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).

# Category: Geometry

Sacred numbers within geometrical forms

## The Crop Circle Phenomenon at Crooked Soley

#### Archive: 26 November 2017

This article explores another modern and highly transient manifestation belonging to the category of **Landforms**, the crop circle. Thought in seventeenth century England to be the work of a mowing devil, and more recently hoaxers, the large crop fields created since the post-war tractor revolution play host to “sacred art” designs that farmers find intrusive, attracting members of the public (to be a-mazed) and aviators to curate these patterns for posterity. The tabloids used to indulge full page colour sections to crop circles but then were dissuaded perhaps by officialdom (we are mad) and apathy (we lose interest) since the cause of the finest examples, such as Crooked Soley, do not appear humanly possible.

## 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).

Continue reading “The Golden Mean compared to PI”## Use of foot ratios in Megalithic Astronomy

The ratios of ancient metrology emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.

The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin Heath) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.

Continue reading “Use of foot ratios in Megalithic Astronomy”## Old Yard’s Mastery of the Square Root of 2

The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules.

So why was this foot and its yard important, in the Scottish megalithic and in later, historical monuments?

If one forms a square with side equal to the old yard, that square can be seen as containing 9 square feet, and each of those has side length 99/100 feet. This can be multiplied by the rough approximation to 1/√ 2 of 5/7 = 0.714285, to obtain a more accurate 1/√ 2 of 99/140 = 0.70714285.

Continue reading “Old Yard’s Mastery of the Square Root of 2”## Models of Time within Henges and Circles

Presenting important information clearly often requires the context be shown, within a greater whole. Map makers often provide an inset, showing a larger map at a ** smaller scaling** (as below, of South America) within a detailed map (of Southern Mexico).

Megalithic astronomy generated maps of time periods, using lines, triangles, diameters and perimeters, in which units of measure represented one day to an inch or to a foot. To quantify these periods, **alignments **on the horizon pointing to sun and moon events were combined with **time counting** between these events,where days, accumulated as feet or inches per day, form a counted length. When one period was much longer than another, the shorter could be counted in feet per day and the smaller in inches per so that both counts could share the same monumental space. In this article we find the culture leading to megalithic astronomy and *stone *circles, previously building circular structures called **henges**, made of concentric banks and ditches.