If the world did end today

The wind is strong
The rain is hard
Strange noises come
from outside my home
Perhaps the world will end tonight
the weather's right
And if it did, I'd leave
so much undone
so much unwritten
but at least
I wouldn't have to face
tomorrow
I took four classes this semester
and failed them all by any measure
If by tonight
I cannot write
several responses to
things I've yet to read
Then another failure
becomes official
unless the first
hasn't yet.
And of the other two
who knows?
I know one failure is deffered
but
can I make it into a success
before that extra time runs
out?
And of the other I haven't heard
Haven't heard for good or ill
which makes the prospect
seem worse still.
I hate uncertainty
it seems
more than certain shattered dreams.
And can I solve things by tonight?
Can I really read and write
what must be done
'fore time runs
down?
All hope seemed lost
long ago
and nothing new
has changed since then
in any way, but wrong.

The wind is strong
The rain is hard
Strange noises come
from the space
beyond my walls
Perhaps the world will end
tonight.
The weather's right.

The end of an Era (image post)

It think we can probably safely assume that people coming here looking for information on the Mayan Calendar will end today, not that the posts on it were ever that popular in the first place, still clearly these have seen their day come and go:

About the Mayan Calendar Part 1: Our Calendar and the Long Count
About the Mayan Calendar Part 2: Hypothetical Calendars
About the Mayan Calendar Part 3: Modular Arithmetic and the the Tzolk'in
About the Mayan Calendar Part 4: The Haab' and the Calendar Round
About the Mayan Calendar Part 5: Prophecy and the Short Count

On the accuracy of the Mayan Calendar,

Yesterday I said all that I'm going to say about the end of the word, so today I figured I'd make a post with images because I've been taking pictures of dawn and sunset for so long that it's about time I used at least one of them for something.

Some people believe that the world as we know it will end today, it will be the end of an era, the sunset of our age:

(Click on smaller image to see it larger) 

Others think that it is instead the dawn of a new era:

(Click on smaller image to see it larger)

And for a lot of people it's some other time of day:

Or night:

[Added:]

Crap, I forgot something, while going through pictures I came across this, I realize it's not plural, but even so:

(And yes, I know it's a mess.)

What to do when the world ends tomorrow/today

So, for some of us the world has already ended.

New Zealand for example, three hours since December 21st 2012 came.  For other's it's yet to come.  Or perhaps I should say that the world ends today for some of use and tomorrow for others.  Either way, doesn't much matter.

What to do when the world is ending?

It's been more than a year since I wrote, "The World Ends Today," but I think the general advice still holds true:

The point is, end of the world. It's kind of a big deal. It would be shameful to allow it to pass unnoticed and unnoted. 

Anyway, when I talk about the end of the world I tend to think of Exit Mundi which is a bunch of nicely sorted descriptions of how the world might end, and Alma Geddon which is a collection of predictions of when the world will end. For example, did you know that the world also ended on Friday, March 25, 970 CE? Well now you do.

And I still have yet to write a novel.  Not going to do it between today and tomorrow, so... damn.

Of course, half the time the world is ending.  These things happen.

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Also, the Mayan Calendar, I've written about that, haven't I?
About the Mayan Calendar Part 1: Our Calendar and the Long Count
About the Mayan Calendar Part 2: Hypothetical Calendars
About the Mayan Calendar Part 3: Modular Arithmetic and the the Tzolk'in
About the Mayan Calendar Part 4: The Haab' and the Calendar Round
About the Mayan Calendar Part 5: Prophecy and the Short Count

On the accuracy of the Mayan Calendar

If you want a, mundane, sign of the world actually ending consider this: that five part series (originally intended to be a single post but it got out of hand) was interrupted (between 2 and 3.)  By snow.  And hardly the first significant snow of the year.  That was more than a year ago, making it earlier in the year than this.

The first significant snow of this year came on Monday, a fair bit later than the interruption of last year.  The rains since Monday have washed most of the snow away.  That means we're not even very deep into Autumn yet.  Winter is nowhere on the horizon.  Temperature wise, I mean.  From a "relationship of the earth to the sun" perspective winter starts tomorrow, when the world ends.  Or today, if you're in certain time zones and the northern hemisphere.  (Only northern hemisphere winter that starts tomorrow, after all.)

Alone that means nothing.  Sometimes seasons come late or don't come at all from a climate/weather perspective, that's why we mark them with an calendar based on astronomical events in the first place.  It'd be damn annoying to have the official seasons shift yearly on the whims of the weather.

The problem is that it's not alone, all the data points in the same direction.  Things are changing, and while the changes might be better for someone, humanity is not counted amoung such.  If you want to be worried about the end of the world, I direct your attention to climate change, not the Long Count of the Maya people.

Of course, the people who say the Mayan Calendar predicts the end of the world tomorrow, do say that end will come by rising waters, and sea levels are rising, so give it some time and maybe it'll turn out that tomorrow was some inflection point.  It's extremely unlikely, but in an infinite universe anything is possible.

On the accuracy of the Mayan Calendar

My watch does not display the correct time.  The time it has is about two and a half minutes ahead of actual time.  Slightly less.

I know this, and since I know this I can tell the correct time fairly accurately.  If I were to reset my watch to the correct time I'd risk forgetting that I reset and assuming it's about two and a half minutes later than it actually is.

