E Pluribus Unum

I have one last parting gift to share. But for you to receive it, you’ll have to work with me a bit.

I’m going to explain a formula—the only one in this book. Fortunately, it’s a simple and elegant. It’s called Euler’s polyhedron formula.

Leonhard Euler, a 19th-century mathematical genius, expressed it as:

V − E + F = 2

It applies to all convex polyhedra. Take a cube as seen in Figure 1. 

• V (vertices) = 8
  

• E (edges) = 12
  

• F (faces) = 6
  
  

Plug those into the formula:
8 − 12 + 6 = 2

This remarkable relationship holds for all convex polyhedra. And yet—this is where Euler and I part ways.

What I want to share with you is a different way of thinking about this formula. In my view, the real value should not be 2 but 1.

Let me explain.

Using structural geometry that I introduced earlier, I’ve reconceived Euler’s formula not as a single equation, but as part of a series of formulas—one for each dimension—each returning the same invariant: 1.

Let’s build a table:

This isn’t just a mathematical curiosity—it has profound implications. Instead of seeing V − E + F = 2 as a formula for polyhedra, I see it as a formula for polygons. Why? Because without S (the number of solids), we’re describing only a polyhedral shell—not a true polyhedron. Without the solid interior, the structure is incomplete.

It’s like saying the emperor has no body, only clothes. In other words, without S, we describe only the outer shell—the surfaces—while S together with the shell gives the true definition of a polyhedron. This resolves a long standing debate as to the definition of a polyhedron; whether it is a solid or a shell and the answer is that it is both. 

Adding S gives the structure substance. And once you see this, you can extend the logic to higher dimensions—4D, 5D, and beyond—where this series continues, and still, each formula gives 1.

These formulas can be applied to any scale as well. For example, lines can be extended to infinity by adding segments to segments. The same applies multiple polygons and polyhedra. For example, it applies to sequences of polygons or planar sheets of polygons as well (Table 2).

As Table 2 shows, this invariant persists across increasingly complex structures.

At first, I didn’t know what this “1” meant. Then a friend suggested: maybe it represents the number of structures present. A single, unified entity.

I have tested these formulas on every geometric structure I could think of, including simple, complex, porous, and entangled structures, and they always produce the same result of 1.

This idea is not limited to geometry. Molecular structures follow the same logic (Table 3).

We have only to convert molecular structure into its geometric analogues to obtain the universal invariant of 1.

This raises an interesting question.

If we could scale molecular structures even to a cosmic scale, would that mean that the universe itself is a structure, would it too have an invariant value of 1?

And if so, might that suggest that beneath the complexity, the chaos, the fragmentation of the world, there is an underlying simplicity, order, and unity?

Does that in turn, suggest that this is a monoverse in which we all participate as both creators and created?

If the answer to these questions is yes, then in the language of myth, we have the following: our round table that I first saw in a ring of six hexagons as a structure with the Grail at its center. Together, they are the fount and seed of creation, and the number 1 is symbolized by Excalibur, and both become symbols of unity.

Thus, with this vision of the monoverse we have a vision for universal structuralism. At Camelot College then we would see the birth of structural geometry, structural chemistry working together along with structural cosmology.

For further information: Albert P. Carpenter