Monday Morning Math: the Winter Solstice

December 12 was going to be the final Monday Morning Math of the semester, but finals had started two days earlier and with one thing and another (well, really just one thing – the aforementioned final exams), it didn’t happen.  

It felt a bit odd to take a break without announcement, however, so here is one final MMM for 2022.

(Arriving on a Monday at least, even though it’s not quite morning anymore.) And the timing is perhaps good for a math-adjacent topic: the Winter Solstice, which happened on Wednesday, December 21.  This is the shortest day of the year in the northern hemisphere, with just under 9 hours of daylight (technically 8 hours, 59 minutes, and 10 seconds) here in Rochester.  But there are two things about the solstice that I find interesting mathematically.

The first is that if you google “When is the Winter Solstice?” you get not just a day but a time: 4:47pm here.  This feels a little weird to me if I think about it being a day, but it has to do with northern hemisphere being tilted as far away as possible from the rays of the sun, as in this tweet from NASA below:

Or, if you want to envision the Earth with its axis vertical, it’s moving along a plane that is not horizontal, as explained on NASA’s blog: 

This is the image that makes the idea of a moment for the solstice make sense to me: it’s at the very peak of the ellipse, and that happens at one particular moment rather than a full day.

The second math adjacent thing is about subtraction.  You might think that the shortest day must have the latest sunrise and the earliest sunset, but actually neither is true: the lastest sunrise doesn’t happen until January 3, 2023, at 7:42am, which is about 3 minutes later than it rose on Dec 21.  So that’s kind of a bummer, in terms of how dark the mornings are.  But that’s compensated by the fact that the earliest sunset happened several weeks ago, back on December 9.  The sun set at 4:35pm that day, about 3 minutes earlier than it set on the solstice.  The sunrises and sunsets don’t quite change symmetrically, and that’s why the shortest day is about halfway in between. 

Happy Solstice and Happy Holidays to everyone! We’ll start again in about a month – see y’all in 2023!

Clocks Around Rome, Part II

A few more clocks to show!

Up in the Pincian Gardens, where all the Math Guys are, is a water clock created back in 1867.  It only worked for about 40 years, however, and then was in disrepair for about a century.  Fortunately, only three years ago the clock was restored and now it totally works. Yay!  Here’s what it looked like when we were approaching it:

And here’s what it looked like when we were standing in front of it:

(Many of the clocks in Rome used Roman numerals, heh heh.)

Here’s a close up of the water portion:

The water pours first on one side, then the other.

Finally, here’s a close up of the plaque, which tells a little about it, if you read Italian:

And a 2007 article here by Brian Barrow which tells even more, including:

The timepiece is the result of the work of two men: Father Giovan Battista Embriaco, a Dominican priest and scientist (1829-1903), and the Swiss-Italian architect Gioacchino Ersoch (1815-1902). Apart from teaching physics and mathematics, Embriaco had the hobby of constructing mechanical water clocks (see box) in which the continuous emptying and filling of containers at the ends of a balanced arm produced the rocking motion which took the place of the traditional pendulum by moving a notched wheel at regular intervals.

 

Despite seeing quite a few neat clocks in Rome, we missed the six-hour clocks.  We’d found information about these on a site of Curious and Unusual things in Rome [a fabulous resource!], where it said:

When clocks finally began to appear on important churches and public buildings, some of them had a dial with only six hours, not twelve as in ordinary clocks, so to divide the day into canonical hours, when the prescribed prayers were to be recited. The bells, instead, rung up to twelve times, despite the dial, and the hours were counted up to 24! For instance, at the 21st hour (i.e. around 4 pm in summer) the dial would have shown III, and nine tolls of the bell would have been heard.

Only two of these dials are still extant, in the main cloister of Santo Spirito in Sassia complex, near the Vatican, and on the façade of Santa Maria dell’Orto’s church, in Trastevere district (pictures on the right).

