Tag: equal perimeters

Easter Aquhorthies

Easter Aquhorthies (i.e. apocathery) has eleven stones in a circle and in between the two south-to-south-west stones is a large (bridging) recumbent stone, more commonly found in Scottish circles  and associated (by Alexander Thom) to lunar observatories because, in Scotland at lunar maximum standstill, the moon can rest upon or be hidden by a raised horizon.

Picture by krautrock, a member of megalithic.co.uk in June 2010.

Figure 1 Alexander Thom’s site plan, with cardinal directions and highlighting the diameter .

It is tempting to assume geometry within stone circles and this one invites that by having eleven regularly placed stones,. However, 11 is rarely found in regular geometries or stone circles. But,

  1. Eleven is associated (geometrically) with the simplest approximate ratio for pi, of 22/7 and, being on a circle perimeter 22 units on the perimeter means 7 as the diameter. However, the diameter here is 7 x 9 = 63 so that 7 units of 9 feet in diameter translates into 22 x 9 = 198 feet on the circumference.
  2. The second symbolism of eleven is found in the solar time cycles of 33 years which traditionally references as the Solar Hero cycle as the age when heroes die. 33 years was signified in this way because of an exact alignment, at which Equinox sunrise should repeat on the distant horizon every 33 years. While the solar year is 365 whole days long it takes an extra 32/132 of a day for the Earth’s orbit to complete, and this fraction (of 32/132) depends upon 4 x 33 = 132 losing its 33, 33 years after an earlier Equinox on the horizon. (Note that one did not need to know the data and then calculate, as we do today: instead, observation of in-exact solar risings or settings could notice near exact repetitions occurring after 8 years, but more perfectly after 33 years.

Returning to the circle, the circumference of this stone circle is 198 feet which is 6 x 33 = 198 so that the 11 stones may be referencing 33 solar years.

The 33-year period is strangely linked to the movement of the moon’s nodes, which take 18.618 years (of the nodal period) to progress once around the path of the sun (ecliptic). The area under this precession of the lunar nodes (pi x 18.618^2 = 1,088.970022752789) relative to the earth it orbits, which closely equals the area of a square of side 33 x 33 = 1,089.

The megalithic and ancient world both made monuments conforming to the equal area and equal perimeter geometries and even though the equal area situation is proven unsolvable, one sees that (before the notion of an analytic solution) a perfect example was found of a workable pair of numbers (33 years and 18.618 years) was available as these two very significant astronomical time periods.

If the circle is 63 feet in diameter, then one divided by 14 and multiplied by 11 to obtains the square of equal perimeter: 63 x 11/14 = 49.5, which is 99/2 feet on the side. This can then establish the radius of the circle as 18.618 of the same units: When divided by 33, 99/2 becomes 3/2 which is the unit for the radius of the circle of equal namely 18.618 x 3/2 = 27.927 or nearly 28 feet. Below is a picture showing the square and the circle, equal in area in red while the perimeter of the stone circle is equal in perimeter to the red square.

One can see that the red circle embraces the recumbent stone
and its altar-like appendages.

Modularity of Seven

Sacred Geometry is metrical, it is based upon the interactive properties of “natural” (that is, whole) numbers and cosmic constants.

We live in a civilization where everything is thought to be functionally due to forces and laws, these all calculated using numbers and algebra. For this reason, it is hard to see the influence of numbers acting directly in situations to reveal that, geometrical forms are only possible due to numbers. One such form is the equal perimeter circle and square: this figuring heavily in my later books, as an ancient model, and in postings on this website (opens in new tab).

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Design of the Taj Mahal: its Façade

The Taj Mahal is one of the most recognizable buildings on earth. It was built by a Moghul king as a memorial for his dead queen and for love itself. The Mughals became famous for their architecture and the Persian notion of the sacred garden though their roots were in Central Asia just north of Persia.

I had been working on Angkor Wat, for my soon to be released book: Sacred Geometry in Ancient Goddess Cultures, where the dominant form of its three inner boundary walls (surrounding the inner sanctum) were in the rectangular ratio of outer walls of six to five. A little later I came across a BBC program about the Mughals and construction of the Taj by a late Moghul ruler, indicating how this style almost certainly arose due the Central Asian influences and amongst these the Samanids and the Kwajaghan (meaning “Masters of Wisdom”). I had also been working on the facades of two major Gothic Cathedrals (see post), and when the dimensions of the façade of the Taj Mahal was established, it too had dimensions six to five. An online pdf document decoding the Taj Mahal, established the likely unit of measure as the Gaz of 8/3 feet (a step of 2.5 feet of 16/15 English feet; the Persepolitan root foot *(see below: John Neal. 2017. 81-82 ). Here, the façade is 84 by 70 gaz.

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The Moon is Key to our Survival

With the advent of many orbital missions, the Moon is threatened with orbital and other changes due to space travel.

