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)
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Use of Ad-Quadratum at Angkor Wat

The large temple complex of Angkor Wat ( photo: Chris Junker at flickr, CC BY-NC-ND 2.0 )

Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root
of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).

The diagonal of a square of unit size is sqrt(2), When a square is nested to just touch a larger square’s opposite sides, one can know the squares differ by sqrt(2)
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Form implied by the Kaaba’s Walls

The Kaaba appears to express a geometrical progression of adjacent odd numbers starting with one and three. This differs from the super-particular ratios found within the right triangles of astronomical time periods formed by the Megalithic, in which the ratio pairs separated by only one rather than by two, between only odd numbers. However, the multiple-square rectangles used by the megalithic to approximation celestial ratios, made use of the three-square rectangle. In one of the smallest of these rectangles, it simultaneously approximates two pairs of ratios: The eclipse year (346.62 days) to the solar year (365.2422 days) and the solar year to the thirteen lunar month year (384 days).


Figure 1 The three-square approximations in a triple series of astronomical periods. Note that the diagonals relative to the base are the result of having three squares in a rectangle then one high and three along – and two different. Two such rectangles geometrically sum (their angles) to give that of the (First Pythagorean) 3-4-5 triangle, 36.8 degrees.
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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

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Politics of Number part 2

The YouTube talk below, from 2012, discusses how numbers are more significant in understanding past and present societies, through the medium of a museum reproduction I made of an object to be seen in the Heraklion Museum. It records a Saturnian calendar that must have existed in the matriarchal societies of the Minoans in Crete and might be a sort of tyranny of time which Zeus-Jupiter (son of Cronos i.e. Saturn) deposed, relegating his father to a golden cave on a smaller island.

This story of Hesiod’s Theogony exactly pollows the harmonic truth that Saturn has a synodic period 16/15 (a semitone) of the lunar year whilst Jupiter’s synod is 9/8 (a whole tone) relative to the lunar year – to very high accuracy. The politics in this is the advent of Indo-Europeans from the North whose patriarchal social structures largely displaced matriarchal structures of the eastern Mediterranean.

Whilst ancient cultures were rooted in stories where numbers were structural rather than merely descriptive, connected to the sky and time as a spiritual superstructure for life on earth, the recent material culture has lost touch with this meaning given to time and numbers as archetypal.