We did sneak over the Santo Spirito, but couldn’t find the clock and there was a wedding just getting out (all the cars had big bows on the antenna; we saw this in another wedding procession the next day) so we didn’t really want to stand and look around.  I’m still not sure where it is.

BUT, as a bonus, we did unexpectedly run across two more sundials in the museum in Ephesus.

 

Ephesus was a Greek city before it became a Roman city before it became a Turkish city, which probably explains the Greek.  (Although it’s interesting that it’s letters instead of numbers.  Unless the letters are also numbers?  And if not, aren’t some letters missing?  I’m so confused.  Most of the stuff in the courtyard was unlabeled, so I couldn’t find out anything additional.)

Next up, even more math in Rome!  Unless I don’t get to it before Friday’s Carnival of Mathematics, in which case the Carnival will be the next up.

Clocks around Rome, Part I

I like clocks, and in planning “How can we cram relaxingly fit many interesting things into just a few days?” I found out that there were a lot of really interesting clocks around Rome.  With very sore feet we managed to see most of them.  In chronological order (heh heh):

This is the Sundial of Augustus.  It’s an obelisk that was originally erected by Psammetichus II (aka Psamtik II) in the sixth-century BCE in the city of Heliopolis by the Nile Delta, but was taken to Rome by Augustus (aka Gaius Julius Caesar Augustus aka Octavian) in 10 BCE, where it became the gnoman (stick-thingy) if a GIGANTIC sundial.

This particular photo was taken (and placed into public domain) by someone named Arpingstone, and it’s much better than any we could have taken, particularly because it was dark when we went to this obelisk.   Which would have been a terrible shame if it were still a working sundial, but it isn’t.  I mean, it still casts a shadow, but I’m not sure if this is the spot it was originally placed on (it fell down for a few centuries); more significantly, the original lines for the sundial, which might have looked like this 19th century painting

but might have just been a meridian [marking noon], seem to be under buildings and stuff.  Rome just doesn’t look like that painting anymroe — it’s a lot more crowded.  So this is only part of a sundial, but it’s still pretty impressive.  (References:  this official sounding page and this Wikipedia article).

Jump forward about 1500 years.  The Baths of Diocletian were built about 1700 years ago and used for over 200 years; part of the remains of the frigidarium (the cold water part of the baths) were turned into Santa Maria degli Angeli e dei Martiri thanks in large part to Michelangelo.  According to Wikipedia,

At the beginning of the eighteenth century, Pope Clement XI commissioned the astronomer, mathematician, archaeologist, historian and philosopher Francesco Bianchini to build a meridian line, a sort of sundial, within the basilica. Completed in 1702, the object had a threefold purpose: the pope wanted to check the accuracy of the Gregorian reformation of the calendar, to produce a tool to exactly predict Easter, and, not least, to give Rome a meridian line as important as the one Giovanni Domenico Cassini had recently built in Bologna’s cathedral, San Petronio.

Here’s a picture from 1703 of how the whole thing would work, from Bianchini’s De nummo:

And this is how the left-hand side of that picture now looks:

See that hole in the wall in the upper right?  Here’s a close-up:

This lets the sunshine in, and there’s a cut in the cornice so that the light shines on the floor.  This is on the floor:

It’s a meridian, and I think the sunlight is supposed to strike it at noon, with “noon” referring to whatever time the sun is as high as it’s going to get that day.  But we were there around noon clock-wise and I looked for sunlight and couldn’t find it.  (In this picture, though, it almost looks like there is some light near the meridian.  It’d be really cool if that was the missing sunlight, but it might just be candles.)

Here’s a diagram that explains it all (click for a larger version).  It’s all in Italian, though.

There was more on the floor — concentric ovals which might have had something to do with Easter, and another meridian-looking thing that was raised in a display box held up by feet:

Despite feeling a little unsure about the details, it was pretty neat to see this sundial.  I’d read about it in this article and was glad to see it in person.

This meridian, by the way, served as an official timekeeper for about a hundred and fifty years.  After that mid-day was marked by another sundial and a cannon fired at noon from the Castel de Sant’Angelo, a tradition that is kept up even today in the form of a cannon fired at noon from the top of the Janiculum Hill.

