First, the Division in Extreme and Mean Ration (**DEMR**) is a proportional relationship discovered by **Euclid** * and documented in his book **Elements **around 300 BC. Simply put, the ratio follows: * ac:ab = ab:bc*.

*Diagram 1*

In other words the whole relates back to the parts. If ab = 1 then bc = .618… (again an irrational number) totaling 1.618. Further, 1 ÷ **1.618**… = .618… also irrational. **ϕ** is the math symbol for the number 1.618… in the same way π is the symbol for 3.14… So two (**1.168** & **ϕ**) of the five options are related to the DEMR.

What if we add .618… to 1.618… (**ϕ**)? 1.618… + .618… = 2.236… which, if you have a calculator, is the √5. That’s 3 for five.

What about that star? Or, more appropriately, **Pentagram**. *Diagram 2*

Well, that’s **4** out of **5**. So what exactly is that first item to the left? It’s an **Icosahedron**. An Icosahedron is one of five **Platonic Solids** which are defined as polyhedron with all sides being regular polygons of the same size. The Icosahedron is composed of 20 **equilateral triangles**.

Time for a quick detour.

Let’s go back to our friend the square, divide it in half, draw a line from the bottom center to the upper right and then swing that line down to the base. *Diagram 3* A square has the proportion of 1:1. The length added to the square from the compass swing is .618… As we know 1 + .618… = 1.618… so we have constructed a **ϕ** rectangle with the proportions 1:1.618…

Turns out that three **ϕ** rectangles fit perfectly within the Icosahedron. One each aligns with the x, y and z axes. *Diagram 4*

So all **5** of the above symbols and numbers relate to the Division in the Extreme and Mean Ratio. And, there’s a lot more to come…

** It’s likely that this concept was understood well before Euclid and that he is just the first to document it. We”ll look at this in more detail later.*