Chapters 5 through 13 in Iconic Arithmetic Volume I, as well as much of the content in Volumes II and III, explore James algebra.

This Boundary Algebra was first developed between 1991 and 1993 by Jeffrey James and William Bricken at the University of Washington Human Interface Technology Lab. The result of Jeffrey’s research was published as his University of Washington 1993 Masters of Science in Engineering thesis, *A Calculus of Number Based on Spatial Forms*. pdf

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JAMES, Jeffrey M., 1993. A calculus of number based on spatial forms. MS thesis. University of Washington. pdf

The abstract from Jeffrey’s thesis provides an excellent summary of the original conception of James algebra.

A calculus for writing and transforming numbers is defined. The calculus is based on a representational and computational paradigm, called Boundary Mathematics, in which representation consists of making distinctions out of the void. The calculus uses three boundary objects to create numbers and covers complex numbers and basic transcendentals. These same objects compose into operations on these numbers. Expressions transform using three spatial match and substitute rules that work in parallel across expressions. From the calculus emerge generalized forms of cardinality and inverse that apply identically to addition and multiplication. An imaginary form in the calculus expresses numbers in phase space, creating complex numbers. The calculus attempts to represent computational constraints explicitly, thereby improving our ability to design computational machinery and mathematical interfaces. Applications of the calculus to computational and educational domains are discussed.