Red Sky

 A weirdly mottled red sky over Crewe this morning. 

I went out specially to catch the image before it faded.

On Sacred Geometry

 On Sacred Geometry

I've been looking at various YouTube sites that have videos on "Sacred Geometry". Some are completely vacuous waffle to me. However a few do contain genuine arithmetic and geometric results that seem of interest from a mathematical viewpoint. In particular there is a series from the "Jain Academy" fronted by an affable Australian lecturer who calls himself "Jain108". His mathematics is mostly correct, except where he obsesses about the "true" value of pi being 3.144... The Jain Academy deals in what I can only call "New Age Eclecticism". In other words it takes bits from everywhere, Hinduism, Kabbala, Christianity, Islam, Buddhism, Astrology, and so on, regardless of dogma. 

The number 108 is apparently ubiquitous in Hindu mysticism. It is said in several sources to be the ratio of distance to diameter for both the Sun and the Moon. For any celestial body this ratio would be near enough the cotangent or cosecant of its apparent angular diameter. The angle whose cotangent or cosecant is 108 turns out to be 31 minutes and 50 seconds to the nearest second. In Patrick Moore's "Atlas of the Universe" (1994), which I happen to have to hand, the mean apparent diameter of the Sun is given as 32' 1" and the mean apparent diameter of the Moon as 31' 6". So the number 108 seems to be a reasonably good estimate. The closeness of the apparent diameters of Moon and Sun is of course why Solar Eclipses can be so spectacular.

Another obsession of the Sacred Geometers is, as might be expected, the dimensions of the Great Pyramid of Khufu at Giza. The Jain Academy makes much of the triangle formed by the pyramid as seen at a distance from a side. Taking its base width to be 2 units it is claimed that its slope length is the golden ratio 1.618 and its height is the square root of the golden ratio. According to Wikipedia the pyramid's original height was 280 cubits and its base 440 cubits. The slope length is thus the square root of (280^2 + 220^2) = root 126800 = 356.09 cubits. The ratio of slope to half base is thus 356.09/220 = 1.61859, and the golden ratio is 1.6180339. So again a plausible approximation. The angle of slope is 51 degrees 5 minutes so the visible angle at the summit would be 77 degrees 50 minutes.

When people make a model of a pyramid they nearly always take the faces to be exact equilateral triangles, but this gives a pyramid whose slope height is half of root three and whose height is half of root two. If the slope height of the pyramid is the golden ratio then this must be the altitude of the triangular face shape. The base angle then works out at 58 degrees 17 minutes and the apex angle as 63 degrees 23 minutes Differing from the equilateral by 3.6 degrees at the apex. 

The symbol used for the golden ratio in Sacred Geometry is phi (though mathematical texts often use tau). The claim about the new value for pi is that is should be 4/(root phi) = 3.144605...as opposed to 3.14159... Or equivalently that 4/pi = 1.2732395... should be root phi = 1.2720196... Whether this means that the Laws of Nature themselves are going to change when the New Age dawns, or only human consciousness of them is not at all clear.