# Mathematical Material

The material linked to on this page is of a fairly technical nature and generally assumes some background knowledge of the Yijing. Some of it is quite mathematical. However, the necessary concepts are fully explained.  The papers are divided into two broad groups:

Ideally, you should probably start with the paper describing the basic formal framework before moving on to read the applications papers.

## Basic Framework

This paper describes the formal logical background to much of the rest of my work.

Boolean Algebra and the Yijing - this provides the necessary background to understand the mathematical techniques used in analysing the structures of the Yi. It takes a formal look at Cleary's Structural Complement and Primal Correlate relations and also provides the algebraic characterization of changing lines that underpins a lot of my subsequent work. An earlier version of this paper originally appeared in The Oracle, The Journal of Yijing Studies, in Volume 2, Number 7 (Summer 1998), pp19-34.

The papers in this section use the formal techniques discussed above to analyse some of the traditional and contemporary structural notions found in the Yijing.

Correctness and Correspondence - this is a follow-on paper to "Boolean Algebra and the Yijing". It applies the mathematical tools introduced in the first paper to provide an analysis of the two traditional ideas of correctness and correspondence. This paper originally appeared in The Oracle, The Journal of Yijing Studies, in Volume 2, Number 8 (February 1999), pp25-37.

## Contemporary Explorations

Flowers and Steps in the Boolean Lattice of Hexagrams - this provides a detailed analysis of Karcher's Steps of Change, Secter's Transition Hexagrams and Change Hexagrams, and Hacker's Hexagram Flowers. It shows how all of these techniques can be understood as special cases of the Boolean lattice structure of hexagrams. This, in turn, is taken to provide further evidence of the utility of an algebraic approach to the structure of the symbols of the Yi Jing.  This article was first published in the Journal of Chinese Philosophy, Volume 31, Number 4, December 2004, pp489-504. © 2004 Journal of Chinese Philosophy. Readers interested in following this work further should refer to the web note showing a worked example of a cubic subspace which includes some illustrative animations.

The Yijing as a Symbolic Language for Abstraction - this paper, which originally appeared in a shorter form in the Proceedings of the 2nd International Conference on I-Ching (Yijing) Studies and Contemporary Civilization, 2005, pp291-305, shows how a formal description of linked chains of substructures within the Boolean lattice can be used to develop a theory of individual perspectives. This theory shows how local perspectives exist in the global context of totality and further, how the focus of the individual determines the scope of their perspective. The following additional material is available:

The Yijing as a Mathematical Metaphysics - this is the full version of the paper which originally appeared in a much shorter form under the title "The Yijing: Metaphysics and Physics" the Proceedings of the 13th I-Ching World Conference, 2010, pp686-695. Another version, intermediate in length, subsequently appeared in the Journal of Chinese Philosophy, Volume 38, Number 3, September 2011, pp412-426. Starting from a discussion of the relationship between mathematics and reality, it discusses the significance of notation in shaping our understanding. It then shows a connection between the traditional philosophical concept wuji er taiji 無極而太極 and the quantum phenomena of vacuum polarization. It then goes on to discuss the connection between the traditional concept of the san cai 三才 and Bohm's ideas of the implicate and explicate order and the role of consciousness. The article concludes with a discussion of divination. The following additional material is also available:

Bipolar Change - following from a conversation with Alain Stalder, this article extends the basic algebraic analysis of change, clarifying the relation between change as a yang energy and change as a yin energy. Extensive examples show how the technique can be applied to represent change through the lattice structure, change in correctness and change of resonance. A parallel is also drawn with the quantum theoretical phenomenon of vacuum polarization. A slightly shorter version of this article appeared in the Journal of Chinese Philosophy Volume 35, Number 2 (June 2008), pp297-317.

Tools for Change - this is jointly authored with Stephen Karcher, and applies the mathematical tools introduced in the Boolean Algebra paper to some of Karcher's techniques of interpretation, showing how this can result in a rich set of images for a divination. It includes a formal analysis of the nuclear hexagram construction and Karcher's Steps of Change technique. This paper originally appeared in The Oracle, The Journal of Yijing Studies, in Volume 2, Number 12 (January 2002), pp20-33.

## Diagrams

Most of the following diagrams are high-resolution images, suitable for full page printing. Note that these diagrams open in a new window. It will probaly help to make sense of the unfoldings in particular if you have read the "Yijing as a Symbolic Langauge for Abstraction" paper above.

The Whole Lattice - this diagrams shows the complete six dimensional lattice structure generated by the Boolean algebra.

Explicate Matter Cubic Unfolding - this diagram shows how the lattice can be unfolded into a chain of three dimensional subspaces, based on the alternations of the lower trigram.

Heart and Mind Planar Unfolding - this diagram shows how the lattice can be unfolded into a chain of two dimensional subspaces, based on the alternations of the middle bigram.

If you want a set of images for displaying all the different individual gua, you may download and use the set that I created for this web site. They are zipped into a single file here.