  A not-too-technical version of the paper: "A Granular Permutation-based
Representation of Complex Numbers and Quaternions: Elements of a Realistic
Quantum Theory" - Proc. Roy. Soc.A (2004) 460, 1039-1055. The phrase
"meteorological butterfly effect" is introduced to illustrate, not the familiar
loss of predictability in low-dimensional chaos, but the much less familiar and
much more radical paradigm of the finite-time predictability horizon,
associated with upscale transfer of uncertainty in certain multi-scale systems.
This motivates a novel reinterpretation of unit complex numbers (and
quaternions) in terms of a family of self-similar permutation operators. A
realistic deterministic kinematic reformulation of the foundations of quantum
theory is given using this reinterpretation of complex numbers. Using a
property of the cosine function not normally encountered in physics, that it is
irrational for all dyadic rational angles between 0 and pi/2, this
reformulation is shown to have the emergent property of counterfactual
indefiniteness and is therefore not non-locally causal.

Quantum Physics