*(Originally published in Fall 2002 issue of JOAC.)*

*Abstract: The traditional emblematic lines are extended in a new model that favors the view that "it's trigrams all the way down". The model of "hypertrigrams" is formally developed and examined. An alternative way of drawing hypertrigrams, called "triangular calligraphy" is proposed as a means to make computing with these figures much easier. This model is supported by an amazing instance of synchronicity.*

This paper is yet another theoretical extension of the emblematic lines to more complex structures, with a view to shedding more light on the underpinnings of the Q-Ching. Like my previous article on HyperEmblematic Lines (see *McFnordland [1998]*), on which the current study is heavily based, this theory of HyperTrigrams is a modern interpretation of ideas that were glimpsed by the old programmer-priests, but which they never developed due to a deficient conceptual basis. In particular, the inspirations for this model are the oft repeated aphorisms from the Book of Trigrams that "All lines are trigrams" and "Hexagrams are trigrams all the way down." Effectively, they intuited the fractal nature of emblematic lines, long before anyone knew about fractals. In the process of fleshing out these notions, a mathematical system of considerable depth and beauty in its own right is revealed, with implications that cross many fields within sacred mathematics and archeocomputing.

The intent of this article is six-fold. First, the model for HyperEmblematic Lines is reviewed. Many of these same intuitions carry over to the Hypertrigram model as well, but there are also some unique features that need to be examined. Following this intuitive overview, the new model is developed more rigorously, including recursive definitions for both the hypertrigrams themselves and the "journey functions" that are used to compute their journeys. Some of the unique qualities of the second order hypertrigrams are explored and the "mystery of missing bits" (quirks of the data compression schemes) is cleared up. An alternative way of drawing hypertrigrams, called "Triangular Calligraphy" is explored, which greatly simplifies the computations and reveals some of the hidden symmetries of these fractals. Finally, further lines of research are sketched out.

The model of HyperEmblematic Lines (HL) was fully developed in my 1998 article. The crucial idea was that very simple types of lines (namely the Basic Lines called "yin" and "yang") can be elaborated on via a recursive process for creating more complicated lines, known as "making a digram". In this sense, a "line" is a function or ordered sequence from some specified domain (usually the set of numbers from 0 to some fixed number n), where all the function values are Basic Lines, either yin or yang. As the domain grows in size, producing an ordered sequence of yin/yang states, the lines start showing an emergent quality called "movement". What distinguishes one line from another is the details of this movement (which I termed the "journey" of a line); "more complicated line" is synonymous with a more complex journey.

More precisely, the HL's are a hierarchy of lines. Using the modern notation of yin = 0 and yang = 1 (instead of the more traditional 2 and 3) for the Basic Line values, there are just two kinds of lines at the bottom of this hierarchy. These sequences are Yin = (0) and Yang = (1), having a domain with one value (usually denoted { 0 }) and thus a single function value. The set { Yin, Yang } is termed the zero order HyperEmblematic Lines, or simply HL(0). When you make a digram out of these HL(0) lines by placing one on top of another, there are 4 possible sequences. These are termed moving yin = (00), static yang = (01), static yin = (10), and moving yang = (11), which form HL(1) or the 4 emblematic lines. Notice the domain now has 2 elements, usually written as { 0, 1 }, corresponding to the fact that the digram consists of 2 basic lines. When one of these HL(1) lines is placed on top of another, a "hybrid digram" is created. These hybrid digrams are actually sequences of length 4 (using a domain of { 0, 1, 2, 3 }), such as (1001). There are 16 of these second order lines, which form HL(2). Putting two HL(2) lines together in a digram creates the HL(3) lines, and so on up the hierarchy to any order HL(n). Notice that the domain doubles in size at each order, making the sequences for HL(n+1) twice as long as for HL(n), for all n; the length is always a power of 2.

This recursive technique of "making a digram" is also the basis for interpretting the movement over time of the HL. Without repeating the details of the construction, the sequence of yin/yang values representing a HL is translated or compiled into a second sequence of values, called the "journey" of that line. The sequence of basic lines in the journey explicitly describes the state of the HL at each step of the reading. Similarly, a hexagram as a whole goes through a sequence of movements or changes as its 6 individual lines change. Whereas traditional hexagrams have (at most) only a beginning and ending value, higher order hexagrams can undergo a large number of intermediate states as well, leading to more complex oracular readings. The compilation process just described actually turns out to be quite straightforward, since there is a direct one-to-one relationship between basic lines in the HL and basic lines in the journey. Thus, each HL contains a complete description within itself of the journey it is about to take (and vice versa).

