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Let us have a little history tour of the division of time, there is mathematical beauty in it.

On what basis has the time been divided? ​

SUN, MOON, STARS

SUN
The succession of day and night is a clear sign for dividing the time. Probably the first word to qualify a fraction of time was a “day”. In many languages it means both, the daylight part of the day and the full 24 hours day. 
Another sign is the difference of daylight and warmth between summer and winter. The recurrence of this cycle and the observation of the agricultural seasons introduce the concept of year. 
A solar year is now calculate to 365 days, 5 hours, 48 minutes, 46 seconds.

MOON 
In many languages the word month, takes its origin from the word moon referring to a complete lunar cycle. A lunar month is about 29 days, 12 hours, 44 minutes. By observation people had to choose between a 29 and 30 days month long. They observed that 12 of these months were fitting in a year.  

STARS
People understood that not only there was a cycle of warmth and cold and a difference of daylight time during a year, but that the position of the stars was slowly moving and coming back to the same place once a year. 
They also noted a special “way” in the sky were the moon, the sun and some special “stars”, the planets, travel on this way. They divided this path into 12 constellations. 

“Let luminaries be in the expanse of the heavens, to make a separation between the day and the night, then they have been for signs, and for seasons, and for days and years.” Genesis 1:14

LUNAR CALENDARS
Based on these signs, people came out with different calendars. For example, we still find in use In Saudi Arabia a very old pre-Islamic lunar calendar. This lunar year is shorter than the solar year by ~11 days. The lunar calendar is very rare, even in antiquity, because of the lack of agricultural and astronomical predictions. It can only satisfy the needs of a nomadic society. 

LUNISOLAR CALENDARS
Much more often, people preffered a lunisolar calendar. They synchronized the “easy to watch” lunar cycle with the real agricultural seasonal solar year. 
Different methods were adopted to this purpose, a really simple and genial one is the calendar given in the Bible to the Israelites when they came out of Egypt: “This month [Abib, is] to you the chief of months -- it [is] the first to you of the months of the year… “Observe the month of Abib”. “Abib,” meaning “Green Ears,” the ears of the barley, the first harvest of the year, being ripe but still soft in the beginning of spring. (Deuteronomy 16:1, Exodus 12:2) At the beginning of the new year, an observation of the first “Green ears of barley” was made and it was decided to add or not a 13th intercalary lunar month (Veadar) to the old year. 
Their three main festivals coincide with three main harvests.

​The position of the sky slightly differ year after year. We call this phenomenon “precession of the equinoxes.” Each 72 years, the sky will differ of 1 degree, so it will take 72 x 360° = 25’920 years to reset the constellations in their first place. (This is the reason why some people speak of: “Age of Aries, Pisces and Aquarius etc…” Each “age” during 72 x 30° = 2160 years. This also means that what we call our “astrological sign” is wrong. The sky already switched of at least one sign from the time they associate the months with the constellations. As a result, all the “horoscopes” are wrong, because Aries month is Pisces and Pisces month is Aquarius...

360 DAYS
In addition, another 360 days calendar of 12 months 30 days each was also in use. Three hundred and sixty is easier to divide and compute and it is an average between the solar and the lunar year: 365.24 (solar) + 354.38 (lunar) AVERAGE = 359.81, nearest integer: 360, which is a highly composite number.

• 360 is the smallest number divisible by every natural number from 1 to 10 except 7.

Turning back in the past, we find the 360 day calendar in the biblical account of the flood were 150 days are equal to five months. (5x30=150; five lunar months should be slightly shorter). The method used at that time to fill the gap between the lunar and the solar years is unknow.

Antediluvian
There are already trace of a 360-day calendar in the Biblical account of the flood were 150 days are equal to five months. (Five lunar months should be slightly shorter). The method used at that time to bridge the gap between the Moon and the solar years is unknown.

Sumerians
Robert K. Englund, Ph.D., professor of Sumerology at UCLA, says in the Journal of Economic and Social History of the Orient, 1988, pp. 121-122 " The calendar and the sexagesimal system have an old relationship…The sexagesimal system of counting … is attested throughout the third millennium ... [The calendar] is attested in its basic form of a twelve-month period, and 360 days year, in archaic documents of the late 4th millennium [BC]." This is probably the origin of the circle divided into 360 degrees.

Egyptians
Herodotus wrote in his stories: “The Egyptians were the first to discover the year and to divide it into twelve parts. They obtained this this knowledge by studying the stars. The Egyptian calendar seems to me to be much more sensible that that of the Greeks; for the Greeks, to make the seasons work out properly, intercalate a whole month every year, whereas the Egyptians, basing the calendar on twelve thirth-day months, (360) intercalate five additional days every year, whereby the cycle of the seasons returns with uniformity.”
  