Knowing the limitations of your time piece can be valuable   It allows one to adjust accordingly.  That's how our own system of leap days came to pass.  They adjust for the limitations of our calendar which left to its own devices would get well out of whack.

A different method for adjusting would be to just keep track of how out of whack it had gotten and adjust one's behavior accordingly.  My watch doesn't really run fast or slow, it's just got the wrong time and so continues to have the wrong time at about the same degree of wrongness as the days go on.  I know it's wrong and it doesn't bother me much.  It might if I didn't know it was wrong.

If I wanted to, I seldom do, I could figure out exactly how far off my watch was and then, via adjustment, tell the correct time very accurately.

In some people's eyes that makes my watch a very accurate watch.  Technically it just has a level of precision and a mostly uniform inaccuracy.

I bring this up in a post with "Mayan Calendar" in the title because of, I suppose, a quote from Silverado: "Life is what you make it.  If it doesn't fit, make adjustments."

I adjust for the inaccuracy of my watch.  Calendars are forced to adjust for the inaccuracy of Calendars because the solar system never got the message that we'd like to have a calendar that consistently reflected the seasons year after year.

We make adjustments with leap days.  Add a day if the year is divisible by four, unless it's divisible by 100, unless it's also divisible by 400 in which case add the day after all.

They Maya people did something more akin to what I do with my watch.  The recognized the inaccuracy  they took it into account, they learned to live with it.  The result was that that their adjustment (which basically involves saying, "If we wait 1508 Haab' years that's the same as 1507 solar years and things line up again") gave them a better approximation of the solar year than ours by a nearly negligible amount.

Presenting that as them having an extremely accurate calendar is like me presenting my watch as extremely accurate because I happen to know how far off it is and can adjust for it.

Their calendar was not accurate.  It was wrong.  They did an impressive job of figuring out precisely how wrong it was and taking that into account, but the calendar itself was not accurate.  The Maya themselves were apparently able to be accurate in their use of it, because they were able to take its creeping inaccuracy into account.

About the Mayan Calendar Part 5: Prophecy and the Short Count

It must be Thursday, I never could get the hang of Thursdays

-Arthur Dent

The mere existence of cycles calls out for us to try to find correlations between the present and the past. Arthur is having trouble with the conversation with Ford so he correctly concludes that it must be Thursday. Garfield has found Mondays unpleasant in the past, therefore future Mondays will be unpleasant. Friday the 13^th is bad. (Though, actually, yesterday seemed just fine to me.)

A solar calendar, such as our own, is designed with this in mind. It's set up to track the seasons so that, “It got cold this time last year,” is actually useful information when trying to figure out whether it is likely to get cold at this time this year. If you happen to know what time it got cold the last six years, even better.

My understanding is that when the Maya made predictions about the future it was often based on the idea that what has happened before is a signpost to what will happen again. They definitely had a firm grasp of the idea of linear time, the Long Count (particularly the occasionally used extensions of it) proves that. None the less they had their cycles and believed that those could shed light on what was to come.

This is, unfortunately, something I know almost nothing about. It's no problem writing about how something like the Tzolk'in works with limited knowledge of the Maya because you don't need to know all that much. If you know that it works on interacting cycles of 13 and 20, and you know a bit of modular arithmetic, you can talk about how it works perfectly well. How it's interpreted is another matter entirely.

I know that there were meanings to the days, I believe that every one of the 260 days in the Tzolk'in had some kind of meaning associated with it, I'm guessing some in more depth than others. I could try to look up those meanings but I'm not sure how comfortable I would be simply relaying that here, unlike the math of the calendar I can't double check it using things I already know. I think taking about that should probably be restricted to people who actually know what they're talking about. I've seen what happens when people who don't know what they're talking about start going off on the Maya. I watch Ancient Aliens. (On occasion. I don't watch it all the time. Don't judge me.)

Also, there are 260 of them, which seems daunting.

Another set of prophecies I've heard brought up a lot has a much smaller number to deal with, by a factor of 20. That seems like a simpler thing to work with. It requires introducing yet another calendar.

If you're like me it probably occurred to you that the Long count could be connected to the Calendar Round to make a really big cycle. Not as big as it at first appears. I had completely forgotten that there were 13 of the largest unit (in part because in the extended version there are 20 of them.) That means that the Long Count is divisible by both 20 and 13 so adding the Tzolk'in wouldn't extend the cylce any. Adding the Haab', on the other hand, would create a cycle of 73 Long Counts.

That didn't happen and there is not a giant Mayan Calendar with a cycle of 377,151 years and change called the Longer Count.

But at some point someone did get the idea to make part of the Long Count cyclical. They threw out the Ba'ak'tuns so that the k'atuns (which last about 19.7 solar years) were there largest unit, and then they differentiated those using the 13 number days from the Tzolkin. Thus the Short Count was born.

For reasons I don't pretend to understand they named each k'atun after the last day in it.

The first one started on 1 Imix. Since the number of days in a k'tun is divisible by 20 the first day of each k'tun was Imix' and the last day was always Ajaw, which is also written Ahau (I think I've been inconsistent with how I have spelled it). What changed was the number day.

