Since i started reading about daoist ideas, i was interested in the meaning of the ba gua trigrams and their various arrangements. I was dissappointed by most of the published information i could find with some notable exceptions.

One was an article in Ted Kaptchuk’s Chinese Medicine, one of the most respected books in its field. This article covered the five element theory. Another source was the reference book from my martial art school, Park Bok Nam’s Fundamentals of Ba Gua Zhang. Ba Gua Zhang is a chinese martial art based on the application of daoist principles to self-defense and health cultivation. The ba gua book described the trinity of natural principles consisting of yin yang theory, five elements, and ba gua. With my maths background i found that i could explain these theories in mathematical terms.

The concept of yin and yang is well known. It appears to represent comparison. Imagine, for example, a function that represents average daily temperature at a particular location. Let’s call temperature on date A, Ta and temperature on date B, Tb. If Ta is higher than Tb we can say Ta is more yang than Tb and Tb is more yin than Ta. So to be yang means having a higher value (hotter), to be yin means having a lower value (cooler). If we set zero as a comparison point to any single temperature value, the way the celsius measurement system uses the temperature of water freezing, then a positive value (hot) can be called yang and a negative value (cold) yin.

The five elements represent change in value. Using the same temperature function we can look at the change in temperature. If Ta is increasing (getting hotter) it is called greater yang and if Ta is descreasing (cooling down) it is lesser yang. Similarly if Tb is decreasing, it is greater yin and if Tb is increasing it is lesser yin. These are four of the five elements. The fifth element, earth, represents the zero point. Hua Hu Ching’s book The Later Teachings of Lao Tzu is one of very few places where i have seen earth put in the middle of the other four elements.

Ba gua, also known as 8 triagrams, represents the above states plus acceleration or change in the change of temperature. For example if a temperature is cooling, then the cooling may be speeding up or slowing down. The most yang case would be a hot temperature that is getting hotter and faster all the time. You can imagine the beginning of an explosion. In math terms, f(x) = x^3 exhibits this behavior for x>0. The most yin case is the opposite, a cold temperature that is cooling is cooling ever quicker. For example the end of summer is like this when “it’s starting to get cold” (compared to the middle of summer). In fact any (piecewise continuous) function f(x) can be expressed as a series of trigrams each representing segments of the function that match to each trigram.

Ba gua are usually shown in two arrangements as shown below. The first arrangement (early heaven) shows continuous change from one state to its opposite. It matches f(x) = sin(x). It shows that in order to go from the most yang state to the most yin state, one must go through all the intermediate states including zero. This sounds a lot like Bolzano’s theorem to me (also known as the intermediate value theorem) except it includes not just the absolute value of a function but also change and acceleration.

For me the profound thing is that once these states of change are expressed they can be applied to any process around us in a qualitative manner (without numbers). The I Ching (book of changes), for example, applies this to our human lives and looks at the change from one of these states toward another (8*8=64 changes). For example going from most yang toward most yin or most positive toward most negative can be seen as stagnation. The opposite as a time of prosperity.

The second ba gua arrangement (later heaven) is the one that really fascinated me. I hope to discuss it in a follow-up article.