Review: Cook’s Classical Chinese Combinatrics

Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence

Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence
by Richard S Cook
University of California, Berkeley: STEDT Monograph Series, Vol. 5, 2006
paperback, 660 pages
ISBN: 9780944613443
Order through Lulu

It seems fitting that a linguist, someone attentive to how patterns combine to produce meaning, would tackle the great puzzle of the Yijing’s hexagram order. The hexagram order, often referred to as the “King Wen order” is the traditional arrangement found in almost all Yijing books; other orders include the Mawangdui order (see I Ching translated by Edward Shaughnessy) and the Zagua order (the Yijing’s Tenth Wing).

Dr. Richard Cook’s Classical Chinese Combinatorics makes use of mathematical theories of Fibonacci, Pythagoros, the Golden Mean, among others. He proposes organizing hexagram by “classes” such as gender (female, male, neuter) based on quantity of yin and yang lines in a hexagram. He uses these theories to try to uncover deeper layers of organization to the traditional hexagram sequence. The merits of Cook’s theories await analysis by those with deeper mathematical skills; in lieu, we offer the publisher’s description:

The first and most enigmatic of the Chinese classics is the Book of Changes, and the reasoning behind its binary hexagram sequence remained an unsolved mystery for some 3,000 years (according to the tradition ascribing it to King Wen of Zhou, d. -11th c.). This Monograph resolves the classical enigma: Richard Cook provides a comprehensive analysis of the hexagram sequence, showing that its classification of binary sequences demonstrates knowledge of the convergence of certain linear recurrence sequences (LRS; Pingala -5th c.?, Fibonacci 1202) to division in extreme and mean ratio (DEMR, the “Golden Section” irrational; Pythagoras -6th c.?, Euclid -4th c.). It is shown that the complex hexagram sequence encapsulates a careful and ingenious demonstration of the LRS/DEMR relation, that this knowledge results from general combinatorial analysis, and is reflected in elements emphasized in ancient Chinese and Western mathematical traditions.

Cook is a linguist associated with the large, ongoing project The Sino-Tibetan Etymological Dictionary and Thesaurus (STEDT) at the University of California, Berkeley. This project began in 1987 with the goal of creating “an etymological dictionary of Proto-Sino-Tibetan (PST), the ancestor language of the large Sino-Tibetan language family. This family includes Chinese, Tibetan, Burmese, and over 200 other languages spoken in South and Southeast Asia.” Cook’s prior work has been in developing Chinese language software, including ancient scripts.

Preview pages, including an abstract, can be viewed at this link. The price of the book, $64 (the symbolism is not lost on us), puts it beyond all but the most serious Yijing students, however copies can be found also in various libraries.

For a deeper analysis of Cook’s theories, we refer the reader to a recent review written by József Drasny for Yijing Dao, which can be read here. Drasny, a retired engineer in Hungary, has created his own interesting theories, including a three-dimensional “Yi-globe.”

Classical Combinatrics is a challenging read, but contains a number of interesting ideas.


Review: Richard Smith on the I Ching

The I Ching: A Biography
Richard J. Smith
Princeton University Press, 2012
ISBN 978-0691145099
Hardcover, 304 pp.
Fathoming the Cosmos and Ordering the World
The Yijing (I Ching, or Classic of Changes) and Its Evolution in China
Richard J. Smith
University of Virginia Press, 2008, 2018 (paper)
Paper, 416 pp.
ISBN 9780813940465
Mapping China and Managing the World: Culture, Cartography and Cosmology in Late Imperial Times
Routledge, 2012
Paper, 288 pp.

Richard Smith, a professor of Chinese history from Rice University in Texas, has written several outstanding books about the Yijing that will be of interest to readers looking to learn about where the Yijiing came from, what it meant, and how it spread around the world.

The first, Fathoming the Cosmos and Ordering the World: The Yijing (I Ching, or Classic of Changes) and Its Evolution in China is an in-depth look at almost every aspect of the Yijing: how it was used and by whom, the schools of thought that used and extended its meaning,  how its use evolved, the key personalities who wrote about it, and how it came to have such a global impact. Fathoming the Cosmos is a key starting point for any English-language Yijing research. Even a cursory look through it will demonstrate that dozens more volumes could easily be written on Yijing topics. The book has recently been reissued in paperback. In a companion volume, The I Ching: A Biography, Smith revisits the material of Fathoming the Cosmos, reworking it to create a volume for a general audience. Both books have thorough notes, indexes, and bibliographies.