A k'atun is 20*18*20 days. Or 7200 days. That's 11 mod 13, so the last day of the first k'atun was 11 Ahau and that is what it is called. 11 = -2 (mod 13) and that'll be easier to work with. To find the order of the k'atuns we just subtract two from eleven repeatedly:

11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2, 13

Just in case anyone was confused by it:

1 – 2 = -1 = 12 (mod 13) and

2 – 2 = 0 = 13 (mod 13)

So that's why it goes from 1 to 12 and 2 to 13. There are various ways to think about why that works, but I think the easiest is to just think of it as a circle where 13 is number before 1. One step before one can be called zero, but in this case it also happens to be 13, and we could call one step before that negative one, but it also happens to be twelve.

I'm never sure how much I should explain the modulo math. On the one hand, saying that one minus two is twelve looks like something that needs an explanation, on the other hand I don't know if people are getting frustrated that I explained it again.

So, moving on, there is this cycle of k'atuns and there are prophecies associated with them. The prophecies are created by looking at what already happened and assuming it will happen again. They're largely negative, probably because text I know about was written after the Spanish showed up and thus the past they were looking to wasn't exactly happy.

So what does it have to say about k'atun starting in December of 2012? (That is what everyone is supposed to wonder about these days, right? The History Channel won't stop making shows about it, after all.)

Well if you look to the internet you'll almost certainly find that it says something like, “For half there will be food, for others misfortunes. A time of the end of the word of God. It is a time for uniting for a cause.” And no one ever seems to say where they got it from. They'll tell you that it's from the prophecies of Chilam Balam, but there are multiple such books. Also, it isn't as if they were written in English, someone had to translate. But the quote always seems to be attributed to Chilam Balam as if he personally dictated it to them in English.

I think it comes from this article from 1996. Its version is, “For half of the katun there will be food, for half some misfortunes. This katun brings the end of the "word of God." It is a time of uniting for a cause,” and unlike most of the rest of the internet, it says where it gets that from. Before laying out its version of the k'atun prophecies the article states that, “The delineations below are a composite taken from the Book of Chilam Balam of Chumayel and the Codex Perez and the Book of Chilam Balam of Mani.”

The Codex Perez is more commonly known as the Paris Codex. It was written in hieroglyphics and I cannot read it. If there's an easy to access translation I do not know of it. You can find photographs of it online, if you want them. It appears to be in poor condition.

In the case of the Chilam Balam book of Mani, just like the Paris Codex there's a problem of access to English translations. I don't see any online, and I'm not going to try to get a book from the library, probably involving inter-library loan, in the middle of writing a post.

That leaves the book from Chumayel. An English translation of that (from 1933) is available online, and is very easy to find at that. Here is a PDF of the translation that was linked to from Wikipedia.

Here is what it has to say:

Katun 2 Ahau Is the twelfth katun. At Maya [uaz] Cuzamil the katun is established. For half <the katun> there will be bread; for half <the katun> there will be water. <It is> the word of God. For half of it there will be a temple for the rulers. <It is> the end of the word of God.

There's nothing about uniting for a cause, but perhaps that came from one of the other sources. If you accept that that water == misfortune then the version from the 1996 article seems to stick reasonably close to the source.

I think that the reason for that is simply because the original is so short. Most of the things are significantly longer and more historical in their nature than the Katun 2 description. As the article's author puts it, “In the several books of Chilam Balam, the influences of the thirteen katuns are stated, usually as a description of historical events that occurred during previous cycles.”

There is actually a lot of history in some of the accounts and it's hard for me to separate the history from the prediction*. The article attempts to do that creating short summaries. The problem is that these summaries then get taken, have the parts that might indicate they are summaries chopped off, and get sent around the internet as if they were the actual texts of the prophecies.

And it's not just the internet. The History Channel in all it's misleading glory has shows where a narrator with an imposing voice reads parts of those summaries as if they were the actual words of Chilam Balam with no indication that they're highly condensed summaries.

The prophecies take up 3451 words in the 1933 translation (which, recall, is stated to be but one of three sources.) The summaries 555 words. Some of those get thrown out. The summary for k'atun 8 is:

This may be the worst of the katuns as both Chichen Itza and Mayapan, the two great ruling cities of Yucatan, were destroyed during its period. The texts speak of demolition and destruction among the governors, an end to greed, but much fighting. It is the katun of "settling down in a new place."

It gets cut down to:

A time of demolition and destruction among the governors, an end to greed, but much fighting. A time to settle down in a new place.

So you can see how little of what was started with must be left in the version you end up with, but that is what is presented as the prophecy. Which is why it made me think that this would be something I could tackle so I could actually share some the the prophecies here. It looks like it would be significantly more difficult than I thought, and once again I find myself running out of day.

So I'll close on this. I may not know exactly what the Maya prophesied, I certainly have no idea if those oft quoted summaries are accurate (I only have access to one of the sources that we're told were used to make them), but I do at least have a grasp on how the prophecy was supposed to work.

The basic idea comes to us from Battlestar Galactica: All this has happened before, and all this will happen again. I'm told the phrase was taken from Disney's Peter Pan, proving that at least in the case of the phrase, it did happen before. The k'atun cycle gives you the schedule. About every 256 years. Closer to 256 and a quarter if you want to be more precise.

So that's the whole idea. Want to know what will happen from 2012 to 2032? Look at what happened from 1756 to 1775. Also try 1500 to 1519. Or the one before that, or the one before that.

