What is a cuboctohedron?!

You’re at a festival, wearing this amazing shirt, and someone asks you.. “what’s that on your shirt?”. What do you say? …  This is a cuboctohedron! As well as being a pretty stunning design, this amazing shape packs some pretty interesting maths!

Geometry has been used to explain not only what we can see and touch (cubes of rock, a triangle drawn in the dirt), but what we can only imagine (complex three dimensional shapes).

While simple shapes such as triangles, squares and circles have been part of mankind’s way of thinking about the world forever, it was the consideration (and naming) of more complex three dimensional shapes which was seen to inspire and develop Western mathematics, science and philosophy from the early Middle Ages (5th century AD).

Polyhedra is the name given to three-dimensional solid shapes based on triangles, squares and circles. These were considered the building blocks of the entire universe, including organic and inorganic matter. So, what is a polyhedra? It is a set of solid shapes which incorporate a number of plane surfaces (or faces) and intersections (vertices). In ‘regular’ polyhedra forms, all of the faces were the same solid shape. There are five of these type of regular polyhedrons, named after Plato (‘platonic solids’).


The five platonic solids are: the tetrahedron 
(consisting of four faces that are equilateral triangles), the hexahedron, also known as a cube (consisting of six square faces), the octahedron (consisting of eight faces that are equilateral triangles), the dodecahedron (12 pentagons), and the icosahedron (20 equilateral triangles).From these five platonic solids, Archimedes theorised the existence of 13 ‘irregular’ polyhedrons (known as ‘Archimedean solids’) in which intersecting faces do not have to be the same.
The cuboctohedron is one of these Archimedean solids and it contains a triangle and a square at each ‘corner’ with 8 triangular faces and 6 square faces. It is effectively the mating of a cube (or hexahedron) and the octahedron.  

The cuboctohedron is considered part of the ‘sacred geometry’ set of shapes which purport to show us the ‘secrets of the creation of the universe’ as it is the only Achimedean solid in which the length of the vertices (or the edges between the faces) is equal to that of the radial distance from its centre of gravity to any vertex. That is, the line from the centre of the structure is the same length no matter which part of the structure it is going to, and all lengths between points are the same as well. Not only that, it has exactly 13 vertices: a central one, plus the outer ones that define 12 directions in space.

In mathematics, 13 is considered pretty special, being a prime number; the smallest prime number which is a prime no matter which way you place the two digits (ie. 13 or 31), and a fibonacci number (see related post on the fibonacci spiral).

There is another way we can represent a cuboctohedon, which is the draw 13 spheres of the same diameter, clustered around a central sphere (or ‘nucleus’) so that each is touching another and the centre of each sphere will be equidistant from all of their adjacent spheres, including the nucleus. If we draw the relationship between each of these sphere centres, we get a cuboctohedron. This diagram (with the spheres) is seen in nature in terms of how cells multiply, which is why the cuboctohedron, as drawn with 13 spheres, is also linked to the seed of life, another typical design you will often see in sacred geometry (see related post).


If we represent that in two dimensions, rather than three, where each of the 13 spheres becomes a circle, this pattern is called Metatron’s cube (see related post about the archangel metatron, the kabbalah and the tree of life as well as the maths behind why metatron’s cube is considered to represent every geometical shape known to mankind). 

So, as well as being trippy to look at, the mathematics and geometry behind the cuboctohedron can be seen to be representative of both sacred (ie. the number 13) and profane (ie. the way cells replicate) elements of the universe. That makes it a pretty spesh!

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