edge length of the cuboctahedron and surround it with 12 spheres of similar size. This was one of the reasons behind Fuller’s fascination with cuboctahedron. The similar size of radius and edge length, who studied it as vector equilibrium and formed jitterbug. You would learn more about it in the next section.

Richard Buckminster Fuller, the popular American architect is known for developing the geodesic dome and also his revolutionary design ideas. He is also known for making the Jitterbug atom. ‘Bucky’ was born in 1895 in New England. Although he was dismissed from the prestigious Harvard University a couple of times, it did not stop him from pursuing his passion for engineering while being in the U.S. Navy. Throughout his lifetime, he designed many things, out of which 28 were patented. Apart from the Jitterbug design, he is also famously known for his Geodesic Tensegrity Sphere.

He went on to design Dymaxion as well, a revolutionary car of the future with a streamlined body. But among the main highlights of his inspiring works, we would be focusing on the jitterbug design.

This design idea paved the way to an entire wave of transformation in the field of architecture and design. Although earlier, the joints were unstable, later special hinge joints were created by Dennis Dreher, his co-worker. This led to easier movement and a smooth look. The image below depicts how the jitterbug opens and when seen in reverse, closes.

Since its inception, the jitterbug design has been used in several tools and models as well. There was also an attempt to make the largest jitterbug design, especially in the science expo held in Zurich, Switzerland in 1991.

From the outside, while closed, the jitterbug looks like an octahedron. And as it is opened completely, it takes the shape of a cuboctahedron.

4.2. How Does it Work?

As the name might suggest, the ‘jitterbug’ atom is based on the swing dancing style by the same name. It is based on the idea of close packing of spheres, as we have discussed earlier. Fuller’s motivation was to see what would happen if the center sphere is dislocated, which means, the spheres are arranged in a manner that there is no sphere in the middle. This is what led him to create the jitterbug atom, which closes and opens in a swing-like motion.

This type of movement from a closed object has been a topic of research for decades, with some designs and conceptual models aiming for motion from the vertex, while some from edges. This is also known as displacive transition. Until the jitterbug design was discovered, the transition was limited to regular polygon shapes and had not been tried in polyhedrons.

As you can observe in the figure above, with movement, the hinges are connected and while the object is closed, it is all tightly bound together. This is where we see that Fuller studied keenly the closest packing of the spheres around a nucleus. He even credits it as one of the most important foundations of his discoveries. It is crucial to note that the jitterbug atom also forms another polyhedron if not expanded completely. If you look closely at the motion in figure 4.1., you will notice an icosahedron.

So, it can be observed that the jitterbug atom’s motion study was based on:

• Transformation through vertex to vertex
• Transformation through vertex to edge to vertex
• Transformation through vertex to the vertex to vertex

Nowadays, the jitterbug atom has been recreated in several objects, with few replacing the traditional hinges with gears and folds. But, the concept and idea of such a displacive transformation have given way to many design artists and mathematical theorists. Its model is also up for sale by a US art and design auction website that provides it in a furnished shiny aluminium model of 18 x 20 x 15 cm.

Let’s now move on to the final topic related to cuboctahedron.

Although the egg of life as a concept has a near-universal application. It is related to biology, music, architecture, natural structure, and several other things. We shall look forward to understanding the concept first

The egg of life concept arises from the symbol by the same name. Notice in the figure above, you would see on the right side the diagram for the egg of life. This is made of 8 spheres, with one in the center. For a two-dimensional figure, it would seem to have 7 circles with one in the middle. Also, notice the figure left to it. It is the flower of life. The egg of life is closely related to the concept of the flower of life as well as it sets the base for it. The flower of life, located at the left side in the given figure, has 19 circles overlapping each other, thus forming a symmetrical flower. They both symbolize an essential process of creation and unity among all things.

In the Figure above you will see different cells in a stage of mitosis. They also create a symbol like that of the egg of life. While mitosis is occurring, and the cells divide until 8, this embryonic division represents the egg of life. It is one of the main reasons why it is called so. Not just this, the egg of life symbol is a part of several religions as well. If you know a little bit about music theory, you should also know that the distance between the spheres in the egg of life represents tone and semitone’s distance.

The Egg of Life Concept in Cuboctahedron

Now, look at figure 5.3., you can see there that the egg of life concept can also be applied in cube and cuboctahedron, with a little different sphere placement. It is the basis for such polyhedrons. Also, tetrahedron and star tetrahedron can be formed from it, by alignment. As we read earlier, also known as vector equilibrium, a cuboctahedron can contain the exact figure of the egg of life. In a three-dimensional shape, if the spheres are arranged as the vertices, with of course having a central vertex, the shape that comes out looks exactly like the egg of life. It can also be seen in figure 5.3.

Hence, you learned about cuboctahedron, its transformation from octahedron to a cube, relation to hexagonal packing and the close sphere arrangement, jitterbug atom design, its history, and the egg of life concept. The scope for research about it under these heads is quite large. The importance of this polyhedron as an Archimedean solid and as a design is also profound, one that accentuates its unique properties as a quasiregular polyhedron.