This post did not turn out at all how I expected it to. I was hoping to be able to share at least some of the prophecy in an interesting way, but it looks like I'm really not equipped to do that. So instead I've given you the tools to make your own. Look at the past and you should be able you make your own prophecy the way the Mayans made theirs.

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*In a certain sense history is prediction here, but I very much doubt that they thought the exact same thing would happen the next time around. I have no idea how similar they expected it to be. Did they expect white men with red beards to come from the east 256 years after the time written about in the prophecies, or did they just expect invaders from somewhere, or did they expect something that would feel like invaders from somewhere? I have no idea.

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[Previous]

About the Mayan Calendar Part 4: The Haab' and the Calendar Round

Last time I talked about Tzolk'in, a 260 day calendar built on cycles of 13 and 20. I'm told that all Mesoamerican cultures had a calendar like the Tzolk'in and that it was the most important of the Mesomerican calendars. That said, I'm told that by Wikipedia and yesterday I had to correct a formatting error in Wikipedia that was screwing up a page on different calendars in such a way as to make it appear that “Poseidon” was a Mayan month. I recommend taking Wikipedia with some salt.

What I can say is that whether or not the Tzolk'in was the most important Mayan calendar, it definitely wasn't the only one they had.

The Haab' is a lot like our calendar. It approximates a solar year, it has months with names, and it doesn't require one to understand modular arithmetic to make sense of it.

The Haab' has 18 twenty-day months. There's a possibility that you might be thinking, “18 twenty-day periods? That's a tun.” You'd be right if not for the fact that it also has a five day period as well. If not for the five day period the Haab' would be just like a Long Count tun and it's months just like Long Count winals.

Anyway, it's basically just a Calendar with months like our own.  There are more of them than our months, and the first day is the Seating of [month] with the day called 1 being the second day, but other than that it's pretty much the same thing. You've got Seating of Pop (which I'm probably going to call 0 Pop from here on out), 1 Pop, 2 Pop, all the way up to 19 Pop, then you're into the next month with 0 Wo', 1 Wo', 2 Wo' and so on.

The names of the months are Pop, Wo', Sip, Sotz', Tzek, Xul, Yaxk'in', Mol, Ch'en, Yax, Sak', Keh, Mak, K'ank'in, Muwan', Pax, K'ayab, and Kumk'u. The five day not-in-a-month period was called Wayeb'. (It went between Kumk'u and Pop.) If I need to refer to them I'll probably resort to numbers, so 1,1 instead of 1 Pop, but we'll see when we get there.

18 months of 20 days makes a 360 day period, and the five extra days makes it 365 which is as close to a solar year as you can get while still using whole days. The Maya knew that it wasn't exact and were content to figure how much it drifted and work with that instead of trying to fix the problem with leap days the way we would.

This brings us to a claim I've often heard which is that the Mayan calendar more accurately reflected the length of a year than ours does. Until I was looking up unrelated things for this series of posts, I never really knew where that claim was coming from. Now I do. It has to do with how we account for drift in our calendars.

It would be completely inaccurate to say that out calendar treats the length of a solar year as 365 days. It doesn't. We know that results in things getting out of whack and if we left it that way things would get way out of whack so every four years we have a leap year to knock things back into whack. If that was all you knew you'd think that our calendar thought that the length of a year was (365*4+1)/4 = 365.25 days. That's also inaccurate. (That's the Julian Calendar, we use the Gregorian one.)

One leap day every four years knocks things too far in the opposite direction, so every hundred years we skip a leap day. Which is why 1900 wasn't a leap year. So in that case you might think our calendar said the length of a year was (365*100 + 25 – 1)/100 = 365.24 days. But it turns out that that pushes things too far back in the first direction. So every 400 years we skip skipping a leap day, which is why 2000 was a leap year after all. And at that point we say, “Close enough” and stop with all this messing with leap days.

So in the end our calendar makes a claim that 400 of our years is roughly equivalent to 400 solar years. Which is to say that it claims a solar year is (365*400 + 100 – 4 + 1)/400 = 365.2425 days long.

The Maya had a completely different method of dealing with this problem. I don't pretend to know their exact reasoning, but their result looks like they said, “Ok, next year's 1 Pop is going to be a little off from this year's, and the year after's 1 Pop will be even more off. If we don't do anything it'll drift further and further in that direction. How long until it comes back around?”

What they figured out was that every 1508 Haab's there were about 1507 solar years. So where our calendar says that one solar year is 400 calendar years over 400, their calendar says that one solar year is 1508 calendar years over 1507. 1508*365/1507 = 365.2422 (I'm rounding down very slightly.)

Now there are two things worth noting here. One is that the claim is accurate, 365.2422 is closer to the length of a solar year than 365.2425. (How much closer depends on the year.) The second is that it's not by much. The difference between the two is less than .0003 days.

I don't know about the rest of you, but for me the claim that the Mayan Calendar is more accurate about the length of a year than our own calendar is a lot less impressive when you look at it in any kind of depth.

One possible reason that they didn't use a leap day is that they wanted to be able to use the Haab' in combination with other calendars. A leap day would have messed that up (unless a leap day was inserted into every single cycle with which the Haab' might interact.) I don't know if that was their reasoning. I do know that they used it in combination.

