Select Page

# 3. Why is the Golden b ‘golden’?

The Golden b shape has the golden ratio (phi = (1 + sqrt(5))/2) embedded into its size and shape. To find out where, let’s start by examining two interlocking Golden b tiles:

One of the properties of the Golden b is that the length of each side is just the length of the next longest side multiplied by a constant factor. Let’s call this factor alpha, and assume for the moment that we don’t know what it is.

In the diagram above, we’ve given the longest side of the combined b-shape a length of 1. That means the second-longest side of the combined b (i.e. the longest edge of the purple b-tile) will have a length of 1 * alpha = alpha.

Following this same pattern, the second-longest edge of the purple tile is simply the longest edge of the purple tile multiplied by alpha, or alpha * alpha = alpha^2.

The next shortest side is alpha * alpha^2 = alpha^3.

And so on, all the way down to the shortest side of the purple tile, which has a length of alpha^6.

This shortest side of the purple tile is the same size as the second-shortest side of the smaller blue tile. Therefore, the second-shortest side of the blue tile is also alpha^6 long.

That means the shortest side of the blue b-tile is alpha^7 long, and the third-shortest side is alpha^5. Following this process, we discover that the longest side of the blue tile is alpha^2.

Now, because the longest side of the combined b has a length of 1, we know that:

alpha^4 + alpha^2 = 1

This is actually a quadratic equation we can use to find the value of alpha.

If we let x = alpha^2, we can rewrite the equation as:

x^2 + x -1 = 0

We can then solve this for x  using the quadratic formula:

x=(-b +- sqrt(b^2-4ac))/(2a), with a = 1, b = 1 and c = -1.

This formula gives us the solutions:

x= (+-sqrt(5)-1)/2

We ignore the solution that would give us a negative value for x, and stick with:

x= (sqrt(5)-1)/2

You might notice that this is very similar to the value of the golden ratio: phi = (1 + sqrt(5))/2.

By multiplying the top and bottom of our solution by (sqrt(5)+1), we get:

x=((sqrt(5)-1)(sqrt(5)+1))/(2(sqrt(5)+1))

And because (sqrt(5)-1)(sqrt(5)+1)=5-1=4, we can simplify this to:

x = 4/(2(sqrt(5)+1))

And then to:

x = 2/((sqrt(5)+1)), which is the same as 1/phi, where phi is the golden ratio!

Remembering that x = alpha^2, we can finally calculate the value for our constant alpha:

alpha = sqrt(x) = 1/sqrt(phi)

Therefore, the ratio between the length of one side of a Golden b tile and the next longest side is 1/sqrt(phi), or approximately 0.78615.

So now you know where the golden ratio is hiding in the Golden b!

Next, let’s take a look at some more of the Golden b’s interesting properties.