More clocks coming up!

Memorial Day 2009

Last year, we posted a brief discussion of the history of Memorial Day, a US holiday of remembrance of Americans who died in military service for their country.

Following up on last year’s post:   Frank Buckles is now 108, and [naturally] is still the last known surviving American veteran of World War One.  Last year’s NPR interview with him is still available on-line.

Photo taken by TwoPi in January 2008 at Fort Rosencrans National Cemetery, on Point Loma, San Diego,  CA.

Why is April 15 Tax Day?

We’re one week away from April 15, and if you’re wishing you had a little more time to finish your taxes then be glad that you don’t live 50 years ago.

In 1913 Congress passed the Sixteenth Amendment to the Constitution, which said that Congress can collect income tax.  They’d already done this before, but were making it all official (although apparently when it was introduced it was expected to fail).  At that point, they declared March 1 to be Tax Day.  March 1, as you may recall, was New Year’s Day for the Ancient Romans, but that’s probably just a coincidence since quite a few days were New Year’s Day at one time and place or another.

After 1918 the Tax Day was changed to March 15, which probably resulted in a host of “Beware the Ides of March” jokes.

Finally, in 1954, the date was changed to April 15 [although that didn’t take effect until the following year].  According to this site (where I got most of the info so far), this was because the IRS was getting swamped with so many returns at the last minute, and they hoped that having more time would spread that out a bit.

In searching around, I also discovered the original 1040 form from 1913, which doesn’t look as simple as I would have hoped.    You can see all of them through the years here, although just looking at all those tax forms doesn’t exactly give one a feeling of peace and relaxation.

The number 2009

Inspired in part by the legendary Hardy-Ramanujan anecdote as well as the post “What is interesting about the number 2009?” at  Walking Randomly, I offer a few arithmetic and demographic curiosities.

2009 is odd, which in particular guarantees it can be written as a difference of squares, most obviously as , but perhaps more interestingly as .

Being a difference of squares is equivalent to being a sum of consecutive odd numbers (in the first instance “2009”, in the second instance “9+11+13+…+89”);  some other curious sums that lead to our new friend 2009 include

Hmmm.  There are our friends 1004 and 1005 again!  Interesting….  [See below for conceptual continuity]

2009 can be written as , which among other things guarantees that all groups of order 2009 are solvable.

Based on data published by Statistics Finland, the 2009th largest city in the world has a population of 207,000 people [a three-way tie between São Leopoldo Brazil,  Ashdod Israel, and Lutsk Ukraine].

Forbes famously publishes lists of the top n individuals, companies, etc… in terms of wealth, corporate size, market share, etc…  Forbes’ list of the world’s billionaires has 1125 entries, so we can’t identify the 2009th wealthiest individual.  However, we can use their data to extrapolate and estimate the net worth of the 2009th richest person.

The Forbes data has a very good fit to a power law distribution (in the graph, the blue rhombi squares represent individual data points, the black line the power law of best fit).  This model predicts that the 2009th wealthiest individual will have a net worth of billion dollars, or $770 million.

Their list of the world’s largest companies, alas, is the “Forbes Global 2000”, and as one might guess, has only 2000 corporations listed.  Unfortunately, Forbes publishes the top 2000 companies as measured by their own internal (unpublished) index, a weighted average of some sort based on sales, profit, assets, market value, etc….  Without knowing how their index is calculated, one can’t easily extrapolate to determine the size of a hypothetical 2009th largest company in the world.  Presumably it is similar to Mitsubishi Gas Chemical, the company ranked 2000th by Forbes in their list.

If these notes inspire you to find more instances or properties of 2009, add them to the comments at “What is interesting about the number 2009?” at  Walking Randomly.

May 2009 bring all of our readers a wonderful new year!

Added 1/1/09: I knew there had to be a rhyme and reason to the recurrence of 1004 and 1005 in both the difference of squares and the sum of fractions deal.  Here’s the scoop:

If x is odd, then.