The Calendar Round was a calendar created by combining the Haab' and the Tzolk'in. 260 days in the Tzolk'in by 365 days in the Haab' would make 94900 potential days but not all of those days can actually happen.  (Remember what I said about combining cycles in post 2.)  Only one fifth of them can. This is because 260 and 365 have a factor of five in common. The Calendar Round is 52 Haab's long, or, if you prefer to think of it in a different way, it is 73 Tzolk'ins long which is 18,980 days.

The current creation is said to have started on 4 Ahau 8 Kumk'u (that's when the Long Count started), and when that date comes back around it's a Calendar Round completion.

I have heard the Calendar Round referred to as being equivalent to our century. Not in length, obviously, but as a concept. When we talk about the '20s, the '60s, '85 and such we don't have to specify which century we're talking about because it's understood which one we mean, I think that's what people are talking about when they compare the Calendar Round to a century.

Haab' years are associated with the Tzolk'in day on which they start, so now seems like a good time to say how the two fit together.

If I just pick numbers at random I risk ending up in one of the four alternate realities with days that are impossible in this one, so I'm not going to do that. I'm told that 1 Ik' (which I've been calling 01;02) falls on 0 Pop in the real world. The Haab' year that starts on 1 Ik' will be associated with 1 Ik' (which is important for Mayan prognostication which I unfortunately know nothing about) and 1 Ik' is known as the year bearer.

There are a couple of things to note. The first is that for the rest of the year the days of the Haab' month and the days of the Tzolk'in month will stay the same. Its not just 0 Pop that will be Ik', every 0 [Haab' month] will fall on an Ik'. Until the five extra days at the end of the year knock the two out of alignment Ik' will be day 0 of every month, Ak'b'al will be day 1, K'an will be day 2, and so on.

The second thing is how the year bearers cycle. 365 = 1 (mod 13).  It is also 5 mod 20. So to find out what the year bearer after 1 Ik' will be we add 01;05 to 01;02 and get 02;07 or 2 Manik'. Do it again and we get 03;12 or 3 Eb', again and it's 04;17 or 4 Kab'an. One more time and it's 05;02 which is 5 Ik'.

In other words, the year bearers work so that the number cylces from 1 to 13 increasing by one each time and the days cycle through the four days of Ik', Manik', Eb', and Kab'an.

Calendar round days are written Tzolk'in date Haab' date. So 1 Ik' 0 Pop if anyone other than I am writing it, and possibly some monstrosity like “01;02_00,01” if I happen to be writing it. I haven't decided yet.

For the Tzolk'in I wrote up a thing on how to figure out how far apart two days are, I suppose I should do something similar with this, but that's going to wait for another time. I don't even think I'll get to it this week.

A full Calendar Round is 0.2.12.13.0 in Long Count notation, if you were wondering.

Next time the Short Count and the very little I know about Mayan prophecy and prognostication.

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[Previous][Next]

About the Mayan Calendar Part 3: Modular Arithmetic and the the Tzolk'in

[I meant to get to the Haab' and the Calendar Round this post, but I ran out of time.]

I should have introduced this last post since that was about concepts rather than actual calendars, but I was tired and for some reason it didn't occur to me that this would be a good thing to introduce. So here's the last thing before we get to the actual calendars: Modular arithmetic.

Modular arithmetic is something you're already familiar with. It's just very basic math on cycles. Hours exist in cycles (be they 12 or 24) so whenever you do math with them you're doing modular arithmetic. Weeks are seven day cycles so if you've ever had to figure out, say, what day of the week ten days from now is then you've done modular arithmetic.

The way it works is pretty simple, doing math on a cycle of a given length, say, “N,” is called doing math “modulo N” which is written “(mod N)” for equations. In math modulo N there are only N numbers, and the way that works is that every number that has the same remainder when divided by N is considered the same number*.

So, for example, the hours in 12 hour time work as modulo 12. We could call the hour before 1, “Zero,” (and in 24 hour time we do just that) but calling it 12 instead of 0 makes no difference in 12 hour time because 0 = 12 (mod 12). It's easy enough to see this because 12 goes into 0 zero times with a remainder of 0, and 12 goes into 12 one time with a remainder of 0. Same remainder when divided by 12 means that they're the same number modulo 12. If you wanted to know what is three hours after Ten, 3+10 = 13, and 13 = 1 (mod 12).

Addition, subtraction, and multiplication all act exactly how you'd expect in modular arithmetic.

Ok, now that we've done that, onto the calendars.

The Tzolk'in is the kind of calendar I was thinking about when I decided to make a post about the Mayan Calendar. It is made by having two smaller cycles run at the same time.

There is a series of 13 days which are simply numbered one to thirteen. There is a series of 20 days that have names. (Imix', Ik', Ak'b'al, K'an, Chikchan, Kimi, Manik', Lamat, Muluk, Ok, Chuwen, Eb', B'en, Ix, Men, K'ib', Kab'an, Etz'nab', Kawak, Ajaw) I'm just going to call them 1 to 20.

A day is written “Number Name”, so 1 Imix or 12 Kawak, I'll be writing those as 01;01 and 12;19 respectively. (Semicolon because a dash looked too much like subtraction and a colon looked too much like time of day.  Using a space to separate them doesn't feel right.)

