21st CENTURY

of the Platonic solids. If universal cause-effect action is representable
as multiply connected circular action, all action
in visual space is fundamentally underlaid by this form of
physical least action .Hence, the self-boundedness of visual
space, as shown by the Platonic solids, must be a constructive
"property" of universal least action of this form. Hence,
the golden section of least action, a construction itself derivable
from nothing but this form of least action, is a sufficient
demonstration of the necessary characteristic of the
self-boundedness of visual space.
The most famous immediate application of this result, by
both Pacioli and Leonardo as collaborators, was their definition
of the form of life: All living processes are distinguished
from ordinary nonliving ones, in respect to morphology
of growth and function, in the respect that that
form is ordered as an harmonic series consistent with the
harmonic series defined by the golden section.
Today, we qualify that discovery. Between the limits of
the very, very large (astrophysics), and of the very, very
small (microphysics), any process which is harmonically
ordered in congruence with the golden section is either a
living process, or is a special class of work done by a living
process. Kepler, who based his founding of a comprehensive
mathematical physics chiefly upon the combined work
Figure 5
LIVING PRC CESSES, PENTAGONAL SYMMETRY,
M D THE GOLDEN SECTION
Living processes, c lant and animal, are characterized by pentagonal
symmetry, while inorganic matter (like the snowflake
shownj is characte ized by hexagonal symmetry (a). As Leonardo
and Pacioli d >monstrated, living processes grow according
to the golden s action ratio. That is, a line will have golden
section proportion when it is divided into two parts such that
the ratio of the firs to the second part is equal to the ratio of
the second part to i um of the two parts.
The pentagon is closely related to the golden section, and
one example ofthii is shown in (b). Beginning with an isoce/es
triangle whose two iqual sides are each 1, and whose two equal
angles are 36°, the longer side will be the golden mean, approximately
1.618, i value designated by the Creek letter phi,
. Identical triang es can be added to form a pentagon, as
shown. Thus a pen agon with sides equal to 1 will have diagonals
equal to the go den mean. The 36" angle is generated when
a circle is divided ir to 10 sections by inscribing a decagon.
of Cusa and P icioli-Leonardo, was the first to prove that the
universe as i whole is governed by the same harmonic
ordering. Soi ie leading scientists among the writer's collaborators
art proving that a Gauss-Riemann correction for
Keplerian law s of astrophysics also rules on the scale of
organization }f atoms and smaller scales of physics. With
that qualifica ion, Pacioli's, Leonardo's, and Kepler's geometrical
(leas -action) definition of living processes is conclusively
dem >nstrated today to be fully as accurate as Pacioli
represented t his to be at the beginning of the 16th century.
Thus, all of the movements and related functions of the
human physi )logy are harmonically ordered least-actionbased
moverr ents of this sort.
This standp oint governed several aspects of the work of
Leonardo. In matomy, he explored the golden section harmonics
of th« physiology of persons, horses, birds, and so
on. In pioneering the principles of design of machinery,
and the desij n and use of weapons, the same principles
predominate I. He revolutionized the science of perspective
by emph; sis upon anomalies of visual space associated
with the perif fiery of vision, rather than an Albertian, linear
vanishing-po nt. This we note in viewing the originals of
such master \ /orks of Raphael as the famous murals in the
papal apartrri' ;nts and the "Transfiguration" in the Vatican.
21st CEN "URY November-December 1988 35