So, I have seen this video circulating around of recent. Go ahead, watch it. It's some high-school level math that apparently reveals some profound secret of the universe. And it is complete crap. The math is accurate enough, but the conclusions are demonstrably bogus. And I intend to lay it to rest below.
The basic premise of the video is as thus: That the reason there are 360° in a circle is not arbitrary, and that the number 9 simultaneously stands for everything and nothing, thus having some "divine symmetry" of some sort. For those who don't want to watch the video, it makes the following claims using the following proofs:
• A full circle is 360°. 3+6+0=9. A circle divided in half is 180°. 1+8+0=9. If you continue these divisions, the pattern holds: 1/4 of a circle is 90°, and 9+0=9. 1/8 of a circle is 45°, and 4+5=9. 1/16 of a circle is 22.5°, and 2+2+5=9. Etcetera. No matter how many times you bisect the circle, the digits always add to 9.
• A similar pattern is seen in regular polygons. An equilateral triangle has 3 angles of 60° each. 60×3=180. 1+8+0=9. A square has 4 angles of 90°. 4×90=360. 3+6+0=9. A regular pentagon has 5 angles of 108°. 5×108=540. 5+4+0=9. A regular hexagon has 6 angles of 120°. 6×120=720. 7+2+0=9. And so forth.
• The sum of all single digits, excluding 9, is 36. 0+1+2+3+4+5+6+7+8=36. 3+6=9.
• Adding any single digit to 9 will produce sum that, when its digits are summed, is the original number you added to 9. 1+9=10, and 1+0=1. 3+9=12, and 1+2=3. 5+9=14, and 1+4=5. 9+9=18, and 1+8=9.
Therefore, 9 is special and has some mystical, divine significance. It simultaneously represents everything and nothing. Yadda yadda.
For the record, I'm not entirely a skeptic when it comes to the whole "sacred geometry" thing. Except for the "sacred" part. There are a lot of genuinely profound and interesting mathematical truths out there, in which are contained the secrets of how the universe works. Like the golden ratio. Study that thing some time.
But there is also a lot of crap out there. To say there is a fine line out there between numerology and genuine mathematics is an understatement – the line is big, bold, and well defined. And that line is when we assign arbitrary significance to something that it doesn't inherently have. Usually because we are unaware of what is arbitrary and what is objective. In this case, we have a clear example of this.
The "proof" demonstrated in this video has a problem. The problem is that it attempts to prove that 360° is not an arbitrarily chosen division of a circle by using an arbitrary number base. That is, it uses the base-10, or decimal, number system as the crux of its proof. And our use of the base-10 system is entirely arbitrary in itself. So we are trying to prove something to be objective using an arbitrary standard.
So, I decided to use the base-16, or hexadecimal, number system, run the math through it, and see what happens. For those not familiar with it, hexadecimal simply means that, instead of 10 possible single digits, there are 16: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, and f. And instead of each digit representing a power of 10, each digit represents a power of 16. So a=10, b=11, c=12, d=13, e=14, and f=15. Once you reach a second digit in hexadecimal, you start counting powers of 16. So 1016=1610, 1116=1710, 1216=1810, and so forth. (Those little numbers mark which system the numbers are in for clarity, but will only be used when necessary. 10 means base-10, and 16 means base-16 respectively.)
In base-16, 36010 is written as 16816. So, here is the same math as above run in hexadecimal:
• A full circle is 168°. 1+6+8=f. (Remember f=1510) A circle divided in half is b4°. b+4=f. If you continue these divisions, the pattern holds: 1/4 of a circle is 5a°, and 5+a=f. 1/8 of a circle is 2d°, and 2+d=f. 1/16 of a circle is 16.8°, and 1+6+8=f. Etcetera. No matter how many times you bisect the circle, the digits always add to f.
• A similar pattern is seen in regular polygons. An equilateral triangle has 3 angles of 3c° each. 3c×3=b4. b+4=f. A square has 4 angles of 5a°. 4×5a=168. 1+6+8=f. A regular pentagon has 5 angles of 6c°. 5×6c=21c. 2+1+c=f. A regular hexagon has 6 angles of 78°. 6×78=2d0. 2+d+0=f. And so forth.
• The sum of all single digits, excluding f, is 69. 0+1+2+3+4+5+6+7+8+9+a+b+c+d+e=69. 6+9=f.
• Adding any single digit to f will produce sum that, when its digits are summed, is the original number you added to f. 1+f=10, and 1+0=1. 3+f=12, and 1+2=3. 5+f=14, and 1+4=5. f+f=1e, and 1+e=f.
Therefore, this isn't some special intrinsic property of the number 9. By switching bases, we can see 15 (represented as f) exhibiting all of the same properties. What do 9 in base 10 and f in base 16 have in common? They are both the final single digit in their respective bases. This is why you can sum all of the other digits in their base together, add the digits of the sum, and get the final digit of that base. It is also why you can add any digit to them, add the digits of the sum, and get the number you added to it. When you try this in other bases, such as base 8, this all works the same way.
However, when you try the math with the angles and the regular polygons in base 8 it doesn't work. And there is a reason for this. 9 and 15 have one more thing in common: 360 is divisible by both of them. 360/9=40. 360/15=24. This is why the math works with them when they are the final digit of their base. The final digit for a base 8 system is 7, and 360 is not divisible by 7. 360/7=51+3/7. However, when you cut a circle into units that are divisible by 7, say, 700 units, the math again works. All of the digits in the bisected circle or the sum of the angles of the regular polygons add up to 14, which is written as 16 in base 8, and 1+6=7. So the pattern still holds.
In conclusion: 360° is entirely arbitrary. It was arbitrarily chosen by the ancient Babylonians (or maybe someone else), possibly because they had a base 60 number system and found 360 easy to work with (360 is also divisible by 60). There is no objective basis for that choice, at least as far as anything intrinsic to math. The only thing that makes it seem special is that it happens to be divisible by the largest single digit in our own chosen base system. And the only thing that makes 9 special is that it is the largest single digit in our chosen base system, which itself is arbitrary. Base 16, base 20, base 60, and other systems have been and are used around the world. I'm a bit partial to base 16 and base 210 myself. 9 is a perfectly lovely number, but it doesn't hold within itself profound secrets of existence. Or at least if it does, this video didn't demonstrate it.
There is some really fascinating math out there. But it is important to recognize where math ends and arbitrary interpretation begins. It is also important to be lucid about what is objective and what is not. There are so many things people take for granted in this world as just "the way things are" as if it were the only possible way it could be, when really, someone simply decided it should be that way, and that decision was entirely arbitrary. I cannot understate the value of becoming aware of the things in our lives which fall into this category. It will change the way you see everything, and for the better.
Bonus: For those who have not already done so, check out Wolfram Alpha. It helped me crunch some of the numbers for this one and is a wonderful tool for all of your math and information seeking needs. Bookmark it. Doooo ittttt!!!!