## Description

This article is from the Calendars FAQ, by Claus Tondering claus@tondering.dk with numerous contributions by
others.

# 5.4. What years are leap years? (Persian Calendar)

Since the Persian year is defined by the astronomical vernal equinox,

the answer is simply: Leap years are years in which there are 366 days

between two Persian new year's days.

However, basing the Persian calendar purely on an astronomical

observation of the vernal equinox is rejected by many, and a few

mathematical rules for determining the length of the year have been

suggested.

The most popular (and complex) of these is probably the following:

The calendar is divided into periods of 2820 years. These periods are

then divided into 88 cycles whose lengths follow this pattern:

29, 33, 33, 33, 29, 33, 33, 33, 29, 33, 33, 33, ...

This gives 2816 years. The total of 2820 years is achieved by

extending the last cycle by 4 years (for a total of 37 years).

If you number the years within each cycle starting with 0, then leap

years are the years that are divisible by 4, except that the year 0 is

not a leap year.

So within, say, a 29 year cycle, this is the leap year pattern:

Year Year Year Year
0 Ordinary 8 Leap 16 Leap 24 Leap
1 Ordinary 9 Ordinary 17 Ordinary 25 Ordinary
2 Ordinary 10 Ordinary 18 Ordinary 26 Ordinary
3 Ordinary 11 Ordinary 19 Ordinary 27 Ordinary
4 Leap 12 Leap 20 Leap 28 Leap
5 Ordinary 13 Ordinary 21 Ordinary
6 Ordinary 14 Ordinary 22 Ordinary
7 Ordinary 15 Ordinary 23 Ordinary

This gives a total of 683 leap years every 2820 years, which

corresponds to an average year length of 365 683/2820 = 365.24220

days. This is a better approximation to the tropical year than the

365.2425 days of the Gregorian calendar.

The current 2820 year period started in the year AP 475 (AD 1096).

This "mathematical" calendar currently coincides closely with the

purely astronomical calendar. In the years between AP 1244 and 1531

(AD 1865 and 2152) a discrepancy of one day is seen twice, namely in

AP 1404 and 1437 (starting at vernal equinox of AD 2025 and 2058).

However, outside this period, discrepancies are more frequent.

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