13 and 20 are relatively prime so they combine to make a 260 day calendar. It's immediately obvious that 260 isn't even close to the number of the days in the solar year. Some speculate that it might have to do with the length of a human pregnancy, or the planet Venus, or a certain season, or it might just be that they liked 13 and 20. It's possible that none of these things are correct. It is also possible that all of them are. (If you were deciding on a new calendar, would you rather have one that had one thing going for it, or twenty seven?) I don't know enough about the Maya, or the other Mesoamerican cultures that used similar calendars, to meaningfully speculate. So, back to the math.

(For the record, I'll be calling the 20 day cycle a month.)

20 is 7 mod 13, so the 20^th day of the calendar would be 07;20. The 21^st would be 08;01, if you were wondering what the first day of the second month would be.

I've mentioned before that it is completely reasonable to add and subtract dates. If you want to know what the 15^th day of the calendar is you can simply add the 5^th day to the 10^th day, for example. It is worth noting that some dates are easier to add than others.

If you wanted to know what was one month from a given day you could add 07;20 to it. Now that 20 might as well be a zero because the second number operates in mod 20. So that's a nice thing to add since you only have to add one number instead of two. Take a date, add seven to the first number, and you get something that's one month later. That's handy, but not nearly as handy as noticing what happens when you add two months.

7+7 = 14 = 1 (mod 13) Which means that to find out what's two months from a given date you can just add one to the first number, to find out what's two months before a given date, just subtract one.

That makes it much easier to figure out how far into the calendar a date is than I imagined. You just take a date, say 08;03, and do the following:

The second number, 3, is how many days into the month the day is. Now that we know that, we want to know how many months into the year we are, so we subtract three days. That puts us at 05;20, but since we'll be dealing with whole months now we don't care about the month number anymore, we just care about the five.

Every two months changes that number by one, so that five represents ten months.

That means that we're ten months and three days into the year, or 203 days into the year.

It is important to remember that if you do it exactly like that sometimes you'll get a number larger than 13 for the number of months. That happens when the you start with is larger than seven, which happens on odd numbered months. If you just subtract 13 from the number of months you do get, everything will work out fine.

To find out how far apart two days are, you subtract the first one from the second, and then do the above to the result. Originally I picked two dates and did needlessly complex math. Those dates, chosen with my closest approximation of randomness** were 05;18 and12;06. I was wondering far after 05;18 12;06 is.  I put myself through way more trouble than I should have to figure out because I had yet to realize the above.

12;06 - 05;18 = 07;08         (because 6 - 18 = -12 = 8 (mod 20))

So now we just do the same stuff as above.

7 – 8 = -1 = 12 (mod 13)

12*2= 24.  24-13 = 11.

12;06 is 11 months and 8 days (228 days total) after 05;18.

That's not nearly as daunting and scary as I expected it to be. If I wanted to figure out how far apart two days are in our calendar I'd probably have to resort to the “Thirty days has September” rhyme. (And that wouldn't work because I've forgotten it.)  But I think most people other than me actually know how many days are in each of the months.

Even though it isn't as hard as I expected, it's nice to be able to look at something and immediately know that it's (approximately) X months away the way we do when see, say, that a date is in August. Also it's nice to have a calendar where the year approximates a solar year. I have no idea if either of these things is the reason that the Maya had a calendar that did just that, but the Maya had a calendar that did just that.

I've run out of day, so next time the Haab' and the Calendar Round.

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* It is actually a little more complicated than that. They aren't considered equal they're considered congruent, and technically I shouldn't be using an equal sign but instead a similar sign with three bars instead of two. For our purposes it is sufficient to say that 12 = 0 (mod 12).  Certainly if you think of them as the hours on a clock face the 0th hour is the 12th our and the 13th is the 1st.

** I'm sure that there are random number generators just a google away, but I have yet to do that googling and thus I just punch keys and pretend the result is truly random.  Part of the reason I can do this is that I'm almost never in a situation where I would actually need a random number.  Certainly if I'd specifically selected the numbers in this post it wouldn't have changed much.

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About the Mayan Calendar Part 2: Hypothetical Calendars

Any day in the year can be uniquely identified with two numbers. For whatever reason I want the number with the smaller range to go first, so let's have an order of Month-Day. The Month number ranging from 1 to 12 and the day number ranging from 1 to 31. In an ideal world that would be all of the information you needed, but there are not 372 days in the year so some of the identifications people might make with those rules, say, “2-31,” are not real days.

But say I didn't care about that, say I didn't care about the number of days in the year at all and I just wanted to make up a calendar. Say I wanted it to be 210 days. 210 is seven times thirty, so I should be able to make a calendar that looks like X-YY where X goes from 1 to 7, and YY goes from 01 to 30. I could make one, so could you.

How would you do it?

I'm guessing that most people reading would create seven months of 30 days each. So you'd have 1-01, 1-02, 1-03, all the way up to 1-30, which would be followed by 2-01, 2-02, and so on. Some people might decide to have 30 weeks instead. So it would be 1-01, 2-01, … , 6-01, 7-01, 1-02, 2-02 and so on.

I'm guessing that because our calendar is built by dividing things up at different scales. We divide the year into 12 months, then divide each of the months into days, and each of the days into hours, and so on. Switching from one cycle to another is like zooming in or zooming out. We almost never have two different things going on at the same scale.

