The Cellular World of Twelve

The foot has twelve inches just as the British shilling had twelve pence. A good case can be made for twelve as a base like 10, since there are 12 months within the year and many ancient monuments can be seen to have employed duodecimal alongside decimal number, to good effect.

Until 1971 the currency in the United Kingdom of Britain was duodecimal, called pounds, shillings and pence.

This old system of currency, known as pounds, shillings and pence or lsd, dated back to Roman times when a pound of silver was divided into 240 pence, or denarius, which is where the ‘d’ in ‘lsd’ comes from. (lsd: librum, solidus, denarius). see historic-uk.com

There were 12 pence in a shilling and 20 shillings in a pound, that is 240 pennies. The change to a decimal (100) pence in a pound caused a lot of inflation during the changeover due to price opportunism, then part lasting recession. In British heads the skill of giving and taking change in a duodecimal arithmetic was soon lost. In the late 70’s my mother, when visiting the US, was amusingly referencing “old money”, alongside the exchange rate between a decimal pound and the decimal dollar, just as Greeks had problems with the Euro.

History of Decimal Measures

Napoleon sought to “rationalize” all the ancient weights and measures of France culminating in the decimal units within modern science, firstly CGS (Centimeter-Gram-Seconds) and then MKS (Meters-Kilogram-Seconds).

The Meter exemplifies the situation: it contains 100 centimeters and 1000 millimeters whereas the root foot for the worldwide and only ancient metrology is that called English which has 12 inches (each a “thumb” in French) and each inch halves, quarters, eighths and sixteenths of an inch, but also 10ths etc., that is a duodecimal system and decimal notation as with 12 = 10 +2.

The metrology of the ancient world had no need for decimalization since it had been formed to employ all the integer numbers, using fractions of a foot – fractions being a combined multiplication (numerator) and division (denominator) operation. That is, there was no base-10 decimal notation when metrology was developed and one can suggest that decimalization was created in the wake of treasuries, mints and central banks.

However, decimal notation emerged much earlier with the alphabetic form of writing languages down. Cuneiform had used compound sounds (called syllables such as “no”) but the new Phoenician alphabets notated the consonants and vowels of specific languages, now called phonemes of sound (for exampple, the phonemes “n” and “o”). This reduces the number of symbols needed to notate speech, and in turn these symbols could then have a decimal function and words could also be numbers, in a code called Gematria:

Gematria is the practice of assigning a numerical value to a name, word or phrase by reading it as a number, or sometimes by using an alphanumerical cipher. Wikipedia on Gematria.

As the name implies, Alpha equals 1, Beta =2, D = 4, J = 10, etc.. Words could then encode a number, as in the Bible where Adam equals the three letters A.D.M whose numerical values in Hebrew/ Aramaic (1.4.40): when added up they “mean” 45. The later letters were values in tens and hundreds so that decimalization probably goes back to the 1st millennium BCE.

Figure 1 Numeric equivalence of Hebrew Alphabet

We are therefore needing to go earlier than the decimal base-10 system or indeed the use of any base at all, to see into the world of the megalithic astronomer and different relationships to numbers.

This previous world which gave birth to a type of math that is not arithmetical but instead used the factors within integers and rational fractions, initially through measured geometrical proportionality but then through sets of measures all rational fractions of the common foot.

Prehistory: Non-Decimal Measures

The earliest number encountered by early astronomers would have been (when they counted) the twelve lunar months within a year. The properties of the number twelve are generally taken to come from its factors (such as 4 x 3), it Platonic solid (the duodecahedron) – see next section. There were no twelve hours in half a day. We will the take a deeper approach, of visualizing the set of numbers within twelve, as {1,2,3,4,5,6,7,8,9,10,11,12}

Factors within Twelve

Twelve does not contain is the prime number 5 nor any higher prime factor. However, in counting to 12, there are two factors containing 5, namely 5 and 10. And there are, of course, the prime numbers and their ennumerated multiples, such as, for 7, {14, 21, 28, 35, 42, 49, 56, …}. This means the number field is made up of empty slots into which the number one greater than the preceding number must then be a prime number. And any prime number can then be doubled, tripled, etc., to become enumerated itself. That is, which we call prime numbers are those that happen to have no preceding number of which it is a multiple of any (previously arisen) number.

