Wed 2021-Dec-15

Tis the Season… of the Analemma

Tagged: Beauty / Math / Obscurantism / Physics / R / TheDivineMadness

It’s that season again: the time of nerdly meditations on the analemma. The shortest day of the year, Dec 21, is yet to come. But for night owls, the day of earliest sunset (at the latitude of Château Weekend) was Dec 8th. How can that be? That’s the tale of the analemma!

Many years ago…

Symbolics 3640 Lisp Machine (Wikipedia) Doug Dodds (LinkedIn)

Many years ago – where “many” may here be understood as meaning ≥ 35 years – I was working for a very strange company in Cambridge called Symbolics, which made very strange (and wonderful) computers known as Lisp machines. One of my main joys in life at that time was that I got to work with not only smart people, but also with people who were at least as strange as me, if not stranger. (In some cases, very much stranger. I could tell you stories, but would first have to check if the statute of limitations is expired.)

One of those smart and interestingly strange colleagues was an excellent fellow called Doug Dodds: an expert in user interface, operating system build procedures, trivia Cantabrigiae, and an inveterate night person. It was from him I learned the axiom: “the only problem with morning is that it comes too early in the day to be useful”.

The analemma

Analemma by Dennis di Cicco As with most people, in deep mid-winter Doug yearned for the return of daylight. (Though I do not: I like dark winter days. See above, re “strange”.)

Now, since Doug came from St. Louis, his freshman year here at MIT was rather hard on him for more than the usual ways in which the first year at MIT is hard on people: northern winters! He spent a fair amount of time working out the orbital mechanics of when sunrise & sunset happened, and noticed a funny thing called the analemma – if you photograph the sun each day at the same time each day for a year, it traces out a figure-8 in the sky, as shown in this now-iconic year-long time-lapse photograph by Dennis di Cicco from 1978-1979 [1].

The source of this peculiarity, if you really work your way down into the orbital mechanics, is an interaction between the earth’s very slightly elliptical orbit and its 23.5° axial tilt.

Mixing analemmas, nerds and the winter holiday season

So back at Symbolics we were talking one day about how the amount of daylight would get better after the Winter Solstice, on Dec 21. But Doug pointed out that for night people who didn’t care about sunrise, the sunsets had already begun to get better! How could that be, that sunsets started to get later before the Winter Solstice?!

The fascinating explanation is part celestial mechanics, part hobby astronomy, and part pretty pictures. (Oh, and some math. [5] [6] There’s always some math!) These factors producing the analemma conspire to create an interesting effect: while the Winter Solstice is always on Dec 21 (a fixed point of the Gregorian solar calendar, regardless of latitude), the earliest sunset is a bit before that and the latest sunrise is a bit after (both depending on latitude). It led to a series of annual emails from Doug, in which he would, each year, alert us to the day of earliest sunset. [2] By now-ancient tradition among the now-ancient Symbolici Cantabrigiae, this is known as “Dodds’s Day”.

For the Weekend Editrix, on the other hand – well, let’s just say she’s a morning person who inexplicably-but-actually cares that the sun should rise before she does. So we call the day of latest sunrise, after which her mornings begin to get brighter, “Weekend Editrix’s Day”. (In the current epoch, it also turns out to be when the earth is nearest the sun, believe it or not.)

This year, though, the Weekend Editrix has told me she cares less these days about sunrise, and more about sunset. So the day she cares about the most is Dodds’s Day, like the rest of us. (We’ll leave the nomenclature in place, though, if only for historical purposes.)

2021: Another year of the dumpster fire

Isn’t it supposed to be against the rules to have 2 dumpster-fire years in a row? What do you mean, there aren’t any rules here?!

Ok… calming down now.

This year, at Boston’s latitude (42.3581° North), Dodd’s day was on December 8th and the Weekend Editrix’s Day will be January 3rd. It’s mainly a function of latitude: if you’re further north, Dodd’s Day and the Weekend Editrix’s Day are closer to the solstice; if you’re further south, they’re farther apart. (If you’re in the southern hemisphere, you’ll have to do the calculation for yourself; please let me know if you do.)

Dodd's Day and the Weekend Editrix's Day, 2021 Boston

Using some data scraped from the web [3] and an R script [4], you can produce the above plot of times of sunrise, local solar noon, and sunset to see the effect.

  • The blue curve is for sunset time, and it obviously reaches its minimum first at 2021-Dec-08, at our latitude.
  • The red curve at the bottom is for sunrise, and obviously reaches its maximum last at 2022-Jan-03, at our latitude.
  • The green curve is for local solar noon, which is just changing gently and slowly.

You can see Dodds’s Day, the winter solstice, and the Weekend Editrix’s Day called out at the appropriate places along the curves.

It’s been a dark couple of years: a pandemic, the march of fascism & authoritarianism, the rise of defiant ignorance to resist vaccines, and the right wing employing direct attacks on previously stable institutions like democracy itself.

Here, then, is a wish for increased literal light, and the corresponding figurative light of intellectual and moral enlightenment for which we are all longing.

Let’s work together – and work hard – to return the world to sanity and to our mutual moral duty of care for each other.

Notes & References

1: D di Cicco, “Exposing the Analemma”, Sky and Telescope, June 1979, pp. 536-540.

2: D Dodds, “Analemma, My Analemma”, once an annual email from Doug Dodds.

3: Time and Date, “Boston, Massachusetts, USA — Sunrise, Sunset, and Daylength, December 2021”,, retrieved 2021-Dec-15. Also used the corresponding files for November and January.

Removed Nov 01 – Nov 06 to avoid complexity of daylight savings time change, flushed a bunch of Unicode troublemakers, and dropped irrelevant columns. See the cleaned-up version suitable for input to the R script, which combines Nov 2021 to Jan 2022.

Use results for other cities at other latitudes. Particularly let me know if you’re in the southern hemisphere and work out the details around your summer solstice!

4: Weekend Editor, “R script for sun times, Dodds Day, and Weekend Editrix Day”,, 2021-Dec-15.

5: CH Holbrow, “Build Your Own Analemma”, arχiv 1302.0765, 2013-Feb-05.

6: P Lynch, “The Equation of Time and the Analemma”, Irish Math Soc. Bull. vol 68, Summer 2012, pp. 47-56.

Published Wed 2021-Dec-15

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