Abstract:
Rational numbers, which correctly describe many recognizable patterns
in the physical world, are often seen to converge in the process to irrational limits
or even singularities. As a common example, atomic numbers are well known
as fundamental parameters in chemistry, but by demonstrating that the periodicity
of atomic matter is simulated by the convergence of rational fractions, from unity
to the golden ratio, the importance of limiting processes and irrational limits in
the modelling of chemical systems and of phenomena such as superconduction is
emphasized. Other limiting formulae feature in atomic spectral series, radioactive
decay, circular measure, absolute temperature, the speed of light, structure of the
solar system and gravitational collapse. In virtually all cases the convergence involves
the irrational golden ratio and the golden spiral, the essential properties of
which are briefly reviewed in summary of the arguments developed in this volume.
The suspicion that molecular shape should have a related number basis could not be
substantiated. Only in the double-helical base pairing of DNA could any correlation
between molecular structure and number theory be demonstrated. It is tempting to
conjecture that the ubiquitous appearance of irrational limits signals the inadequacy
of the R3 number system to provide a detailed account of the four-dimensional
world.
