# Eigenform

Louis H. Kauffman

Mathematics Department, Unversity of
Illinois at Chicago, Chicago, Illinois 60607-7045

## Abstract

This talk will review and extend the concept of eigenform due
originally to Heinz von Foerster. An eigenform is a fixed point
of a transformation or collection of Transformations. Such a
fixed point may or may not exist in the original domain in which
these transformations are defined. For example, the mapping F(x)
= -1/x has no fixed points in the real numbers, but F(i) = i when
i denotes the complex square root of negative one. Another
example is G(x) = <x> where this denotes the formal
bracketing of x. Then J = <<<<...>>>> (an
infinite nest of brackets) is a solution to the equation G(J) =
J. The concept of eigeform is closely related to cybernetic
notions of objects and observers and their interplay, to
incompleteness of formal systems and attendant paradoxes, to the
patterns of quantum mechanics and generalizations of quantum
mechanics to systems including observers and participators. This
talk will cover these relationships and raise more questions than
it can answer.

*Keywords*: eigenform, fixed point, recursion, object,
observer, cybernetics