Smith is author of several other highly regarded books that look at aspects of imperial Chinese history, culture, ritual, and ordering of the world. Fortunetellers and Philosophers: Divination in Traditional Chinese Society (1991) was his first foray into the Yijing, and also looked at topics such as fengshui, mediums, and face-reading.

Smith’s recent book, Mapping China and Managing the World (2013) revises a number of his lectures and articles. Several of these, updated into full chapters, are directly about the Yijing. “The Languages of the Yijing and the Representation of Reality,” is a thorough overview of Yijing studies. “Divination in the Qing” shows how involved the Qing dynasty rulers were with Yijing study and divination (remembering that the Qing themselves were Manchurians, not Chinese), while “Jesuit Interpretations of the Yijing in Global Perspective” covers the intriguing and complex interactions between European Christians and the Chinese during that dynasty. Smith’s introduction to the book also provides a good summary of current questions and issues about Chinese studies in general, and explains how he came to be interested in the Yijing. Central to this was comprehending that the Yijing is deeply connected with all aspects of Chinese culture and society.

Richard Smith’s books are excellent resources for anyone wishing to trace the Yijing’s history, whether for a class paper or for personal enrichment.

Review: Cheng Yi’s Yi River Commentary of the Book of Changes

The Yi River Commentary on the Book of Changes
by Cheng Yi, edited and translated by L. Michael Harrington
Introduction by L. Michael Harrington and Robin R. Wang
Yale University Press, 2019
ISBN: 9780300218077
Hardcover, 576 pages
$85 USD

Cheng Yi (1033–1107) lived during the Song dynasty, and was one of the era’s great thinkers. His Yijing commentary greatly influenced subsequent generations. Understanding the Yijing was important, but difficult, it was by then already a 2000-year-old book. Translator L. Michael Harrington, a professor of philosophy, gives a succinct overview of Cheng Yi’s thoughts and his influence on subsequent generations, including our own. He notes an important feature of Cheng’s commentary, that it offers full discussion of the hexagram components, rather than the snippets of explanation and terse definitions common to other commentaries. In addition, Cheng Yi creates a “coherent conceptual structure: the principle that governs the interaction between different capacities and functions in any state of affairs.”

The Yi River Commentary is an excellent companion to I Ching editions of Richard Wilhelm (translated by Cary Baynes), and Wang Bi (translated by Richard John Lynn). These are all dependable translations that introduce readers to particular strains of Yijing interpretations. In Cheng’s presentation, we can, in particular, see the roots of Richard Wilhelm’s work eight centuries later, both using a “teaching voice” that seeks to engage the student in depth.

Translating the Yijing is a difficult task, and Harrington clearly describes methods used in his effort to see the Yijing through Cheng’s eyes. He explains translation choices, as well. One interesting choice, for example, is the name “Ebb” for Hexagram 59 渙, which is “Dispersion/Dissolution” in the classic Wilhelm/Baynes I Ching. Harrington includes an introduction (written with colleague Robin Wang), notes on Cheng’s quotations of other material, extensive glossary, and a thorough index.

Avid Yijing readers will undoubtedly want to compare this translation of Cheng Yi’s work to others. Harrington’s is a model of transparency, detailing sources, methodology, and what material is included. Another English translation of Cheng Yi, by Thomas Cleary, is titled I Ching: The Tao of Organization. As with his other Yijing translations, The Taoist I Ching, The Buddhist I Ching, and I Ching Mandalas  (all from Shambhala Books) Cleary has succeeded in his effort to reach an educated popular audience. However, in each translation, he offers little clarity about his source material or methods, and, in this particular case, interpolates modern ideas (e.g., sociology). In addition, Cleary severely edited the original material, leaving out what Harrington estimates to be more than half of Cheng’s comments, in particular, on the Ten Wings as related to each hexagram. An interested reader would have to do their own comparative reading of the original to figure out what Cleary included or omitted, and what of Cheng’s text is actually quotes of other material. Harrington, in contrast, includes thorough information, greatly enriching the reading experience.

