The 33-year spring

The calendar with no leap rule at all — a year that is as long as it has to be, a new year decided by the March equinox at a Tehran noon, and a 33-year rhythm that only looks like a formula. ← Calendar Contraptions

Most calendars are formulas. The Gregorian adds a day when the year number divides a certain way; the Jewish calendar runs its moon on fixed arithmetic; even the Maya were just gearwork. The Solar Hijri calendar — the official calendar of Iran, and the everyday calendar of Afghanistan and the Persian-speaking world — refuses the whole idea. Its rule is one sentence: the year begins on the day whose Tehran noon falls closest to the March equinox. If the equinox happens before noon, that day is 1 Farvardin. If it happens after noon, that day is 30 Esfand and the next day is New Year. That is the entire law. Whether a year has 365 or 366 days is not decided by arithmetic — it is decided by the sky, one year at a time.

Look at the results from far away, though, and a rhythm appears: leap years arrive every four years, with one five-year breath every 33 — a pattern so steady that whole computer libraries just hard-code it, and one mathematician spent years freezing it into a 2,820-year formula that doesn't work. This page shows both the sky and the rhythm, and where they part ways.

Twelve names, twice

Every Solar Hijri month has two names, because the calendar has two homelands. Iran uses the old Zoroastrian month names — each one a divine figure of the Avesta, in the genitive case, since the month literally belongs to that yazata. Afghanistan (and Iran before 1925) uses the Arabic names of the zodiac signs instead. Same machine, different labels:

The equinox machine

Here is the whole calendar law, running live. The machine computes the March equinox for the year, converts it to Tehran true solar time, and asks one question: before noon, or after?

The 33-year wheel

Because the equinox drifts about 5 hours 49 minutes later each year, the before/after-noon verdict cycles through a pattern: roughly two equinoxes after noon, then two before, then again — and once every 33 years the equinox lands so close to noon that a pair collapses, leaving a five-year gap between leap years. The wheel below is the 33-year cycle the picked year lives in, leap years in gold. The remainder rule (y mod 33 ∈ 1, 5, 9, 13, 17, 22, 26, 30) predicts the sky almost everywhere from 1178 on — the computed exceptions are the knife-edge years 1502 and 1503, where a 2124 equinox misses the Tehran noon by 2.8 minutes, and from 1634 the drift wins one cycle after another (1634/35, 1667/68, 1700/01…, a pair every 33 years). The wheel shows the computed truth, with rule-breaking years flagged.

Two calendars named Hijri

Both this calendar and the Islamic lunar calendar count from the Hijrah — Muhammad's journey from Mecca to Medina — but one counts solar years and the other lunar ones. A lunar year is about 10.88 days shorter, so the two year numbers slide apart by one roughly every 33.6 years: the same 33.6 years it takes the wandering Islamic year (its own contraption, see the year that wanders) to circle the seasons once. Today the gap is 43 years; in the year they were named, it was 1.

The arithmetic year that wasn't: Birashk's 2,820

In 1993 the Iranian mathematician Ahmad Birashk proposed — as Behruz had half-proposed in 1952 — that the leap pattern really ran on a hidden 2,820-year cycle: 21 grand cycles of 128 years (split 29 + 33 + 33 + 33) plus one of 132 (29 + 33 + 33 + 37), for 2,137 common years and 683 leap years. It is beautiful: 683/2820 gives a year of days, matching the tropical year to about 2 parts in 10,000,000. It is also wrong, twice over. First, the calendar follows no cycle at all — its year length is decided year by year by the noon rule. Second, the year that matters is not the tropical year but the vernal-equinox year, which is 15.82 seconds longer. Over its 2,820 years the arithmetic calendar accumulates 1,029,983 days while the sky needs 1,029,983.46 — the scheme gains half a day per cycle and drifts. The astronomer M. Heydari-Malayeri, checking against IMCCE ephemeris data, found it "runs into trouble many times during a period of 2025 years, for example 21 March A.D. −1, 21 March A.D. 1600, and 21 March A.D. 2025". The suspiciously perfect year length had long been attributed to Omar Khayyam himself; the fraction 683/2820 behind it is almost certainly where that legend came from, and no historical record connects the 2,820 cycle to Khayyam.

