Equation of the Day #10: Golden Pentagrams

Ah, the pentagram, a shape associated with a variety of different ideas, some holy, some less savory. But to me, it’s a golden figure, and not just because of how I chose to render it here. The pentagram has a connection with a number known as the golden ratio, which is defined as

\begin{aligned} \phi &= \dfrac{a}{b} = \dfrac{a+b}{a} \text{ for } a>b\\[8pt] &= \dfrac{1+\sqrt{5}}{2} \approx 1.618\ldots \end{aligned}

This number is tied to the Fibonacci sequence and the Lucas numbers and seems to crop up a lot in nature (although how much it crops up is disputed). It turns out that the various line segments present in the pentagram are in golden ratio with one another.

In the image above, the ratio of red:green = green:blue = blue:black is the golden ratio. The reason for this is not immediately obvious and requires a bit of digging, but the proof is fairly straightforward and boils down to a simple statement.

First, let’s consider the pentagon at the center of the pentagram. What is the angle at each corner of a pentagon? There’s a clever way to deduce this. It’s not quite clear what the interior angle is (that is, the angle on the inside of the shape at an individual corner), but it’s quite easy to get the exterior angle.

The exterior angle of the pentagon (which is the angle of the base of the triangles that form the points of the pentagram) is equal to 1/5 of a complete revolution around the circle, or 72°. For the moment, let’s call this angle 2θ. To get the angle that forms the points of the pentagram, we need to invoke the fact that the sum of all angles in a triangle must equal 180°. Thus, the angle at the top is 180° – 72° – 72° = 36°. This angle I will call θ. While I’m at it, I’m going to label the sides of the triangle x and s (the blue and black line segments from earlier, respectively).

We’re nearly there! We just have one more angle to determine, and that’s the first angle I mentioned – the interior angle of the pentagon. Well, we know that the interior angle added to the exterior angle must be 180°, since the angles both lie on a straight line, so the interior angle is 180° − 72° = 108° = 3θ. Combining the pentagon and the triangle, we obtain the following picture.

Now you can probably tell why I labeled the angles the way I did; they are all multiples of 36°. What we want to show is that the ratio x/s is the golden ratio. Looking at the obtuse triangle inside the pentagon, the ratio of x/s is equal to the long side of the triangle divided by one of the short sides. This smaller triangle (with sides ss, and x) is similar to the larger obtuse triangle extending to the point at the upper right (with sides x, x, and x + s), which we can tell by seeing that the three angles in each triangle are equal. That means the ratio of sides in the larger obtuse triangle will equal the ratio of sides on the smaller obtuse triangle. Taking these ratios and setting them equal gives us

\dfrac{x}{s} = \dfrac{x+s}{x}.

But this is just the same equation we used at the top to define the golden ratio with the replacements of a with x and b with s. Therefore, x/s is indeed the golden ratio! Huzzah!

The reason the pentagram and pentagon are so closely tied to the golden ratio has to do with the fact that the angles they contain are multiples of the same angle, 36°, or one-tenth of a full rotation of the circle. Additionally, since the regular dodecahedron (d12) and regular icosahedron (d20) contain pentagons, the golden ratio is abound in them as well.

As a fun bonus fact, the two isosceles triangles are known as the golden triangle (all acute angles) and the golden gnomon (obtuse triangle), and are the two unique isosceles triangles whose sides are in golden ratio with one another.

So the next time you see the star on a Christmas tree, the rank of a military officer, or the geocentric orbit of Venus, think of the number that lurks within those five-pointed shapes.


Edit (16-Feb-2021): I’ve updated the logic at the end to be purely geometric. The original algebraic argument is pasted below.

By invoking the law of sines on the two isosceles triangles in the image above, we can show that

\dfrac{x}{s} = \dfrac{\sin 2\theta}{\sin\theta} = \dfrac{\sin 3\theta}{\sin\theta}

This equation just simplifies to sin 2θ = sin 3θ. With some useful trigonometric identities, we get a quadratic equation which we can solve for cos θ.

4\cos^2\theta - 2\cos\theta - 1 =0

Solving this equation with the quadratic formula yields

2\cos\theta = \dfrac{\sin 2\theta}{\sin\theta} = \phi,

which, when taken together with the equation for x/s, shows that x/s is indeed the golden ratio! Huzzah!