The Yi River Commentary is a well-made book physically, with a sturdy hardcover binding, though it weighs in at over two pounds. Unfortunately, the price for this book is set for libraries and specialists, not for the average Yijing reader, who would undoubtedly find this book of interest. The book is missing two key items for general readers: a hexagram finding chart and instructions for how to use the Yijing. Fortunately, these are easily found elsewhere (see our Basics page for such).

The Yi River Commentary is highly recommended for Yijing readers who enjoy pondering the deeper meaning of this ancient book and what it meant to the philosophers of imperial China, and for those readers who may be interested in the sources of modern interpretations.


Portraits of Fortunetellers

fortunetellerHong Kong is home to many fortunetellers. This article presents “The Fortune Market” a collection of portraits by photographer Kris Vervaeke of fortunetellers next to a Daoist temple.

“The fortune tellers have different backgrounds,” Vervaeke explained. “For some the job is passed on from each generation to the next within a family. Others learn it from a Master or through other means. The future of the shops is uncertain, however, as finding someone interested to take over is getting more and more difficult.” He also added that the rise of online fortune telling puts the physical institutions’ futures in jeopardy.

Divination in China Thesis

cropped-yi-hex-29-30-laiA new PhD dissertation related to the I Ching is available, “Divination And Deviation: The Problem Of Prediction And Personal Freedom In Early China” by Yunwoo Song, from the University of Pennsylvania. This interesting dissertation explores ancient methods and meaning of divination in early China through early imperial times. The dissertation can be viewed at UPenn’s site.



I Ching Diagrams Now Available

Curious about how people in past centuries envisioned the I Ching hexagrams? Visit our RESOURCES page to see some examples of charts organized by trigrams or  received order.

These handy diagrams were used to study characteristics and relationships between hexagrams, as mnemonics to help memorize hexagrams, and as hexagram finding aids.

The Nuclear Hexagram Chart by Qian Chengzhi (Qing dynasty) to the left is one example. Around the outside are the sixty-four I Ching hexagrams with their names, arranged by lower trigrams in the “Pre-heavenly” order (the same as the Fuxi/Shaoyong chart). The second ring inward are the nuclear hexagrams for the outer ring, formed from the inner four lines of the original hexagrams. The third ring repeats the process, in each case yielding one of the four “core” hexagrams.

Book Review: Yijing, Shamanic Oracle

Yijing, Shamanic Oracle of China: A New Book of Change

Richard Bertschinger, translation and commentary
Singing Dragon 2012
Hardcover/ebook, 336 pp.
US $30 / £25.00
ISBN: 978-1-84819-083-2

Yijing, Shamanic Oracle of China is a nicely presented contemplative book. Richard Bertschinger, an acupuncturist and student of  Gia-fu Feng, has done an engaging job of presenting his thoughts about the Yijing. With occasional cross-cultural philosophical comparisons, Bertschinger presents a translation of the core Zhouyi and selected Wings, enhanced with his comments, and including a helpful glossary.

“The Yijing can reveal the seeds of the future, but above all it teaches reverence for those small inklings in the present which determine the science of change. It encourages us all to be alive to the world, and take a stand on destiny.”

Bertschinger’s translation emphasizes finding one’s path. “You can transform those observing below, by observing the causes of your own life. Observe yourself within and contemplate the evens of your own life. At no time neglect to seek the middle path and at all times act properly” (from Hexagram 20).

This is a very accessible Yijing book, drawn from the author’s reading of traditional Chinese texts, accessible to beginners, but meaty enough for experienced users.

Oracle Bones Crowd-sourced

Chinese scholars are asking for help with deciphering thousands of oracle bones, and are offering rewards:
“The National Museum of Chinese Writing in Anyang, Henan province, has issued a worldwide appeal for help to decipher thousands of esoteric characters cut into bones and shells dating back more than 3,000 years to the Shang dynasty.
The inscriptions, resembling modern characters, are the earliest written records of Chin­ese civilisation and shed light on life and society at a time.
They were carved by fortune-tellers on turtle shells and ox shoulder blades known as oracle bones, and record questions on everything from weather to taxes.
So far, scholars have managed to crack the code to less than half of the roughly 5,000 characters found on excavated oracle bones. Around 3,000 of them remain a mystery.” Read the whole article here.