The Long Count I discussed in the last post is also built on that kind of thinking with each unit divided up by a unit below it, used to divide up a unit above it, or both. Never were there two things happening side by side on the same scale. As I said, I don't really find that interesting.

The Mayan calendars that I do find interesting work entirely differently.

If you wanted to make the calendar I ordered above in the style of one of those Mayan calendars you wouldn't divide and then divide the divisions. You'd simply start counting seven day weeks and thirty day months. You wouldn't have one number stay fixed while the other cycled through, you'd have both numbers moving at the same time. 1-01 would be the first day, 2-02 the second. After 7-07 would come 1-08. The thirtieth day would be 2-30, and the thirty-first 3-01.

Each day still has a unique identifier, you've still got 210 days mapped out so that you can find any day in it by being told it's X-YY, but the way we arrived there is completely different. Instead of naming months or numbering weeks or something like that, you've just got months and weeks and are letting them work alongside each other to create a larger system than either makes on its own.

In this system staying something happened on Friday the 13^th narrows it down to one day every seven months, because Friday the 13^th comes on a regular schedule. Which means that if 210 days is the period you're concerned with, you don't have to name the months. And if you did want to name them, you could just name them after the day they start on since each month in that period starts on a different day.

We could also imagine a system where the months were exactly the same as we're used to, but the there was no leap year (Or if leap day didn't count as a day of the week that would work too.) In that case, weeks would fit together with years such that, say, Sunday the first of January only came once every seven years. It would actually fit together quite nicely because on the next year the first of January would be Monday, then Tuesday the year after and so on. The result would be a seven year cycle. Years within the cycle wouldn't need to be numbered because the year that starts on Wednesday clearly comes two years after the one that starts on Monday.

These probably aren't the best examples, in part because I'm trying to use intervals we're familiar with, a seven day week, a thirty day month, a 365 day year.

That said, I think they're at least somewhat instructive. They show how you could set things up, they also, I think, how confusing this whole thing seems. I don't know about anyone else, but I definitely can't quickly figure out if two dates are close together or far apart. It would be pretty straightforward to know what day tomorrow or yesterday is. But if you pick two random dates in the month and week calendar I described, say Monday the 1^st and Friday the 13^th, it isn't immediately clear to me if they're very close together or very far apart. (I had to derive a formula for converting between a week-month calendar and one where the days were numbered 1 thorough 210 to figure it out.)

On the other hand if I say the 1^st of month 5 and the 13^th of month 1, you've got a much better idea of how far apart they are (though do remember that there are seven months here, not twelve.) So there's definitely something to be said for the way we do things. And when I get to the Maya's actual calendars you'll see that they seem to have realized that. Unfortunately it doesn't look like I'll be getting to it today.

So I want to close this with some other hypothetical calendars just to illustrate some points.

Seven and thirty make a calendar with 7*30 days because they don't have any factors in common. If I'd chosen six and thirty it wouldn't have worked at all. The thirtieth day would have been 6-30, and the thirty-first day would have been 1-01, right back at the beginning. The six day cycle would function as a subdivision of the 30 day month and nothing more. This is, obviously, because six divides 30. There's no remainder so the six cycle is back to the beginning after 30 days.

If I'd decided that I wanted to get closer to the number of months in a year and chosen 12 instead of seven, that wouldn't have worked quite right either. It would have been a perfectly legitimate calendar, but it would have been a much shorter one. It would have been 60 days instead of the 360 you might expect if you thought there would be 12*30 days. The thirtieth day would have been 06-30, which means that in another thirty days you'd find yourself at 12-30, and the day after would be 01-01.

I haven't mentioned this, but it's probably worth noting that you can feel completely free to add dates. If X days into a calendar is A-C, and Y days in is B-D then X+Y days in is (A+B)-(C+D). And now I'm regretting using a dash, which looks an awful lot like a minus symbol, to separate the numbers in the dates. Anyway, that works. The only thing is that the number you get might be too high. As in the last example if you add 06-30 to itself you get 12-60, but you can't have a number higher than 30 in the second slot. So you just subtract a 30 and you get 12-30, which is the actual date you're looking for. (If the number in the first slot had been to high you'd subtract a 12.)

Anyway, digression over, there is in fact a rule for how many days are in cycle created by using smaller cycles. There are as many as the least common multiple of the smaller cycles. The least common multiple of two or more numbers is just the smallest positive number (zero is not a positive number) that is a multiple of all of the numbers in question. 210 is the smallest positive number that is a multiple of both seven and thirty. 60 is the smallest positive number that is a multiple of both twelve and thirty. 30 is the smallest positive number that is a multiple of both six and thirty.

So if we decided to expand our 210 day calendar by adding a 12 cycle to go alongside the 7 and 30 cycles, the result would be a 420 day calendar, as that is the least common multiple of 7, 12, and 30.

So, right now I'm going to sleep but next time I plan to get to actual Mayan Calendars that use this sort of system.

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About the Mayan Calendar Part 1: Our Calendar and the Long Count

I've written and rewritten this post several times because there's so many possible places to start and so many ways to approach this. I'm still not sure what the best one is.

I've heard it said that our calendar is linear while the Mayan calendar is cylindrical. That's an oversimplification in various ways.