Furthermore, , so in fact

,

and similarly since , we also have

.

3, 2, 1, 1 Happy New Year!

Ahh, the perennial problem:  what to do with the pesky fact that the year isn’t exactly as long as we want.   The Romans initially handled this by having the season of “winter” be of varying length.  Of course, those early Romans weren’t exactly known for their accurate record keeping: the pontifex maximus (Calendar Guy) would add an extra month from time to time to keep things sort of on track, but since a calendar year was also the same as the term of office in politics, he would also add days to a year when his allies were in power and subtract them when his opponents were in charge.  In other words, it pays to be on good terms with the person in charge of the calendar.

So we probably shouldn’t turn to the ancient Romans for calendar advice, though to be fair Julius Caesar is the one who authorized giving the calendar a much-needed overhaul and began the regular adding of leap days by having a second February 24 every 4 years (seriously:  February 29 didn’t happen for more than a thousand years after that).

But an extra day every 4 years isn’t exactly right either:  you’re off by a day every hundred or so years.  So with the Gregorian calendar the leap day is skipped at the turn of a new century.  Well, it’s skipped 3 out of 4 centuries:  1700, 1800, 1900, 2100, etc. skip their leap days, although 2000 (and 1600 and 2400) still have them.  And that got the alignment mostly on track, but it’s not perfect.

The problem is that the imperfection is inconsistent.  The length of a year actually varies a bit:  According to The Guardian:

The snag is that the rotation of the Earth is not so reliable. It is gradually slowing down and factors such as disruptions in the Earth’s core, extreme weather, volcanic eruptions and earthquakes can all influence the precise length of the astronomical day. From time to time, the rotation-based clock — UT1 time — and UTC [Coordinated Universal Time] need to be brought back into line.

So occasionally, like tonight, an extra second is added [or in theory subtracted, although that’s never actually happened].  Really, it’s like a Roman Intercalary month second since it doesn’t happen on a regular or predictable fashion.  And for most people it means nothing except perhaps an extra kiss at midnight, but people who pay very close attention to time get some extra work in by making sure that everything is aligned perfectly world-wide.

And that causes its own problems, because what with new-fangled technology like computers, being off by a second requires all sorts of overtime and problems.  The extra second is actually added at midnight Greenwich Time: according to a bulletin from the U.S.  Navy here, the official sequence will be

             31 DEC 2008 23 HOURS 59 MINUTES 59 SECONDS
             31 DEC 2008 23 HOURS 59 MINUTES 60 SECONDS
             01 JAN 2009 00 HOURS 00 MINUTES 00 SECONDS

but that’s actually 7pm here in New York and is mid-day in other parts of the world.  In other words, if you’re working 9-5 in California, you might want to ask for that extra second in overtime pay.

Note 1: Did you notice that UTC, not CUT, is the abbreviation for Coordinated Universal Time and assume that the abbreviation must be from another language? If so, you’d be wrong but for an amusing reason:   according to the National Institute of Standards and Technology, the abbreviation was specifically chosen not to stand for anything.  The English Coordinated Universal Time would be reduced to CUT and the French Temps Universel Coordonné would be abbreviated as TUC, so the International Telecommunication Union decided that everyone in the whole world should use the initials UTC as a compromise.

Note 2: The photo of the Times Square Ball is actually from last year, and is licensed under Creative Commons Attribution 2.0 by Clare Cridland.  But if you want to be current, Dave Richeson has a really cool post about the 2009 Times Square New Year’s Eve ball here on his blog Division by Zero.

