Fibonacci in Jupiter’s 12-fold Heaven

The Fibonacci series is an ideal pattern, widely found within living systems, in which the present magnitude or location of something is the product of two previous magnitudes or locations of it. The next magnitude will again be the sum of the last two magnitudes in what is, an algorithmic pattern producing approximation to the Golden Mean (designated by the Greek letter φ,’phi’). As the series gets larger, the ratio (or proportion) between successive magnitudes will better approximate the irrational value of φ = 1.618033 … – which has an unlimited fractional part whilst the virtue of the Fibonacci numbers within the Series is that they are integers forming rational fractions.

Jupiter taken by the Wide Field Hubble Telescope by NASA, ESA, and A. Simon (Goddard Space Flight Center)
Continue reading “Fibonacci in Jupiter’s 12-fold Heaven”

The Golden Mean compared to PI

In reviewing some ancient notes of mine, I came across an interesting comparison between the Golden Mean (Phi) and PI. They are more interesting in reverse:

A phi square (area: 2.618, side: 1.618) has grown in area relative to a unit square by the amount (area: 0.618) plus the rectangle (area:1 ). This reveals the role of phi’s reciprocal square (area: 0.384) in being the reciprocal of the reciprocal so that in product they return the unity (area: 1).

On the right, the phi squared square showing how the reciprocal of phi and its square uniquely sum to unity (area: 1), a property that is scale invariant between structures who share the same units and grow according to the Golden Mean.
Continue reading “The Golden Mean compared to PI”

Venus: Planet of Harmony: part 1

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 synodic 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)

The Venus cycle of eight years in which a morning or evening star has five manifestations, dividing the zodiac into a pentagram (Figure 1.6 Sacred Number and the Origins of Civilization)

Part 1: A Nearly Golden Mean

Continue reading “Venus: Planet of Harmony: part 1”