Recently I found this article on crop circles:

It seems that every once in a while, some crop circle appears in a field, then some UFO conspiracy theorist with a wild imagination studies it, and jumps to wild conclusions.

One of the recurring themes seems to be complicated mathematical properties showing up. The underlying argument of course is that this "proves" the circles were made by some masterfully intelligent being, which of course implies that we're being visited by some technologically advanced alien species.

There are several problems with this:

- Humans have been able to make crop circles themselves, without the help of aerial devices. All you really need is a some rope and plywood to flatten out the crops. And if you want to make circle shapes, tie a rope to steak and sweep the radius. The two guys who created the whole hoax came out and admitted it in 1991. If pranksters can make them, then there's really no need to bring in notions of spaceships and Martians to explain them.
- Even the more complicated-looking crop circles can be done by repeating the same geometric patterns or shapes over and over, in a circle. This is how toys such as Skedoodle work to draw designs. (If you don't remember this toy from the 80s, click on the link and see the video clip.) So you don't need to be master artist or mathematical genius to make such shapes.
- Similarly, just because you can derive mathematical numbers from a shape doesn't mean that the shape was designed with that in mind. A child can trace the base of a coffee mug on a piece of paper to draw a circle; that doesn't mean she knows anything about pi.
- This all begs the question of why such advanced beings would be leaving us messages this way. Why the ambiguity? Why only in the middle of the night? Why in the middle of fields of crops, where the population is so sparse? The old claim used to be that the crop circles were marks where spaceships landed. Obviously, the conspiracy theorists have changed that idea.

Lucy Pringle, the "renowned crop circle researcher" quoted in the article, painfully reveals her ignorance when she claims "Historically over the years, crop circles have been associated with diatonic scales (white notes on the piano)".

- Again, "connections" to things like musical intervals are really just speculative, and not all that hard to "find". Crop circles are overlapping geometric shapes, which means they have some whole-number ratios that you can pick out and compare (e.g. "the lengths of this line and this other line are in a 3:2 ratio"). Years ago, we used to construct all musical notes out of whole-number ratios too. For example, if hear a note when you pluck a string it makes a note, and plucking a string that's half the length gives you a note that's one octave up. But there are lots of other things that use whole-number ratios: baking recipes, mixed alcohol drinks, the approximates we use for statistics we report, etc. Just because you can measure and compare lengths or areas in different parts of a crop circle, doesn't mean there are aliens encoding musical sequences.
- Another problem with the musical ratio argument is that the numbers aren't so special. If you pick ANY two single-digit numbers and take the ratio of one to the other, there's an over 75% chance that you'll get a "diatonic ratio" (1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1) when you reduce it and adjust for octaves. Yes, I did the math to get the "75%".
- The biggest problem with the musical crop circle claim is that our modern music system simply DOES NOT USE these ratios. For the past several centuries, all the music of the western world (classical, hip-hop, metal, jazz, you name it) has used what's called equal temperament. This is where the ratio between consecutive notes is always the 12th root of 2 (approximately 1.0594).

Now let's actually get into the math (or "maths" as it's called on the other side of the Atlantic), and the claims about the "Complex Maths Circle":

Their use of "complex" here is a bit misleading. In mathematics, "complex" is a specific term used to refer to a particular type of number. It doesn't necessarily mean complicated or intricate, which is what "complex" means in general English usage.

It took me a while to understand how they were deriving the formula from the crop circle picture. Part of the reason for this, is because the photo is incorrectly labeled. The outside characters (e, ^, (, h" etc.) should appear at the end of each radial line, not in between. This is because each of the 12 lines represent a character. The picture on the other hand makes it look like the crop circle is broken up into 12 pie wedges, and that each wedge is a letter.

There are additional mistakes in the photo. The "Ao" and "Th.c" marks on the axes are gibberish, or more likely (ironically enough) bad ASCII character renditions. And the formula is supposed to be "e^(i pi)+1=0". Actually, it's supposed to be written with the "pi" as the Greek letter pi (π) and both the i and the pi superscripted. But this is the way you'd write out Euler's formula if you evaluate it with theta=pi, and could only type it out instead of normally being able to write it with a pencil. It's not e^(i)pi+1=0, as the article incorrectly put it. That gives you an incorrect formula.

Regardless, I get it now. Here's how you supposedly derive the formula from the photo:

- Start from the center of the circle, and pick a line.
- As you move across this line from the inside to the outside, you'll see perpendicular bars on either side. Each bar falls on either the left side or the right side. If it falls on the left, write down "0". If it falls on the right, write "1".
- When you're done with the line, you'll have a total of eight numbers. This gives you a binary number.
- Translate the binary number into a decimal number. This corresponds to an ASCII code for some keystroke.
- Starting with the top number in the photo and moving clockwise, you get the following twelve characters:
**e ^ ( h i ) p i ) 1 = 0**

This certainly resembles "e^(i pi)+1=0", but why would somebody who came up with such an elaborate system, get 25% of the characters wrong?

Hey, I love both music and mathematics. They're my two big passions in life. So I'd love for stories like this to have some validity. But the math has to add up. And it doesn't.