Many of these intuitions carry over to the trigram based model. The lines created by this model are variously called "hypertrigrams", "trigrammatic emblematic lines", or just the "TL hierarchy" (where TL stands for "trigram lines"). Again, we start with the Basic Line values to create functions or ordered sequences. There are 2 basic lines at the bottom of this hierarchy, which are still named Yin = (0) and Yang = (1); these lines form a set called TL(0). The recursive technique for forming new lines is "making a trigram", that is, stacking 3 lines from the previous order on top of each other. The sequences triple in length each time we move to the next higher order. There are 8 trigrams in TL(1), from K'un = (000) to Ch'ien = (111), just as in the traditional approach. The next order, TL(2) or "trigrams of trigrams", has 512 figures, most of which have no counterparts to the original 8 trigrams. TL(n) grows exponentially with increasing n.

The process of translating a TL into its corresponding journey of states requires an entirely new approach from the HL lines, however, indicative of some novel phenomena with trigrams that aren't present in the digram method. We see these differences emerging even at the first order lines, the traditional 8 trigrams vs. the 4 digrams, which led to the famous 6-7-8-9 encoding used in the Q-Ching.

In a digram, the upper line describes the starting value of the emblematic line, while the lower line (negated) gives the final state -- pretty simple. Things are more complicated with trigrams, however. No one line seems to be in control of any property of the TL, due to the details of the data compression mechanism. Unlike the digrams, knowledge of the yin/yang state of any basic line in the trigram doesn't in general give you **any** information about the starting or final state of the journey for that trigram. In fact, knowledge of two lines in the trigram still doesn't determine the trigram's behavior, since only one of the starting or final states is now determined, but not both of them. Somehow, 2 bits of information are distributed across 3 lines, giving each line an information content of just 2/3 of a bit! Only by looking at all 3 lines, but from 2 different perspectives, can the full picture of the trigram's journey be seen. It only gets worse with higher order figures. This is very similar in principle to a hologram, where information about a picture is non-locally spread across the entire photographic image, and several different pictures can be decoded from the same image by shining the laser beam (or "reference beam") at different angles onto the hologram. This non-local quality of hypertrigrams is one of the main reasons why they are so confusing to most people, although the hologram analogy should lessen the initial shock.

When "interpretting" a HL, we focused on the process of manipulating the HL into its equivalent journey. With hypertrigrams, we focus on the creation of "interpretation functions" that analyze the TL from different perspectives. This series of functions, applied to the TL, generates a sequence of basic yin/yang states, which defines the journey of the TL. The definition of these interpretation functions is a relatively simple recursive definition, although most of these functions are pretty hard to describe in English. One consequence of using these functions to define a journey is that tremendous amounts of information initially present in the TL are simply lost and unused. Unlike the interpretation technique for HLs, which is reversible (you can go from HL to journey and back again, without getting lost), the interpretation technique for TLs is highly irreversible. Many different TLs have the exact same behavior (or journey), making them indistinguishable simply by observing their movements.

What we perceive as moving or still, yin or yang,

Is indeed a trigram of three lines.

All lines are trigrams and all trigrams are lines.

A hexagram is but six lines together,

And each line is composed of three changes.

Each change is also a trigram of changes.

Like the tortoises that hold up the corners of the world,

The hexagram is trigrams all the way down.

-- Book of Trigrams

We will first put some meat on the bones of these intuitive notions by putting these ideas on a firm mathematical foundation. Since many of these ideas are similar to the 1998 article, I won't be quite so formal this time around, but no substantial details will be left out. After defining the hypertrigram hierarchy and how these figures are analyzed to reveal their corresponding journey, we'll look at some of the intriguing mathematical patterns in this formalism. Finally, I'd like to introduce a new way of drawing trigrams that not only hints at their fractal nature quite explicitly, but also makes the calculations of journeys much easier.

First of all, let's establish some basic definitions related to sets and functions that will make the discussion easier. A set, of course, is simply a collection of items enclosed in curly brackets. For instance, the set of the first 6 numbers is written { 0, 1, 2, 3, 4, 5 }. Given 2 sets, D and R, a "function" f from "domain" D to "range" R is a particular way of matching items from D with items from R. In particular, for each item x in D, the function f associates a unique item from R, usually denoted f(x). The following sets and functions will be important for our presentation:

Def: The set of "basic line values" = BL = { yin, yang } = { 0, 1 }. Notice we are using the modern mapping of yin = 0 and yang = 1, instead of the traditional mapping of yin = 2 and yang = 3 that the ancient Chinese scholars used. BL is commonly used as the range of many of our functions, that is, these functions will have values of either yin or yang.

Def: The set 2^{N}equals { 0, 1, 2, ... top }, where top = 2^{N}-1. The corresponding value half = 2^{N -1}. For instance, if N = 4, then top = 15 and half = 8; the set 2^{4}= { 0, 1, 2, ... 15 }. If there is possible confusion as to the appropriate value of N being refered to, the notations top(N) and half(N) will be used.