The modern theory of relativity has joined the worlds of space and time, now called spacetime. As beings we live in space while moving through time, and both these are assumed to be neutral dimensions having mere extension. However, spacetime is distorted by the massive gravitational objects found in solar systems, these exerting an attractive force on all objects including ourselves. As humans, we are therefore locked onto the surface of the earth by gravity, viewing a solar system of eight orbiting planets seen in the sky from the surface of the third planet from the sun. The earth has an unusually large moon which has fallen into resonance with the planets, a resonance then belonging to time.

The Moon was formed 4 to 5 billion years ago and this affected the Earth’s geology, stabilized its tilt (giving stability to the seasons) and providing tidal reaches on coasts. But apart from such direct physical changes to the earth, the moon has now developed resonances with the solar system, especially its outer giant planets, and this has given time on earth a highly specific resonant environment, based upon the lunar month and year of 12 lunar months. This resonant network appears to be numerical when counting days, months and years, in between significant events in the sky.

The structure of time is numerical because of these resonances between the moon, the sun and the planets. This resonance came to be known by previous civilizations and was thought meaningful in explaining how the world was created. Time was deemed spiritual because its organisation allowed human beings to understand the purpose of life and of the earth through the structure of time. In particular, the moon was a key to unlocking the time world as a link to a higher or spiritual world, a literal sky heaven organised according to numbers.

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A House that is your Home

This graphic demonstrates how the inner geometry within numbers can point to significant aspects of Celestial Time or here Space regarding the relative sizes of the Earth and the Moon, namely 11 to 3 according to pi as 22/7.

In some ways one cannot understand numbers without giving them some kind of concrete form as with seeing them as a number of identical units. Sixteen units can make a square of side 4 since the square root of 16 is 4 and 6 is factorial 3 (3! = 1 x 2 x 3 and 1 + 2 + 3) which is triangular, so together they make 22, and if the triangle to placed on top of the square, like a house and its roof, then the house is 7 tall. If you want an accurate approximation to pi of 3.14159 … (pi is transcendental), the 22/7 is good and the house defines it.

This adds another mystery to this form of pi often used in the ancient world where numbers were best handled as whole numbers and ratios of these. This pi allows a circle of diameter 11 to be set within a square of side 11, whose perimeter is then 44. This can be seen in the diagram as made up of 16 yellow squares and 6 blue ones, centered on the circle and making 22 squares in all.

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The Metonic Period at Ushtogai Square

If one takes the figure of 940 feet (that is, 286.512 meters) as the side length factorizing 940 gives 20 x 47 and 47 (a prime number) times 5 gives 235 which is the number of lunar months in 19 solar years: the Metonic period. image by Google Earth

This is the larger of three bounding periods for the sun, moon, and earth. The lower boundary is exactly 19 eclipse years, called the Saros eclipse period of 18.03 solar years. . Within that range of 18-19 years lies the moon’s nodal period of 18.618 years, this being the time taken for the two lunar nodes, of the lunar orbit, to travel once backwards around the ecliptic. It is only at these nodal points that eclipses of sun and moon can occur, when both bodies are sitting on the nodes.

The first article on Ushtogai showed how, by daily counting all the tumuli in a special way, the 6800 days of the nodal period would keep a tally in days, to quantify where the nodes were on the ecliptic as well as predicting the lunar maximum and minimum standstills.

It now seems that, if the absolute size of the monument’s perimeter was able to count the 19-year Metonic, not by counting days but rather, counting the 235 lunar months of the Metonic period. The lunar month would then be 16 feet long. And, within that counting, one could also have counted the 223 lunar months between eclipses having the same appearance. The diameter of a circle drawn within the square would then have a diameter of 235 (lunar months) divided by 4 = 58.75 lunar months which, times the 16 feet per month, is the 940 feet of the square’s side length.

Figure 1. The size of Ushtogai Square, side length 940 feet, is 235 x 4 feet, making its perimeter able to count 235 lunar months of 16 feet.

In Cappadocia, present-day Turkey, this type of geometrical usage can be seen within a rock-cut church called Ayvali Kelise, only then in miniature to form a circular apse, just over 100 times smaller! The church was built in the early Christian period (see figure 2).

Figure 2 The Apse of Ayvali Kelise in Cappadocia, which presented the same geometry in miniature. [part of figure 7.5 from Sacred Geometry in Ancient Goddess Cultures.]

The Ushtagai Square has the basic form for the equal perimeter geometry. If so, that would form a tradition at least 10,000 years old. As a counting framework for the 18-19 solar year recurrences of aspects between the the Sun, Moon, Earth, eclipses and nodes the Square appears to be both a tour-de-force in a form of astronomy now largely forgotten.

Figure 3 Showing the circle equal in perimeter to the Ushtagai Square, the size of the Earth (in-circle of diameter 11) and Moon (four circles of diameter 3.)

As an earthwork where tumuli punctuate geometrical lines, it is a highly portable symbol of great time and a highly specific astronomical construction. It was an observatory and also a snapshot within celestial time, built just after the Ice Age had ended.