*Forma,* Vol. 19 (No. 4), pp. 315-334, 2004

*Original Paper*

## Fibonacci Form and Beyond

Louis H. Kauffman

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607-7045, U.S.A.

E-mail address: kauffman@uic.edu

(Received February 25, 2005; Accepted March 15, 2005)

**Keywords: **
Fibonacci, Fibonacci Form, Reentry, Eigenform, Recursive Form, Golden Ratio, Golden Rectangle, Marked State, Unmarked State, Mark, Fibonacci Particles

**Abstract. **
This paper develops a context for the well-known Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) in terms of self-referential forms and a basis for mathematics in terms of distinctions that is harmonious with G. Spencer-Brown's Laws of Form and Heinz von Foerster's notion of an eigenform. The paper begins with a new characterization of the infinite decomposition of a rectangle into squares that is characteristic of the golden rectangle. The paper discusses key reentry forms that include the Fibonacci form, and the paper ends with a discussion of the structure of the "Fibonacci anyons" a bit of mathematical physics that relates to the quantum theory of the self-interaction of the marked state of a distinction.

[Full text] (PDF 804 KB)