## Mathematics of the Kumbha Mela

The math of the frequency of Kumbha Mela is similar to the math of adding leap days to a year which involves approximation of a non-integer real number with multiples of an integer.

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The earth takes around 365.2425 days to go around the sun once. Our usual year has 365 days, an underestimate for 365.2425. Over four years, this creates a difference of almost one day, as (365.2425 – 365) * 4 = 0.97 ≈ 1. So we add one day every four years as a leap day. Now this effectively makes a year last 365.25 days, which is an overestimate for 365.2425. Over four hundred years, this one-leap-year-in-four rule creates a difference of three days, as (365.25 – 365.2425) * 400 = 3. Therefore we take out three such leap days (added once every four years) over 400 years. This is done by making a year divisible by 100 a leap year only if it is divisible by 400 also. This is why 2000 was a leap year but 2100, 2200, and 2300 will not be leap years.

The Kumbha Mela is held, usually, once every 12 years at a place. The Ardha Kumbha is held halfway, 6 years after one Kumbha and 6 before the next. Why 12 years? The Kumbha is held when the Sun and Jupiter are in specific zodiac signs. The Sun takes one year (365.2425 days) to transit once through all twelve signs. Jupiter takes almost twelve [earth] years. Hence, after almost twelve years, Sun and Jupiter are in the same respective zodiac signs. This is why, usually, the Kumbha Mela is held once every 12 years.

But Jupiter takes slightly less than 12 years to transit once through all twelve signs. The exact period is 11.8618 years, and the usual frequency (12) is an overestimate for it. Over seven transits, this difference becomes almost equal to one year (12 – 11.8618) x 7 = 0.9674 ≈ 1. Seven times 12 is 84 but seven times 11.8618 is almost 83 (83.0326). So we have to take out one year every 84 years to keep the Kumbha Mela cycle and Sun and Jupiter positions in sync. This is why, after six Melas held in 72 years (each 12 years after the previous one), the seventh is held 11 years after the sixth. E.g. last Haridwar Kumbha was in 2010, next would be in 2021. This gives us seven Melas in 83 years.

Seven Melas in 83 effectively gives an estimate of around 11.85714 years for Jupiter’s transit through the zodiac. This is a slight underestimate for the actual number 11.8618 (up to four decimals). So this Kumbha Mela cycle of seven in 83 would also be out of sync with positions of Sun and Jupiter at some point, just like the 1-in-4 leap year cycle gets out of sync by three days over 400 years. For example, this 7-in-83 cycle will create a difference of one year in around 29 iterations, as Jupiter will complete 210 transits of zodiac in almost 2,491 years (11.8618 x 210 = 2490.978) but 30 iterations of the 7-in-83 cycle will give us 2,490 years. So something like this would need to be done: after 29 cycles of 7 Melas in 83 years, we will need one cycle of 7 Melas in 84 years. This will result in 210 Kumbha Melas in 2,491 (29 x 83 + 84) years.

The more precision we use, the more refined these rules will become.

At the end of the day, the Kumbha Mela is all about mathematics, as is everything else in the world.
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