# Eigenform

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