Yin-Yang Challenge

Published on Sunday, April 30, 2017 in , , ,

yin yang symbol, drawn mathematicallyFor this post, I'd like to turn to a variation of a classic Henry Dudeney puzzle, from his book Amusements in Mathematics. It can also be found in Martin Gardner's October 1960 Scientific American column, or his book, New Mathematical Diversions, as the 5th puzzle ,“Bisecting Yin and Yang”, in chapter 12, “Nine Problems”.

As you've no doubt guessed, this puzzle involved the yin (dark) yang (light) symbol. For this puzzle, I've drawn it in a very mathematically precise way over at Desmos. The outside is a unit circle (so, the radius is 1 unit), the main semicircular divisions of the design have a radius of 12 unit, and the opposite-color dots have a radius of 16 unit. Here's the challenge: What's the equation of the line that divides the design so that each side of the line contains exactly equal amounts of the dark and light portions?

As with many puzzles, this one seems hard, until you break it down into simpler steps. Let's start with a much easier puzzle: If the top half of the puzzle were dark, and the bottom half light, as in this rendering, where would you draw the line? The answer is easy. It should be a vertical line, so the equation would be x=0.

Next, we change the design a bit, so as to get closer to the yin yang symbol. Starting with the previous design, we cut a half circle (12 unit radius, remember?) of the dark portion from the right side, and add that to the left side, giving us the design below. The vertical line obviously won't work anymore, and we'll need to rotate that line by some amount to compensate, but how much?

Again, the secret is to take steps slowly. If you remember your high school geometry, you remember that the formula for a circle is A=πr2, and that our design as a whole, being a unit circle, has an area equal to π.

The formula for the semicircle we've moved, then, is A=12πr2. Plugging in 12 for the radius, we get π8 units. So, to compensate for π8 units out of a full circle with an area of π-units, we simply rotate the formerly-vertical line counterclockwise by 18 of the circle, or 45°! The upper left quadrant completely dark, so that makes this adjustment simple. The blue line is the dividing line for this design:

This, in fact, is the answer to the original puzzle as posed by Dudeney and Gardner. This is NOT the answer to the problem I posed above. When I first ran across this puzzle, it annoyed me that it wasn't done with the full yin yang symbol. The dots are a symbol of how, in nature, nothing is purely one thing or the other, and are a very important part of the design.

It's time to go back to the full design. Compared to the previous step, we're going to be removing some of the dark area from the right side to the left side. This means that we'll end up rotating our dividing line some distance clockwise this time, and we need to figure out by how much. Yes, once again, we'll be using our area formulas to work out the adjustment. We even know that the result should be easy to interpret, since the result will π over something, and this comes out of a π-unit circle.

The dots, of course, are full circles, so we use the formula for the area of a full circle once again. The dots have a radius of 16 of a unit, and plugging that into the formula, we get π36. In other words, the line needs to be moved back clockwise 136 of a full circle, or 10°. That brings the line to being 35° off of the original vertical line.

A 35° line must be our answer, right? Wrong. Go back and look at the original question: Here's the challenge: What's the equation of the line that divides the design so that each side of the line contains exactly equal amounts of the dark and light portions? We need to work out enough details for the line equation y = mx + b, where m is the slope, b is where the line crosses the y-axis, and y and x remain as variables. The line, of course, crosses the y-axis at 0, so b = 0. That reduces the equation to just y = mx, so we need to figure out the slope.

First, angles are usually measured in relation to the positive x axis, so we're actually talking about a 125° (35° + 90°), or 25π36 radians (Confused? Read Intuitive Guide to Angles, Degrees and Radians). In geometry, we'd say we were trying to calculate rise over run (rise ÷ run). In trigonometry, we're trying to calculate the opposite site of the angle by the adjacent side (Confused? Read How To Learn Trigonometry Intuitively), and that means we need to use the tangent!

So, the equation is y = tan(25π36)x, or y = tan(125°)x, if you prefer. The actual slope is an irrational number which is roughly equal to -1.428148. There you have it, the equation to a line which divides the design into equal parts of light and dark, as shown below.

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