Quantifying Change: Introduction
In this series of posts, the author purposes to develop a mathematical framework for studying Change, the constant, unvarying. This paper assumes a practical familiarity with I Ching and its internal structures.
In presenting this treatment of Quantifying Change, the author has spared little effort in making it accessible to the casual student of I Ching, or even to the interested layperson by reducing unnecessary jargon. Regretfully, everyday language often falls short of providing what is required for this endeavor. This is especially true where mathematics are concerned. Clear expression of even straightforward math concepts requires verbal precision that everyday language seldom provides. The author, where specialized terms have been employed, has attempted to make the meanings clear and relevant to the application of the terms.
Graphic descriptions (far more useful in this regard) have been employed where words proved inadequate. Future treatments of this topic will make broader use of graphical explanations.
The philosophy espoused by commentators is that study of the Changes can produce a penetrating understanding of the processes of Change as they manifest in our personal lives, in the cosmos, and at the various scales of dimension. It is reputed that mastery of this discipline enables passage through the world essentially without misstep since one will have developed the ability to reliably discern the appropriate course of action in any situation.
'Change' is capitalized, above and elsewhere, to distinguish common usage from our use in the sense of the philosophy of I Ching. Generally speaking, reference to 'Change' indicates the eternal process of transformation, creation and destruction, formation and disintegration.
Herein, reference to 'Changes' or the 'Change operation' will also indicate what were previously called “expanded first-order difference integers*.” The Changes are a product of the King Wen sequence of hexagrams. The King Wen sequence is the likeliest form in which one will encounter the Changes. The King Wen sequence is arranged in pairs of hexagrams, beginning with Ch’ien and K’un, the Gates of Change, represented by hexagrams #1 Heaven and #2 Earth. It ends with hexagrams #63 After Completion and #64 Before Completion.
NB: *First-order differences is a term coined by the late Terrence McKenna in his presentation of Timewave (elsewhere discussed). Our treatment of the Changes is a fuller development of McKenna's nascent concept.
It is unclear (and dubious) that any comprehensive explanation of the ordering principle behind the Wen sequence has been published. This is to say that the sequence has not, to the knowledge of the author, been proven computable or 'deterministic' in the sense that the sequence may be generated from a reasonably small set of inputs and rules. Compare this to say, the first 64 digits of π, or the of any common mathematical function or value.
Before Change and the nature of the King Wen pairs are discussed, some terms:
- xiang: literal meaning: symbol; in this context, indicates the primordial reality (represented by HEAVEN, EARTH, FIRE, and WATER) that is believed to have emerged after the “separation” of YIN and YANG
- line: (also: , place, position, or yao) any of the six steps or stages of a situation or process as given by I Ching, ordered from bottom to top. Lines take on two basic forms: YIN (open, divided), and YANG (closed, undivided). Through Change, a line may move to a different position within a hexagram, or transform into its opposite; that is, YIN may alternate to YANG, and conversely.
- ritual number: (closely-related to xiang) a product of the yarrow-stalk oracle which determines the specific type of each line in a hexagram. Ritual numbers may have the following values: [6, 7, 8, 9].
- digram (also bi-gram): (related to xiang) any of four combinations of two lines (places, positions, or yao) taken as a group. Digrams are figurative representations of xiang, and have traditional ritual number assignments.
- trigram: eight discrete trigrams exist; each is a ordered collection of three contiguous lines that compose half of a hexagram. Two trigrams are traditionally oriented in superposition. They are named HEAVEN, EARTH, THUNDER, WIND, WATER, FIRE, MOUNTAIN, LAKE.
- hexagram: any of sixty-four distinct figures composed of six places, lines, positions, or yao. Hexagrams may also be interpreted as comprising three digrams or two trigrams.
- trigram exchange: an operation where the inner/lower trigram exchanges positioning with the outer/upper trigram such that the inner state becomes the outer state, and conversely.
- pantonggua: place-wise or positional negation (to/from YIN from/to YANG), applied to an entire hexagram. YIN lines will alternate to YANG, and conversely.
- fangua: figure-wise inversion of a six-place hexagram that re-orders it from topmost place to lowest); applied to an entire hexagram. Line 6 changes places with position 1, line 5 changes places with line 2, line 4 changes places with line 3, and so on.
- King Wen pairs: (also: anti-pairs) sixty-four of these exist. They individually comprise a unique hexagram (H) and its mate (H’). In general, the mate (H’) is defined by trigram exchange. Exception: in eight cases this operation produces the self-same hexagram; for those eight cases, the mate is defined as the pantonggua (the positional negation).
- Changes: (similar to hexagram) An ordered sequence of sixty-four six-line figures reflecting the state-change occurring within King Wen pairs as the sequence is fully-traversed (from beginning to end and back).