## Reflected Heart Yin-Yang Symbol

I have long been fascinated by some of the more mathematical Chinese philosophical writings such as the binary complications of the I Ching and Tao duality. Specifically in this case, the Yin-Yang symbol of the cyclic dependence of dual principals- night to day to night; winter changing into summer, summer back to winter (my home in Minnesota is particularly prone to this cycle), and the basic notion of the Yin-Yang- when the dark is darkest and most triumphant, that only means that the ascension of the light has begun.

But the mathematical side of me has always been bothered, deep inside, by the symbol itself. Because it doesn’t really show what they say it shows. As we see in the next figure, while it may be 100% white at the top, it doesn’t start to change to black until almost one-quarter of the turn is complete. So even though there is a nice dot at the top center to show that there is always black even in brightest white, that promise is then broken for half of the turn to darkest black.

So we can fix this. We can insist that after 1/4 of the turn from white to black, that 25% be black. At 1/2 the turn, half is black. And at 3/4 of the half turn to black, 75% of the radius should be black. Mathematically, we can describe this quite simply using polar coordinates: $r=\theta/\pi$ where $r$ is the amount of black and $\theta$ is the angle of turn. Within mathematics, this is known as Archimedes spiral.

Archimedes Yin-Yang

The resulting figure is, to my eye, quite striking. It no longers needs the smaller dots, because even when white is completely triumphant, white is about to be overwhelmed by a wave of black.

This variation on the Yin-Yang is not new. One of the earliest known versions by Zhao Huiqian (1351-1395 CE), in Tiandi Zhiran Hetu (Heaven and Earth’s Natural Diagram of the River) has much the same appearance. And there is a common (0n the web) historical explanation of the symbol that relates it to the annual variation of the length of a shadow at high noon as the sun goes through its seasons in which the drawing is an Archimede’s spiral.

Fermat Yin-Yang

There is an even more extreme version of this known as Fermat’s spiral, which to a mathematician has an intriguing set of extra symmetries. See Roice for an elegant video demonstration.

But both of these still suffer from a flaw. If we look at the graph of the amount of white as the symbol turns for the two variations, we get the following abrupt curves.

An ideal cycle would more resemble the graph below. In fact, as regards the “traditional” explanation involving shadows, the sun’s ecliptical does follow this graph. This would be the case if we used a polar equation of $r=(1-cos(\theta))/2$, known as a a cardioid. The next figure shows the resulting yin-yang symbol.

Reflected Heart Yin-Yang

The result is very elegant, to my eye. It reaches the 25% mark at 60 degrees (instead of 45 degrees as in the Archimedes spiral). It’s length is exactly the diameter of the circle itself. It smoothly blends into the outer circle. So it is my Model Yin-Yang symbol.

But even though in this version also has the overhanging wave to ensure that one does not forget, even in the throes of full light, that darkness will loom, I miss the smaller dots. It just looks nicer, more elegant, more aesthetic.

The Heart for Yin-Yang

And by reflecting the right half vertically, we can see the reflected heart, the cardioid, of this Yin-Yang symbol.