Babylonians
The Babylonians, although the finest astronomers used a lunisolar cycle very similar to the Hebrews. They probably synchronized it with the equinox. They are the inventors of the 19-year cycle, later called Metonic, to anticipate the 13th additional months. Respectively, the year 3, 6, 8, 11, 14, 17, and 19 were those with 13 months. During their exile in Babylon, the Hebrews adopted these methods and also the names  of the Babylonian months.

Maya
Their tun was a year of 360 days. Twenty tun equaled a katun and twenty katun equaled a baktun which is equal to 144,000 days, so 400 x 360 days. Also if they had different and interrelated calendars; to identify a real date in history they used the 360-day calendar.

Aztecs
The Encyclopedia of Time, p. 49, said of the Aztecs, "they had different and complex calendar systems ... they have nonetheless continued to pay tribute to a year of 360 days.

Indians
The Indo-Aryan encyclopedia G. Thibaut, says: "The Vedas speak uniformly and exclusively of a year of 360 days. The passages where the length of the year is directly stated are found in all the Brahmana’s .. "In addition, the Vedas contain four ages, such as the Golden Age, the Silver Age, each of which are divisible by 360. The silver age (Treta Yuga) is 360 * 3600 years long.

PersiansThe Zoroastrian calendar year was of 12 months of 30 days, plus five days. It became official under the Sasanian dynasty, from about C.E. 226 until the Arab conquest in 621. The Arabs introduced the Muslim lunar year, but the Persians continued to use the Sasanian solar year, which in 1079 was made equal to the Julian year by the introduction of the leap year.
Greeks
Herodotus wrote "These seventy years represent, excluding intercalated months, 25,200 days" so using this calculations, a year would be of 360 days. 25,200 / 360 = 70
The Antykithera mechanism shows not only that the Greeks managed to create a "clock" that can predict the Metonic cycle, and also eclipses!
 
Romans
The life of Plutarch's Numa explain: “they had been irrational and irregular in their fixing of the months, reckoning some at less than twenty days, some at thirty-five, and some at more; they had no idea of the inequality of the annual motions of the sun and moon, but held to this principle only, that the year should consist of three hundred and sixty days." 

​For more informations about the use of 360 days calendars in ancient history, follow the link
http://www.webexhibits.org/calendars/calendar-ancient.html

Modern 360-day calendar
Today 360 is still used in financial markets and in computer models. In France this year is still called Lombard year in memory of Lombardy bankers who in the Renaissance use this calendar to simplify their divisions in their interests rates.

Beauty of 360
This fascination through millennia of 360 takes us to explore its beauty a little more closely. By dividing or multiplying by 2 we continuously obtained the same digital root of 9: 3 + 6 + 0 = 9; 1 + 8 + 0 = 9; 9 + 0 = 9; 4 + 5 = 9; 2 + 2 + 5 = 9; 1 + 1 + 2 + 5 = 9; etc ad infinitum.
The sum of the angles of regular polygons, derivatives division 360⁰ produces the same result: 180 ° 360 ° 540 ° 720 ° 900 ° 1080 ° 1260 ° 1440 °.
This property is the one of the number 9, number highly mathmagic that produces this result in every one of its multiple and continuously when we divide it by 2.

The “Gezer calendar”, is an ancient inscription in Hebrew found in 1908, which confirms the synchronization of the lunar months with the solar agricultural year in the antiquity. It shows the lunar months in connection with the agricultural events.

PROPHETIC 360 IN THE BIBLE
In revelation 11:2, 3. Forty-two months equal 1260 days = three years and a half (of 360 days). The same period is mentioned in Daniel 12:7 and revelation 12:6, 14.
Daniel 12:11, 12 speaks about 1290 days and 1335 days. Respectively three years and seven months and three years and eight and a half months if we use 30 days months. These periods confirm a 360 days calendar with 12 months of 30 days . 
Daniel 4:16 use the expression "Seven times", meaning seven periods of 360.
It seems that these prophetic years were not mean to be synchronized with the solar year.

The Antikythera mechanism
​​“The most sophisticated mechanism known from the ancient world.”
The front dial indicated lunar phases and positions of the sun and the moon. It also showed the day and month according to the solar calendar and movement of the sun (and the visible planets) in relation to the constellations of the zodiac
The upper back dial indicated the relation between lunar months, solar years, (Metonic cycle) and the times of the Panhellenic Games
The lower back dial predicted solar and lunar eclipses
Archimedes of Syracuse could have invented it.

If you are interested to reproduce this mechanism by yourself, you should follow the work of greekgears.com you will find all you need to build it.

Like the sequence of 60 digits, 24 gives also very interesting symetries. The digit sum of each fibonacci number gives another ever-reproducing sequence of 24 numbers.
Multiples of 3 fit the hexagon.
​The sum of all the opposites gives 9

Digit sums of the digits on the hexagons: 9

9 is such an interesting number. As you can watch in the following video 360 and 9 are beautifully related.