Def: Similarly, the set 3^{N}equals { 0, 1, 2, ... top3 }, where top3 = 3^{N}- 1. The corresponding value third = 3^{N - 1}. So if N = 3, then top3 = 26 and third = 9; 3^{N}= { 0, 1, 2, ... 26 }.

Def: A "line" (of the TL hierarchy) is a function f from 3^{N}to BL. If N = 0 (so that the domain { 0 } has only a single item and the function has only one yin/yang value), the function is called a basic line. There are 2 basic lines: Yin (where Yin(0) = 0) and Yang (with Yang(0) = 1). If N = 1, the line is sometimes termed a simple trigram; these 8 functions correspond to the 8 traditional trigrams.

Def: Assuming the 3 functions f0, f1 and f2 are all from the same domain 3^{N - 1}with values in BL, there is a function g with domain 3^{N}that is called the "trigram of f0, f1, f2" or g = (f0, f1, f2) defined by:

if x < third(N), then g(x) = f0(x),In the trigram g, f0 is called the "lower line" of g, f1 the "middle line", and f2 the "upper line". We will sometimes write f0 = lower(g), f1 = middle(g) and f2 = upper(g).

if x >= third(N) and x < 2*third(N), then g(x) = f1(x - third(N)),

if x >= 2 * third(N), then g(x) = f2(x - 2 * third(N)).

Def: A function f from { 0, 1, 2, ... n } to BL will often be written as the ordered sequence ( f(0), f(1), ... f(n) ), that is, the function values are listed in order between parentheses. In particular, Yin = (0) and Yang = (1). If g = (f0, f1, f2), then g can be written as a sequence by listing all the values of f0, then f1, and lastly f2 between parentheses; each sub-sequence is exactly one third of the full sequence.

The TL hierarchy is now defined recursively as:

(It is easily seen that all members of TL(n) are functions from 3

Def: TL(0) = { Yin, Yang }.

For n > 0, TL(n) = { (f0, f1, f2) : f0, f1, f2 are in TL(n - 1) }.

When we move to the hypertrigrams in TL(2), we form trigrams of 3 simple trigrams, which can be symbolized as a sequence of 9 basic lines. Since each of these lines can be yin or yang, there are 512 such figures. The hypertrigrams in TL(3) consist of 27 lines each; there are 134,217,728 of these figures. As we move up the hierarchy, the number of hypertrigrams literally explodes.

As with the HL lines, the next step is to define how hypertrigrams move, that is, what is the relationship between a hypertrigram and its journey. This is commonly known as the "interpretation problem" for a figure. The journey of a TL is simply the sequence of yin/yang values that the TL moves through at each step of the reading. This journey can be written as ( j0, j1, j2, ... jn ), where each j has a value of 0 or 1. Such a journey will be specified via a sequence of "interpretation functions" (also known as "state functions" or "reference functions", due to the hologram metaphor), one for each step of the journey.

The journey of a basic line from TL(0) is pretty simple: the journey of Yin is just (0) and of Yang is (1).

The interpretation of the simple trigrams in TL(1) is specified by the classical 6-7-8-9 encoding that was just summarized. The initial value of the trigram is determined by the gender of the figure. So, father and sons start out yang, while mother and daughters begin yin. However, the final state uses a different criterion. This time, it's mother and sons that end up yang, while father and daughters finish yin. Examining the 8 possible outcomes, it appears the gender depends on whether there are an odd number of yin or yang lines in the trigram. If the 3 lines of the trigram are called x, y and z, then odd(x, y, z) = 0 if there's an odd number of yins, or else equals 1 if there's an odd number of yangs. Further, the distinction between trigrams that end up yin as opposed to yang depends on which kind of lines are in the minority. Mother trigrams have no yang lines, sons only 1: yang is outnumbered, but these trigrams become yang. Daughters have 2 yangs, while fathers have 3 yangs: yin is in the minority, so these trigrams become yin. We'll use the function fewer(x, y, z) to tell us whether yin or yang is outnumbered. Hence, the journey of a trigram (xyz) is simply ( odd(x, y, z), fewer(x, y, z) ), a digram that can also be interpretted as an emblematic line.

(A warning about notation here. We are writing the sequence for journeys from left to right in this article. In the 1998 article, they were written right to left. When represented in the traditional calligraphy, journeys will be a stack of lines that are read from the top down in either context. Please be aware of these differences.)