The fixed dates of the year

Nowruz is the great festival, and it owns thirteen days: the thirteenth, siz-dah bedar ("outdoor 13th", formally Nature Day), is spent outside, shaking off the bad luck the first twelve days are said to accumulate. Fixed in the Solar Hijri year, they wander a little on the Gregorian one:

Convert the other way

A short history of the spring

Khayyam's observatory, 1079. The story that ends in this calendar starts on 15 March 1079, when the Seljuk sultan Malik-Shah adopted a reformed solar calendar at the recommendation of a committee of astronomers working under the grand vizier Nizam al-Mulk, with Omar Khayyam among them — observations made in Isfahan (the capital), Rey and Nishapur. The Jalali calendar (named for the sultan, Jalal al-Din) began each month at the sun's actual transit into the next zodiac sign, a system with roots in the Indian Surya Siddhanta — so its months could be 29, 30, 31 or even 32 days, and it needed no leap days at all, because the sky did the adjusting. It ran for eight centuries. What it cost was computability: every month required real ephemeris work, and the year's start was entangled with tax season. Historians still argue about exactly which day the Jalali epoch was anchored to — the "enigma of its radix date". Its average year, 365.2422 days, gains a day on the old Julian calendar about every 128 years.

The 1925 law. The modern Solar Hijri calendar was adopted by the Iranian parliament on 31 March 1925 — 11 Farvardin 1304. The law fixed what the sky had left loose: month lengths locked at 31/31/31/31/31/31/30/30/30/30/30/29–30, the Zoroastrian names restored, the epoch set at the Hijrah, and the new year defined by the equinox-noon rule. Afghanistan had adopted the same calendar in 1922 but kept the Arabic zodiac month names — Hamal, Sawr, Jawza, Saratan — which is why the 1978 Saur revolution is named for Taurus. (The Taliban's first government swapped the whole calendar for the lunar Hijri in 1996 — the year jumped from 1375 to 1417 overnight — and as of 2024 Iran is the calendar's only official user, though it remains the everyday calendar of Afghanistan.)

The year that got renamed. In March 1975 the Shah's parliament renumbered the calendar overnight from 1354 to 2534, counting from Cyrus the Great instead of the Hijrah — chosen so the Shah's own accession fell in the round year 2500. The "Shahanshahi" era lasted three and a half years; in August 1978 the Hijrah count came back. The days of the week had kept counting through it all: Saturday shanbeh, then yekshanbeh, doshanbeh, seshanbeh, chaharshanbeh, panjshanbeh — literally one-day through five-day — and Friday jom'eh (assembly day, from Arabic; the old Persian name is adineh), the day of rest. The week starts on Saturday here.

Older than all of it. The month names are three-thousand-year-old prayers: Farvardin from the fravashis (guardian spirits of the righteous), Ordibehesht "best truth", Khordad "wholeness", Tir from Tishtrya — the star Sirius, whose rising brought the rain — Mordad "immortality", Shahrivar "desirable dominion", Mehr the covenant-god Mithra, Aban the waters, Azar the fire, Dey the creator himself, Bahman the good spirit, Esfand holy devotion. A reform of 487 BC fixed this pantheon in place and parked its five spare days just before Nowruz; the land-tax year then slipped a whole month every 120 years to stay honest with the harvest. When Quintus Curtius Rufus watched a Persian ceremony in 333 BC, he counted three hundred and sixty-five young men in purple robes, "equal in number to the days of a whole year."

Where it cheats

Almost nowhere in its own terms — the beauty of the noon rule is that the calendar can't drift: the year start is pinned to the sky by definition, which is what the "most accurate calendar in the world" brag is really measuring. But the honest seams exist. The rule's working form is a published table: for 1975–2041 this page's astronomy reproduces the published table exactly (all 66 year-starts and leap marks, cross-checked with two independent implementations), yet the deciding margins are occasionally minutes — for 1635 the computed equinox lands about eight minutes from the Tehran noon that decides the year, closer than ΔT models can promise, and a the 33-remainder rule that libraries hard-code predicts the sky almost perfectly for three centuries before 1502 and then walks away from it a pair of years every 33. Outside the checked era, treat every computed date as approximate; inside it, the machine and the official tables agree to the day. The other seam is invisible from inside: days start at midnight here, by a long Iranian tradition — the equinox moment itself (tahvil-e sal) is celebrated to the second, but no sunset and no moon decides anything, which makes this the calmest machine in this whole corner.