With Halloween upon us I thought it would be fun to tell you about a fact noticed by Steve Wang more than 20 years ago and published in the Journal of Recreational Mathematics (pdf of the paper here). While most of us think of Halloween as a harmless night of dressing up and turning up our noses at candy corn, some associate it with Satanism and witchcraft. The Book of Revelation tells us that the Number of the Beast, the Sign of the Devil, is 666. If you have a calculator handy enter 666 and hit the sine key (make sure you're in degree mode!). My smartphone tells me the answer is -0.809016994374948..., a rather random-looking number and certainly not one that's immediately recognizable.
Except that it is, once you know what to look for, but to get there we have to take a diversion. Supposedly, the ancient Greeks, being master aestheticians, liked their geometric figures to be pleasantly proportioned (there's no real evidence for this, but it's one of those things we like to repeat anyway). In the case of rectangles, they preferred the side lengths to be in proportion as the golden ratio.
Here the green rectangle on the right has the correct proportions: a/b = φ, the golden ratio. This number is defined by the property that if you place the blue square of side length a adjacent to the green rectangle, the resulting rectangle has the same proportions; that is, (a+b)/a = a/b. This equation allows us to find the value of φ exactly. Take b=1 and then simplify to get the equation a² = a + 1. Via the quadratic formula, we then get two solutions, one positive and one negative. The positive solution is what we call φ:
So what? Divide this by 2 and you get 0.80901699437494..., which agrees with our calculation of the sine of 666°, up to a sign. That's not a proof of course, since we only have 14 decimal places of accuracy, but this would be a pretty big coincidence if there were not an exact equality between the sine of 666° and -φ/2.
How can we prove it? First, let's reduce the angle to something smaller. The sine function is periodic with period 360° (the calculus teacher in me is struggling with using degrees instead of radians). That means that we may as well use the angle 306°, but even better we can use -54° (666-720) and the fact that sin(-α) = -sin(α) for any angle α. So we've reduced the question to showing that sin(54°) = φ/2.
The trick now is to consider the isosceles triangle ABC shown below, whose angles are 36°, 72° and 72°.
Bisecting the top angle yields a triangle BCD, which is similar to ABC. If we denote the length of the segment CD by a, and the segment BD by b, then we may fill in the lengths of the remaining sides as indicated. Since similar triangles are proportional we have (AC/BC) = (BC/BD), or (a+b)/a = a/b. In other words, a and b are in proportion as the golden ratio and we call ABC a golden triangle.
Now drop a perpendicular from point D to segment AC; this bisects the angle at D since triangle ADC is isosceles. We then have the triangle below, with the indicated angle measurements.
We can now unravel what we need: sin(54°) = AE/AD = 2AE/2AD = (a+b)/2a = (1/2)(a/b) = φ/2. Pretty cool.
The golden ratio shows up in a number of contexts, most notably in connection with the Fibonacci sequence, but this Satanic link is certainly the most entertaining. Perhaps St. Augustine was correct when he warned us, "The good Christian should beware of mathematicians. The danger already exists that mathematicians have made a covenant with the Devil to darken the spirit and confine man in the bonds of Hell." He really meant numerologists and astrologers, but, whatever; it makes for a good quote. Happy Halloween!
