“360 Degrees in a Circle” – A Load of Hooey

“360 Degrees in a Circle” – A Load of Hooey

Written by Dr. Seshadri Kumar, 31 December, 2014

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There is a video circulating on facebook that is titled “Why there are 360 degrees in a circle” that owes its origin to a facebook page calling itself “Architecture and Design Magazine.”

It tries to take perfectly normal mathematics and try to make it “divine.” There is even a claim that some unspecified “vortex mathematics” is the explanation for the stuff in the video.

This is a masterful attempt at woo, with statements punctuating the video like:

“Is there  a divine code embedded in our number system?”

“Vortex-based mathematics says, yes.”

“Meaningless numerology? Or divine symmetry?”

Just High School Math and Trig

This is high school math, not “Vortex Mathematics” as the video says. There is also no “divine code embedded in our number system.”

A couple of my friends, who are engineers by training to boot, have circulated this video on facebook and seem to be terribly impressed with it. The video shows some geometric facts and then talks about “vortex mathematics” and “divinity in numbers.” The strange thing is that these engineer friends of mine do not even use their high school geometry to see that there is nothing “divine,” “cosmic,” or “supernatural” about this. It is mundane trigonometry.

Here’s the deal that explains the two “divine” phenomena in the video:

1. Someone a few thousand years ago arbitrarily decided that a circle should be divided into 360 “degrees” for convenience. This was not purely accidental - this derived from the fact that the Babylonian mathematicians used a sexagesimal system - a base 60 system, unlike our present base 10 system. They found certain calculation advantages in the base 60 system (see here for details.) Because of this, we have 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. Nowadays the mathematical/engineering unit is radians, which is nothing similar. It so turns out that
   1. 360 is a multiple of 9, and
   2. The sum of the digits of any multiple of 9 is divisible by 9
   3. This basic fact (elementary arithmetic) is the reason for the “ooooh” in the video.
2. If you bisect 360 degrees again and again, you will get numbers that are multiples of 9. There is no mystery in this. It happens simply because, for any number x, x/2 = x*5/10. So if x is a multiple of 9, its digits sum to a multiple of 9, and so x multiplied by 5 is also a multiple of 9, and so its digits add to 9 eventually. And the division by 10? That just moves the decimal point, so it doesn’t affect the sum of the digits. That’s why, when you keep bisecting a circle, you will keep getting multiples of 9 in the angles: 360, 180, 90, 45, 22.5, 11.25, 5.625, and so on. No woo stuff here.
3. As for the angles in a polygon summing to multiples of 9, again this is basic geometry. The reason for this is that the sum of the internal angles of a polygon is always 180*(n-2), where n is the number of sides of the polygon. You can prove that this HAS to be the case, because you can decompose the polygon into triangles, and the sum of internal angles of a triangle is always 180 degrees. And that’s because…you guessed it, because the number of degrees in a circle is (arbitrarily taken to be) 360. And since 180 is a multiple of 9, you find that the sum of internal angles of any polygon is (n-2)*180 = a multiple of 9. Again, no “divine” explanation is needed, merely a refresh of your high school trigonometry.
4. And, BTW, the sum of digits of multiples of 9 being a multiple of 9 is nothing magical. It is a result of the fact that we are using a digital system (a system to the base 10). So, 9x2 = 18 = 9+9 = 10+8. (8 = 10 - 2 removed for multiplying by 2). The zero doesn’t add anything, so 1+8=9. Similarly, 9x3=27 = 9+9+9 = 2*10+7. (7 = 10 - 3 removed for multiplying by 3). Again, the zero doesn’t add anything, so 2+7=9. BTW, there are repeating sequences for ALL numbers, not just 9; just that they are not so obvious. See here for those sequences.

So, all of the “mystery” in the video is because of two things: 1. The sum of the digits of any multiple of 9 is a multiple of 9, and 2. The total number of degrees in a circle was arbitrarily taken to be 360 long ago, a multiple of 9, because the Babylonians found it convenient to work in a base 60 system.

So, for 9’s sake stop forwarding this video! I don’t mind the “isn’t that cool?” aspect of it, but the video claiming that there is some “divinity” here or that there is some “vortex mathematics” is offensive to me (especially as I am a fluid dynamics expert and know a fair amount about vortices – and 9 has nothing to do with them!)

Of late, there has been a lot of stuff being circulated on the internet claiming that there are supernatural phenomena. I can safely ignore most of them, but when religion tries to hijack mathematics or science itself to propagate superstition, someone has to raise their voice to debunk it.