Venus has played a strong role in mankind’s imagination, being a bright object in the sky in the evening sky and then the morning sky, whilst also viewed as the primary female goddess of the Ancient Near East. To recent astronomers, she is covered in impenetrable clouds, whilst the invention of radar revealed a rocky sister planet to Earth but with no life as we know it. It is perennially associated with the pentagon, because its synodicThe recurring time cycle of a given celestial phenomenon seen from the Earth. periods draw out a pentagon within the zodiac in 8 solar years. The reasons it does so are intriguing to say the least, and we explore the unusual numerical characteristics of Venus seen from Earth.

(adapted from a 1994 text, using 2020 hindsight)

## Part 1: A Nearly Golden Mean

The Golden Mean or Golden Section is a most
extraordinary number approximately 1.618 in value. If you square it you get 2.618:
that is the square is the number itself plus one. There must be and can only be
one number that has this characteristic, and it may occur between any two
lengths, areas, volumes or *celestial time periods*. If you make a
reciprocal of the Golden Mean you get 0.618: that is the reciprocal is the
number minus 1. This behaviour makes equations where the algebra get very
simple. This is clear in the fact that the square of 0.618 is 0.382: that is
the square or the reciprocal is equal to one minus the reciprocal. In fact,
given the previous relations, this relation has to be true.

So what has the Golden Mean got to do with Venus? The answer is in its sidereal period. Venus orbits the Sun in 0.615 years, a value close to the Golden Mean as 8/13. This value is of course only related to the Earth, because it is in Earth years.

The sidereal period of 0.615 years (224.701
days) automatically makes its synodic period 1.618 years (583.92 days) because its
synodic period is based upon both the sidereal period and the solar year. The
0.615 value is interesting because it is a usable Fibonacci approximation to
the Golden Mean, available through the fraction 8/13. If the Venus orbit is 8/13^{ths}
of a year, then the remaining fraction of a year is 5/13^{ths}. If this
is divided into 8/13ths then the thirteens cancel out and 8/5^{ths}
remains, i.e. 1.6 years, which is the Venus synodic period.

This synodic period is the time taken for Venus to be in the same relation to the Sun, which it clearly orbits. Thus, if the maximum Morning Star phenomenon occurs at one time, then it will recur after 1.6 years when the Sun has travelled one and three fifths times around the zodiac. This will appear as two fifths further west (in the sky-ecliptic) to a real observer on the surface of the Earth. Whilst proximity to the Golden Mean appears unlikely, the orbit of Venus comes under the laws of integer residues in the context of orbits that are re-entrant (re-enter themselves). just The Fibonacci series of numbers {1 1 2 3 5 8 13 …} are all integer and it is remarkable that Venus, in her synodic period and sidereal orbit, are a living demonstration of the earliest part of the Fibonnacci series where good approximations of Phi (adjacent members) and Phi squared (next-door-but-one ) appear, namely in the range {5 8 13}. These three numbers enable the sidereal 8/13 to be the reciprocal of 8/5.