Countdown to the Olympics

Nowadays we have fancy clocks that countdown to the Olympics, like this

Beijing clock; posted under creative commons license by topgold

or this

Vancouver clock by Makaristos

And these are nice and all, but they’re not exactly portable. For that, you need to go back 2100 years. In 1901, spongedivers found an ancient shipwreck near southern Greece. It had all sorts of stuff with it, including a bronze clock that was somewhat beat up but, in its heyday, looked something like this:

Olympic Digital Camera; creative commons copyright by Marsyas

The clock was named the Antikythera Mechanism, in honor of the island Antikythera near where the ship sunk, and it got its very own website here. For a long time no one knew exactly how it worked or what it measured. But then in 2005 Britain sent a 3-dimensional (tomography) x-ray machine down to Athens, where the pieces of the Antikythera Mechanism are on display, and they found out a couple things. First, it used gears, well before the Swiss watchmakers. Second, it kept track of the Olympic games. That’s particularly interesting because, with the games occurring every 4 years, a clock wasn’t strictly necessary. But as the folk of Beijing and Vancouver know, that doesn’t mean it’s not special.

The news articles that talk about this clock (e.g. here and here) also draw a connection to Archimedes. Different parts of Greece used different names for the months, and the names used on this clock were used in Syracuse, where Archimedes had lived 100 years earlier. (They were also used in other cities, but that’s not part of this story.) So Alexander Jones (from the Institute for the Study of the Ancient World at New York University) said:

We know Archimedes did mechanical astronomy here 100 years earlier and this could be from his home city, it could have been inspired by his work, or it could have been a local tradition that he started.

That’s a lot of “could”s. Nonetheless, whether or not Archimedes had anything to do with the clock or not, it’s still a pretty neat object.

Thanks to Pam for bringing my attention to this story!

Happy Pi Day!

Happy Pi Day!!!

22 July is celebrated throughout (much of) the world as Pi Day, for the ratio 22/7 is a reasonably accurate rational approximation to the number π.

Pi Day is also celebrated on March 14, in those parts of the world who would abbreviate today’s date ( July 22, 2008 ) as 7/22/2008, since March 14 becomes 3/14 under such a scheme. According to Wikipedia (“So you know it’s true!”™), only a handful of countries follow this scheme. Most would abbreviate using either a little-endian scheme ( 22/7/2008 ) or a big-endian scheme ( 2008/7/22 ). The amount of space on Wikipedia devoted to a flamewar discussion about the relative merits of each scheme is astounding.

There are many days when I’m happy to be a mathematician, and not a copy editor for an international open content network based encyclopedia.

The Calculus of Crabbing

TwoPi and I, still traveling around visiting family, were just on the Oregon coast for a few days. While we were there, my brother-in-law Ken took us out crabbing. The crabbing turned out to be a wash dinner-wise since they were too small, but the day was beautiful, the beaches calm, and as a bonus there was some cool math.

It seems that the best time to go crabbing is when the currents are weakest. To find out those times, Ken used a Tide Predictor:

The weakest tides are right at High Tide and at Low Tide. Why? That’s when the change in sea level — the derivative, in other words, is close to zero. Calculus in action! (Indeed, I imagine that current could be viewed as a kind of derivative of the sea height, since it is strongest when the slope of the water levels is changing the fastest.) As an aside, high tide is apparently better for catching crabs than low tide, but that has less to do with calculus and more to do with crustaceous personalities.

Edited 7/2 to add: I just realized that the second derivative also plays a role!  If the second derivative is closer to zero as well, it means that the current isn’t changing as quickly (in addition to not being very strong) so that gives a longer time period to check the traps and put them out again before the current gets strong enough that the crabs run back to the river sides or ocean.

If we hadn’t had the tide charts, we could have used this fancy Tide Clock on the wall:

Except it wasn’t working.

The tide itself leads to all sorts of other math problems. One of the neatest has to do with the cycles of the tides. The high tide peaks changed by almost 25 hours each day, not 24, so high and low tide cycle through different times of the year. It turns out that it takes 18.6 years for the pattern of high/low tide times to repeat itself. Apparently people used to be hired to take careful measurements of the tide and once the record stretched back 18.6 years, it was considered complete for that particular area. Not the most exciting job I suspect, but certainly important.

Memorial Day

Today is the day in the US for remembering those who died in US wars and conflicts.