It's easy to see why people say our calendar is linear, just look at the years. 2011 will not be coming back around. No matter how much time passes it won't come back. It just gets further in the past. But Thursday will come back around.

So will 5 AM, and half passed the hour, and the '20s. These things all happen in regular and predictable intervals. We have a Thursday every seven days, we have a 5 AM every day (assuming you limit yourself to one time zone), we have a half passed the hour every hour, and we have a '20s every century. There are definitely cycles.

I bring this up in part because I think it's important to note that cyclical and linear things kind of go together. If you look at a cycle (say the days of the week) it looks perfectly linear until it starts repeating. If you look at the units that make a linear count they're almost certainly going to be cyclical.

The digits of numbers are cyclical, for example. You can count up and up forever and never repeat the same number, but the ones' digit is going to cycle through the same ten numbers over and over again.

Our method of time keeping is largely like that, but with a truly bizarre hodgepodge of bases. If we wrote the time like we write a number, with the smallest unit on the right and the largest on the left, we'd write it as:

millennium, century, decade, year, month, day, (AM-PM), hour, minute, second.

Whether AM or PM is included depends on whether you prefer 12 or 24 hour time.

Part of the reason that I'm bringing this up is because, for everyone who uses the same calendar and clock system as I do, this is the system you're used to. This is something that probably makes sense to you without even thinking about it. This is probably what you consider easy. Yet here is how it works:

A millennium is 10 centuries, each of which is 10 decades, each of which is 10 years, each of which is 12 months, each of which is [28, 29, 30, or 31] days, each of which is 24 hours each of which is 60 minutes, each of which is 60 seconds. If you use 12 hours with AM and PM then it's slightly different and I'm not sure what you call the periods designated "AM hours" and "PM hours" perhaps?

If you work on 24 hour time you're essentially working in a mix of bases 10, 12, 24, 28, 29, 30, 31, and 60. If you use 12 hour time it's 2, 10, 12, 28, 29, 30, 31, and 60.

Given that I'm just planning on talking about the Mayan calendar, not their way of dividing up a day, perhaps I should leave out the hours minutes and seconds. In that case it's just 10, 12, 28, 29, 30, and 31. Still an odd mix.

I bring this up because when I say that the Long Count is incredibly simple to the point that I don't find it interesting, I want you to have our system fresh in your mind.

So, the Long Count calendar. It's probably the most famous of the Mayan calendars, it's the one that ends in 2012 and thus something that you've probably heard a lot about lately. I don't find it interesting.

I find it interesting that it's in base 20 and 18, I find it interesting that they do have an apparently linear calendar, I find it interesting that they decided to date it's beginning to a time that might have been about 3,000 years in their past when they created the calendar. I think it's interesting in that, according to certain sources that might not be reliable, it might have been set up to correspond to certain astronomical alignments. I don't think it's very interesting as a calendar.

That might be a complement. The interesting things about our calendar are the ones that make it harder to use. Screwy things are interesting. Difficult things are interesting. Well ordered simple ones are not.

The long count has five pieces to it, so you could think of it as being equivalent to “century, decade, year, month, day” in our calendar but where in our calendar you end up sort of working in bases 10, 12, 28, 29, 30, and 31 to make sense of those things, with 28, 29, 30, and 31 all applying to the same digit, in the Mayan calendar you only need to to work in bases 20 and 18. And you never have something like our months where the length varies depending on which one you're talking about.

The five things are “B'ak'tun, K'atun, Tun, Winal, K'in” where a B'ak'tun is 20 K'atuns, a K'atun is twenty Tuns, a Tun is 18 Winals, a Winal is 20 K'ins, and a K'in is a day. Other than the unfamiliar names, that's immediately easier to understand than our our calendar.

A Winal is 20 days, where a month for us is … how many again? Just not having cause to have the, “30 days has September,” rhyme is a sign of how much simpler it is. A Tun is 18 Winals where a year is 12 months. I'd say the switch to 18 is confusing, but it's not as if the numbers of months in our years are the same as the number of days in our months. After that you switch back to 20.

Now, as it turns out, they only have 13 B'ak'tuns. After that it's like the odometer runs out and everything sets back to zero. (Except that the B'ak'tun's don't have a zero so they go to 13, sort of like how in 12 hour time midnight and noon are represented by 12 instead of 0.) And that, they say, is how the world will end. Except for the fact that the Maya had ways of indicating dates that occurred after that and none of those indications came with footnote stating, “By the way, there won't be a world when this happens.”

From what I'm seeing it looks like the way that the Maya would indicate a date after the Long Count had cycled through if they wanted to is the same way we'd indicate a date after 9999, they'd add another digit. Actually, if I'm reading things right, first they'd expand the B'ak'tuns so there were 20 instead of 13. Then, once they finished all of those, they'd add another digit. (And at that point there would be a zero in the B'aktuns' place.)

So it's all pretty straightforward and simple. And long. They're not kidding when they call it the Long Count. It starts in the 32^nd century BCE, the normal version of it ends this year. That's a long time. According to the internet the Maya once used an extended version to indicate a date 400 million years from now. If they ever had need to add another digit after that, I'm sure they could have managed it.

Sorry to ruin the end of the world.

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I'll talk about the calendars that I do find interesting in the next post third and fourth posts.

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