Numbers Within Twelve

Twelve does not contain is the prime number 5 nor any higher prime factor. However, in counting to 12, there are two factors containing 5, namely 5 and 10. And there are, of course, the prime numbers and their ennumerated multiples, such as, for 7, {14, 21, 28, 35, 42, 49, 56, …}. This means the number field is made up of empty slots into which the number one greater than the preceding number must then be a prime number. And any prime number can then be doubled, tripled, etc., to become enumerated itself. That is, which we call prime numbers are those that happen to have no preceding number of which it is a multiple of any (previously arisen) number.

Figure 2 The inner structure of Twelve

Figure of (top) the first twelve numbers, four of which divide by three, making the even numbers (orange) alternate with the odd numbers in serpentine fashion. Numbers dividing by 5 then alternate down then up, every two threes.  (bottom) the color keys used. (One could show primes with italics)

In a following post, the consequences of this inner structure reveal Twelve’s cellular structure within the number field.

Double Square and the Golden Rectangle

above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangle existed.”

Here we will continue in the mode of a lesson in Geometry where what is grasped intuitively has to have reason for it to be true. It occurred to me that the square in the top hemisphere is the twin of a square in the lower hemisphere, hence this has a relationship to the double square rectangle. So one can (1) Make a Double Square and then (2) Find the center and (3) a radius can then draw the out-circle of a double square (see diagram below).

The diagonal from the centre would be the square root of 5 if the top square is seen as two double squares of unit size, that is (4) Identify the units as nested double squares. One can then see (5) a cross within the circle holding 12 squares, but when (6) the root 5 comes down to the right horizontal then the familiar formula (root(5) – 1)/2 = 0.618 so there are many transcendent (not Fibonacci) versions of the Golden mean within in the diagram as shown below.

The in-circle of the cross, radius 2, shows how one can divide that circle into twelve equal portions as with the Zodiac, matching the twelve squares. The out-circle shows Dan’s insight as eight golden rectangles which, overlap over the four “missing” squares of the 16 square grid, which is a simpler framework for generating this geometry as a Whole.

Twelve: determining Time and Space on the Earth

ABOVE: South rose window in Angers Cathedral of Saint Maurice. Stained glass by Andre Robin created after the fire of 1451. At centre, Christ of the Apocalypse, in glory (Revelation 21:5). At bottom, 12 radial windows showing 12 elders, crowned and playing musical instruments, rejoicing, indicating the remade world (the heavenly Jerusalem). At top, circular ends of 12 radial windows showing the 12 signs of the Zodiac, indicating the incarnation of Christ as a man on earth under the stars. Sequence from left to right has last 2 signs before first, i.e. Aquarius/Water-bearer (grey), Pisces/Fish (grey), Aries/Ram, Taurus/Bull (yellow), Gemini/Twins, Cancer/Crab (red), Leo/Lion (yellow), Virgo/Virgin, Libra/Scales, Scorpio/Scorpion, Sagittarius/Archer, Capricorn/He-goat (blue background)
photo: Chiswick Chap for Wikipedia Foundation.

The Moon was the means by which a 12-fold harmony became established on the Earth. This harmonization occurred through the lengthening of the lunar month until 12 months fitted, in a special way, within the solar year. The excess of the solar year over the lunar year of 12 months became 7/19 lunar months, causing seven extra whole months over 19 years. This 19-year (235 month) Metonic period was well-known to the ancient world, and it leads to the remarkably short cycle for the pattern of similar eclipses, we call the Saros period, which repeat every 18 years (235 minus 12 months = 223 months). And eclipses are highly visible because the disk of the Moon has come to be the same angular size as the disk of the Sun, causing total solar eclipses.

Continue reading “Twelve: determining Time and Space on the Earth”