top of page

Dualing Platonic Solids

Updated: Jul 6, 2023

There is a movement within Metatron’s cube that most are unaware of and I have never seen it mentioned anywhere else. In the last blog, we focused our attention on the most obvious movement that Metatron’s cube makes and that is rotation. Each and every one of the Platonic solids is capable of rotation while nested within Metatron’s cube. There is another type of movement that is equally important. The Platonic solids are capable of expansion and contraction.

Each Platonic solid has a special relationship with one of the other Platonic solids. This is where projective geometry comes in. If you are unfamiliar with projective geometry, you are not alone. I was unaware of it until it was discussed in one of our sessions. Most of us learned Euclidean geometry in high school. This is perhaps the oldest form of geometry dealing with planes and solids. Projective geometry deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface.

Projective geometry deals with perspective. It gives rise to two point and three point perspective in works of art. It is currently being used to help teach computers how to “see”. There are two concepts within projective geometry that are of particular interest for us and they are duality and polarity. For our purposes in describing the movement of the platonic solids, I am just going to focus on duality. Simply put, the dual of something is its opposite.

When to comes to the Platonic solids, there are three sets of duals. The hexahedron or cube is the dual of the octahedron. The dodecahedron is the dual of the icosahedron. And interesting enough, the tetrahedron is a dual to its inverted self. How is a hexahedron the opposite of an octahedron? From a projective geometry perspective, we are concerned with the faces of the shapes and the corners or vertexes of the shape. The face of a hexahedron is a square. Each face can also be called a plane. Each plane has a middle point to it. The face of an octahedron is a triangle. When you place an octahedron inside of a cube, each vertex of the octahedron touches the cube at the midpoint of each plane. See the image on the left below.

When you place the cube inside the octahedron, you get the same result. Each vertex of the cube touches the midpoint of each plane of the octahedron. Because of this unique relationship with the other, they are referred to as duals of each other. The same is true for the dodecahedron and the icosahedron.

The figure on the left shows the dodecahedron inside the

icosahedron. You will notice how each vertex of the inner dodecahedron touches the plane of the icosahedron right in its midpoint. Likewise, the right figure shows the icosahedron inside the dodecahedron. Each vertex of the icosahedron touches the midpoint of the dodecahedron in its midpoint.

This leaves us with the tetrahedron. As I mentioned earlier, the tetrahedron is unique in that it is dual to its inverted self. You can see in the picture below that each vertex of the inverted tetrahedron touches the midpoint of the plane of the larger tetrahedron.

Those are the dual relationships of the Platonic solids. Now for the movement within Metatron’s cube. They actually expand and contract within each other. The tetrahedron is a little bit different as I mentioned above. Its dual is an inverted tetrahedron. Click on the video below to see an animation of what the tetrahedron expansion and contraction looks like. You may notice a very familiar shape. If you pause the video in just the right place you will see a star tetrahedron, a merkaba.

I realize that it may be extremely difficult to visualize this expansion and contraction motion within a rotating Metatron’s cube. Don’t worry. I am only including this ebb and flow motion in order to more accurately describe the complex motion within Metatron’s cube. I have not seen this type of motion described anywhere else in reference to Metatron's cube.

There will be more on how to use Metatron’s cube to achieve personal goals in upcoming blogs.

Recent Posts

See All


bottom of page