Precession and Root Two Geometry

There is an interesting but hard to grasp relationship between the precession of the Earth over around 26,000 years and the natural geometry of the square. This means there are possibilities for representing precession numerically and symbolically using the quadrature squares commonly used in the medieval period and before, into prehistory.

Precession is the movement of the Earth's poles relative to the polar stars. Polaris is only an occasional pole star that punctuates this round. The axial movement has the effect on the celestial equator that where it crosses the Sun's yearly path, the line of the ecliptic, changes over this period. These two crossing points are the places where the Sun lies at the spring and autumn equinoxes, then day equals night all over the Earth, whatever the latitude.

The movement of the whole frame over one cycle has also been called the Great Year, because it echoes on a grand scale the phenomenon of the year as a cycle within a cycle.

The physical causes of precession are that the equatorial bulge of the spinning Earth rises above and below the plane of the solar system (i.e. the ecliptic again). This pulls the Earth in a way that would cause circle of the equator to become the same as the plane of ecliptic. However, the earth is spinning and no spinning object can be directly made to change its axis of rotation without causing an equal force at right angles - a force which then drives the endless circulation of precession than then never normalises the Earth to the solar system plane.

However, the axis of the Earth would not be stable without the presence of the Moon which tends to damp the wilder oscillation found in similar planets without a large moon.

That being the briefest background, the actual time taken by precession is not exactly known because there are many factors that will make each round slightly different. Two estimates are 25,725 years (fixed ecliptic) and 25,784 years (moving ecliptic).

In the 1990's I was working in days and not degrees, as a way of normalising all such matters to time viewed from Earth. In this a DAY in angle is the angle the Sun moves in a day, which most books cite as being "nearly a degree per (tropical) day", the degree being lesser because there are 360 degrees in a circle versus 365.242 DAYs in a "year circle".

There was a repeated tendancy to use year circles and squares in megalithic times, where the units of measure seem to incorporate the number of days in a year around the perimeter. One could argue that there is no more natural way to divide a circle when it represents the sun's motion in particular.

Taking 25800 years for precession, it is significant to divide this by 365.242 so as to find out how many years it takes for the equinoctal sun to move by one DAY on the ecliptic. The result is 70.638097 years per DAY. I noticed that this was very close to the reciprocal of the square root of two. This inference leads to an interesting way of viewing the numerocity of precession relative to the solar year (where numerocity involves focussing on the numerical content independant of the powers of ten involved).

Like most significant irrationals the square root of two has important powers. Firstly, it allows the unit right angled triangle to have a hypotenuse root two long, thus expressing the relationship between two dimensions at right angles. Secondly, the reciprocal of root two multiplied by two becomes root two, and this leads to the quadrature phenomenon in which squares can be doubled by inscribing successive squares within squares, the intermediates being squares involving root two.

This behaviour is noted in Honnecourt's notebook, one of the great architects of the Gothic style as below [from Patterns of Thought: The Hidden Meaning of the Westminster Pavement by Richard Foster, Cape, London, 1991]:

 

 One can see the inscribed squares-within-squares being put to good use in the construction of a sacred monumental form.

What precession indicates is that

• 100 times 365.242 / root two = 25826.5  and therefore,
• 50 times 365.242 times root two = the same
  

and this second version can be shown visually as

 

It is interesting that the Great Pyramid seen by Piazzi Smythe had 36524 "primitive inches" around it. Though such a verdict was based on inaccurate data the idea was valid, the Aroura was an established concept in ancient buildings, where the perimeter was expressing, in some way the perimeter. A similar relationship is however found within the Sarsen ring of Stonehenge, as a circular form [Sun Moon and Stonehenge, Proof of High Culture in Great Britain by Robin Heath, Bluestone Press, Cardigan, 1998].

 

However, it is the square form (as one might achieve in a pyramid base) that appears to combine with the idea of the solar year perimeter to naturally generate the accurate root two approximation to precession, both as 25826 years and as 70.71 years per DAY of movement of the equinoctal point.

It is unusual that this should be the case. One suggestion I came up with about ten years ago was that the pull on the equatorial bulge was being rectified since when the Sun or the Moon are on the equinoctal point, they exert no precessional force: thus the force itself varies, over these different periodicities, in a sinusoidal way. The power transmitted by a sinusoidal force is less than its peak value and the usual formula is to equate this to an average (Root Mean Squared) equivalent force, its peak value divided by root two.

Even if this were true, then there would still be a difficulty in constructing a Sun Moon Earth system that could normalise the resulting precessional period to being numerically similar to the solar year. but I guess that is where the mysteries with these things really lies. Our culture says they cannot be designed, they have to be coming together with only general laws being satisfied and not specific, new ideas entering such as the creation of the world according to some intent or meaning.

This is not the only such phenomenon like this that I have found. Matrix of Creation was quite full of them and also, an early paper is available (SYNCHRONICITY OF THE EARTH’S ROTATION WITH MOON AND SUN) showing that the Moon's nodes display behaviour uniquely normalised to the numerical nature of the year and hence to the rotation of the Earth. 

Update on Thursday, January 24, 2008 at 03:41PM by Richard Heath

I did some more work on this and eventually got back into being able to make a pdf by replacing my word processor with www.openoffice.org Writer - so here it is.

The Precessional Design of the Earth by Richard Heath