Whether we are generating forecasts for tomorrow or for 20 years from now, the calendar plays a crucial role because human activity generally and energy consumption specifically are impacted by days of the week, holidays and seasons. But do we really understand the calendar as well as we think?

The primary civil calendar used globally is the Gregorian Calendar, which replaced the Julian calendar in 1582. The main difference between the two is the treatment of leap years. 

First, why do we have a calendar at all? We use a calendar to reckon the passage of time. The Gregorian calendar is a solar calendar. That is, it is based on earth’s orbit around the sun. Lunar calendars are based on the orbit of the moon around the earth. Lunisolar calendars are based on both the sun and the moon. 

Second, why do we have leap years? We use leap years because earth’s orbit around the sun is not exactly 365-days. In fact, earth’s orbit is approximately 365.2425 days. If we were to use a calendar with exactly 365-days, the errors would accumulate over centuries, and the seasons (as defined by the equinoxes) would eventually shift away from their calendar definitions; this is precisely what has happened with the Julian calendar, which is primarily used for liturgical purposes in the Eastern Orthodox Churches. 

In the Julian calendar, leap years occur every 4 years. I can already hear you saying, “But leap years do occur every 4 years!” Well, yes and no; it’s more complicated than that. In the Gregorian calendar, leap years occur on years that are divisible by 4, except for years that are divisible by 100, unless they are also divisible by 400. For example, 1900 was not a leap year because it is divisible by 4 and 100 but not 400, whereas 2000 was a leap year because it is divisible by 4, 100 and 400. This rule results in 97 leap years every 400 years (i.e., three fewer than in the Julian calendar), and this helps keep the Gregorian calendar aligned with the Earth’s orbit around the sun. 

January 1st can occur on any day of the week. Each year also belongs to one of four types: a leap year, or the first, second or third year following a leap year. Combining these two factors (the seven possible starting weekdays and four year-types) creates a repeating 28-year cycle in the structure of the calendar. 

The following depicts the 28-year pattern with each vertical group representing 28 years. The columns indicate the year, the day of week of January 1st of that year and the day of week of December 31st of that year. The first row of data highlighted in green indicates the years 2000, 2028, 2056 and 2084, all of which are leap years starting on a Saturday and ending on a Sunday. The second row of data highlighted in purple indicates the years 2001, 2029, 2057 and 2085, all of which are non-leap years that start and end on Monday. Aligned this way, the point of the figure becomes clear: the start and end day of the year form a repeating pattern. However, this 28-year cycles persists until 2100 where it breaks because the other years on the same row (2016, 2044 and 2072) are leap years while 2100 is not – see the rule above. The pattern then resets. The same will occur in 2200 and 2300 because they are not leap years. In fact, the full cycle is, perhaps not surprisingly, 400 years. 

Note: While I built the above figure in Excel for presentation purposes, I have built the same logic in a MetrixND project file, which you can download here. 

There are a few interesting observations. First, a non-leap year starts and ends on the same day of the week (e.g., Jan 1st and Dec 31st are Mondays in 2001). 

In other words, there is 1/7th of a week left over, which corresponds to 1 extra day. Thus, in this example, the week starts on Monday, and we complete 52 full weeks that end on Sunday, but there is one more day left over to complete the year, which ends on Monday.

Second, a leap year starts on one day of the week and ends on the following day of the week (e.g., Jan 1st is a Saturday and Dec 31st is a Sunday, as in 2000).

In other words, there is 2/7th of a week left over, which corresponds to 2 extra days. Thus, in this example, the week starts on Saturday, and we complete 52 full weeks that end on Friday, but there are two days left over to complete the year: Saturday and Sunday.

We can actually compress the combinations of days of the week and year types even further for practical purposes. We can think about the years as being either leap years or non-leap years. Thus, a non-leap year such as 2021 where January 1st is a Friday has the same calendar as 2027 where January 1st is also a Friday (see the figure above). The downside is that there are exceptions: Easter, which occurs on the first Sunday after the first full moon after the vernal equinox (autumnal if you’re in the southern hemisphere), the Jewish and Chinese holidays which are based on lunisolar calendars and the Muslim holidays which are based on a lunar calendar. The upside is that if you like your wall calendar featuring puppies, desert landscapes or covered bridges in New Hampshire, you can use it again if you are mostly concerned about the occurrence of Memorial Day, Labor Day or Victoria Day. In 2027, you can use your 2021 calendar once again!