*Figure 1. Calculating the Journey of (111 010 011).*

Let's take a typical member of TL(2) and see if we can extend these basic intuitions. Assume we are looking at the hypertrigram g = (111 010 011), which can be broken up into the 3 sub-trigrams f0 = (111), f1 = (010) and f2 = (011). f0 is moving yang, with journey (10). f1 is a son, represented by static yang or (11); f2 is static yin or (00). (Remember, these are journey sequences, not traditional digrams, that we're talking about here.) But f0, f1 and f2 are also "lines" within the higher order trigram. As the 3 sub-journeys indicate, this is a "moving trigram", changing from (110) to (010). So during the first half of g's movements, it looks like (110), while during the last half it behaves like (010). (110) is a daughter or static yin, which moves in the sequence (00); (010) is a son or (11). Hence, the entire sequence of journey steps for g is (0011). This is illustrated in Figure 1. The 3 sub-trigrams are in the first column of the figure, while the second column has the journeys associated with these sub-trigrams. The upper lines and lower lines of each sub-journey are consolidated into 2 separate trigrams in column 3. These higher level trigrams are then converted to the upper and lower parts of the journey for the entire hypertrigram g. As another example, consider g = (111 001 000). The lower simple trigram, f0 = (111) is a father/moving yang figure or (10). The middle trigram is f1 = (001), a son/static yang or (11). The upper trigram is f2 = (000), a mother/moving yin or (01). So the first half of the journey resembles (110), which behaves like static yin or (00); the second half resembles (011), which is also static yin. So the complete journey is (0000), meaning g stays yin the entire journey (analogous to static yin), despite the fact that 2 of its "lines" or sub-trigrams are moving. This type of analysis can be extended to TL(3) and beyond, so let's formalize it now.

Trigram | xyz | Values | Class | Odd() | Fewer() | Journey |
---|---|---|---|---|---|---|

000 | 0 or 6 | Moving Yin | 0 | 1 | 01 | |

100 | 1 or 7 | Static Yang | 1 | 1 | 11 | |

010 | 1 or 7 | Static Yang | 1 | 1 | 11 | |

001 | 1 or 7 | Static Yang | 1 | 1 | 11 | |

011 | 2 or 8 | Static Yin | 0 | 0 | 00 | |

101 | 2 or 8 | Static Yin | 0 | 0 | 00 | |

110 | 2 or 8 | Static Yin | 0 | 0 | 00 | |

111 | 3 or 9 | Moving Yang | 1 | 0 | 10 |

First of all, Table 1 summarizes all the numerical and symbolic correspondences for the simple trigrams, as well as the definitions of odd() and fewer(). These 2 functions are used to define the first and last halves of the journey, respectively, when applied to the sub-trigrams of the figure. Of course, the journeys of the sub-trigrams also depend on the trigrams that in turn make them up, and so on all the way down to the basic lines at the bottom level of the figure. All these calculations can be captured by a collection of interpretation functions operating on the original hypertrigram, telling us what the state (yin or yang) of the figure is at a particular step of the journey. More precisely, if g is a hypertrigram of the n'th order (i.e., in TL(n)), then the function h[n, j] will allow us to calculate the state of g at the j'th step, namely h[n, j](g) = the j'th step of the journey of g. These interpretations are defined recursively, based on the recursive definitions of both the TL and HL hierarchies. The recursion goes as follows:

Assume g = (f0, f1, f2) is an n'th order hypertrigram.

Def: h[0, 0] is the interpretation function for TL(0). A zero order TL has only one line in it, so it can't be broken up into sub-trigrams. The value of h[0, 0](g) is simply g(0). Thus h[0, 0](g) = 0 if g is Yin and has a value of 1 if g = Yang.

Def: h[n, j] is the j'th interpretation function for TL(n), where n > 0 and j is in the set 2^{n}. If g = (f0, f1, f2) is in TL(n), then h[n, j](g) is defined to be:odd( h'(f0), h'(f1) , h'(f2) ), if j is even, orHence, each h[n, j] is defined in terms of the interpretation functions for the next lower order, all the way down to the basic lines.

fewer( h'(f0), h'(f1) , h'(f2) ), if j is odd,

where h' is the function h[n-1, j/2] defined for order n-1.

Def: The journey function for TL(n), denoted H_{n}, will convert a hypertrigram into its equivalent journey, mapping TL(n) onto HL(n). The journey of g is defined as H_{n}(g) = ( h[n,0](g), h[n,1](g), ... h[n, top(n)](g) ). Intuitively, this is simply a sequence of yin/yang states, one for each step of the journey, where each step is defined by the action of an interpretation function on g.

Summing up, in the hologram model, the trigram is the hologram itself, while the various interpretation functions are different reference beams of laser light that reveal distinct images in the hologram. Each of these images is a single yin or yang line that tells us the state of the trigram at a specific step of its journey. And the sum total of all these images is the full journey of the trigram, a moving picture of how the trigram changes during a reading. Finally, since the entire hexagram produced by the oracle is simply 6 of these hypertrigrams, we can plot the journey of the full hexagram for the reading.