Memorial Day has always been in May, although the date has changed over time. It began as a holiday to commemorate those who had died in the Civil War, which ended in 1865. There were several such ceremonies in the year after the war, but the “official” beginning (as declared by Congress and President Lyndon B. Johnson in 1966) was one in Waterloo, New York on May 5, 1866, which honored local veterans who had died in the Civil War.

Two years later, Major General John A. Logan of the Grand Army of the Republic announced what was originally known as Decoration Day.

The 30th day of May, 1868, is designated for the purpose of strewing with flowers, or other decorating the graves of comrades who died in defense of their country during the late rebellion, and whose bodies now lie in almost every city, village and hamlet churchyard in the land. In this observance no form of ceremony is prescribed, but Posts and comrades will, in their own way, arrange such fitting services and testimonials of respect as circumstances may permit.

The first (official) Decoration Day ceremony was held at Arlington National Cemetery.

While the ceremony was originally intended to honor only those who died in the Civil War, after World War I it was changed to honor all who had died in US wars. The date remained May 30 until 1971, when the Uniform Holidays Act changed it to be the last Monday in May.

One interesting note: Frank Woodruff Buckles was honored in a special ceremony this weekend. At age 107, he is the last known US veteran of World War I (having lied about his age in order to enlist in the Army). You can read more about his story here, and more about the history of Memorial Day here.

Happy New Year (Again!)

So, did anyone head out to any New Year’s Eve parties last night? Not surprising, I suppose, since the New Year has been happening on January 1 in many parts of the world for quite some time. Exactly how long depends on what country you’re in.

The early Romans considered March 1 the first day of the year, which is why September, October, November, and December mean 7^th, 8^th, 9^th, and 10^th month respectively. In 153 B.C.E, however, the New Year was set at January 1. Even though that’s the same date that many people use today, its adoption (like that of the Gregorian Calendar) wasn’t completely straightforward.

Easter and The Gregorian Calendar

Happy Easter! Since Easter played a key role in moving away from the Julian calendar, it seems fitting to talk about the adoption of the Gregorian calendar today.

As mentioned earlier, the Roman calendar had been having all sorts of problems until the time of Julius Caesar, who right before his death got everything back on track by adding a leap day (a second February 24^th) every four years. This worked really well except for one small problem: the solar year isn’t exactly 365.25 days long, it’s about 11 minutes/year less than that. But 11 minutes is hard to notice, so everything seemed hunky dory for a very long time.

More on Daylight Savings Time

Back in November, on the 6th day of existence of this blog (aahh, it seems so long ago!), I wrote a post giving a brief history of Daylight Savings Time. There was a pop quiz at the end: “Do you know which Department controls time laws in the United States?” and I had every intention of answering, but, well, I forgot.

So for all of you who have been waiting four months for the answer, here it is! (Drumroll, please). It’s the Department of Transportation. From “Saving Time, Saving Energy”:

In 1918, the U.S. Congress made the U.S. rail zones official under federal law and gave the responsibility to make any changes to the Interstate Commerce Commission, the only federal transportation regulatory agency at the time. When Congress created the Department of Transportation in 1966, it transferred the responsibility for the time laws to the new department.

Speaking of Daylight Savings, you can find a world-wide overview of DST on this webexibits site. Interestingly, in the US, Canada, and Mexico most of the country observes DST but there is a portion of each country that doesn’t. Likewise, different portions of Antarctica have different rules: Rothera (a British base) does not use DST, but McMurdo and Amundsen-Scott South Pole Station (US bases) do.

Happy Leap Day!

As mentioned earlier, January and February were added to the end of the previously ten-month-plus-winter Roman Calendar by Numa Pompilius around 713 B.C.E. From the start Februarius (February) had fewer days than the other months. Indeed, since the Romans didn’t like even numbers it was the only month with an even number of days at all: Martius (March), Maius (May), Quinctilis (July), and October all had 31 days, while Ianuarius (January), Aprilis, Iunius (June), Sextilis (August), September, November, and December had 29 days.