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## Calendrics

At this time of year, my thoughts often turn to….

…the sinusoid?

The sinusoid is a fancy name for a mathematical relationship that is periodic – like the tides or length of day.  Sinusoids are also useful in modeling alternating electrical current, light waves, and sound waves.

Below is graph of function y=sin(x)

I became interested in calendrics when I investigated the occurrence of Friday the 13th.  Keeping track of time is an almost universal cultural concern.

The Mayan civilization produced a sophisticated mathematical system that was used for both astronomy/astrology and calendrics (which are obviously closely related).

Moon phases and eclipses and the transits of Venus are all contained in the Dresden Codex, one of the few pieces of Mayan knowledge that survived the destruction of their libraries put in motion by Bishop Diego de Landa in 1562.

I studied the Mayan calendar briefly a few years ago because of its complexity.

I was intrigued by the combination of a 365-day solar calendar with a 260-day ceremonial calendar.

The Mayan 365-day solar calendar was comprised of 18 20-day months and 1 short 5-day month.  The short 5-day month (Wayeb or Uayeb) may have been used as an intercalary period to make up for the fact that the solar year is not exactly 365 days.

The structure of this solar calendar puzzled me.

Why would the Mayans choose a solar calendar in which there is almost no difference in the calendar from one month to the next?

Granted, they had the ceremonial calendar to denote important ceremonial days.  But, for some reason, I was bothered by the structure of the Mayan solar calendar.

I decided (quite without evidence of any kind) that this calendar was a result of the fact that the Mayan homeland on the Yucatan peninsula is situated in the tropics at about 20° north latitude.  Consequently, the length of day is fairly constant, never varying more than an hour or so from the 12 hour equinoctal day and night.

Above about 40°N latitude, the change in the length of day from winter to summer solstice is quite noticeable and above about 45°N, the extended sunlight of the summer solstice and darkness of the winter solstice has an undeniable effect on human beings living in or visiting these regions.

I decided that a good calendar for people living above 40°N latitude would acknowledge the importance of the length of day.

The sinusoid is a good function to model the change in the length of day throughout the year.  If we make certain adjustments to the original formula y=sin(x), the result models the length of day for a particular region quite nicely.

The equation for the graph below is

y=12+3.5sin((2*PI/365)*(x-80))

In this graph, the x values represent the number of days since January 1st for a particular year and the y values represent the number of hours of sunlight for that day.

The first thing that is important about this relationship is that it is NON-LINEAR.  This means that the rate of change of the length of day is not constant.

In this particular application this means that we gain more light from March 1 to March 7 than we do from June 1 to June 7.

Around the winter and summer solstice, we are hardly gaining or losing any light at all.  At the Spring and Fall equinoxes, we gain and lose the most light we will gain or lose all year.

Based on this information, I started to look at where the changes in amount of light gained or lost are happening.

If we start with the Winter Solstice, we see that for two weeks on either side of the solstice, there is almost no change in the length of day.  After this period there are four weeks of gradually increasing daylight followed by another four weeks of slightly greater increasing daylight.

This brings us to the end of February on the standard Gregorian calendar, which corresponds to an x value of about 59(=31JAN+28FEB) on the graph above.

Notice that, at this point on the graph, the curve nearly straightens out for about 40 days.  This is the six week period that lasts from early March to mid-April.  Three weeks previous to the equinox and three weeks following the equinox, we gain the most light that we will gain all year.

After this six week period, we continue to gain daylight, but at a slightly reduced rate.  Four weeks of gradually reduced gains followed by another four weeks of even smaller gains in length of day bring us to the end of the first week in June.

Two weeks before the summer solstice.

For these two weeks prior to the summer solstice and for two weeks following the solstice, we hardly gain or lose any daylight at all.  Although these are the longest days of the year, we are hardly gaining or losing any light at all.

Two weeks after the solstice, the loss of daylight begins to slowly pick up speed for four weeks, and then we lose even more daylight in the following four weeks.  This brings us to the end of August, three weeks before the Autumnal Equinox.

For three weeks before the equinox and three weeks after, we will lose the most daylight that we lose all year.

Then, four weeks of gradually declining loss of daylight, followed by four weeks of of even smaller losses of daylight bring us back to the end of the first week in December, which is two weeks before the winter solstice.

Right back where we started.

SO –

Two weeks*Winter Solstice*Two weeks

Four weeks*Four weeks

Three weeks*Spring Equinox*Three weeks

Four weeks*Four weeks

Two weeks*Summer Solstice*Two weeks

Four weeks*Four weeks

Three weeks*Autumnal Equinox*Three weeks

Four weeks*Four weeks

Two weeks*Winter Solstice*Two weeks….

If we add all this up, it comes out to 364 days.  Like all calendars, this division of time requires intercalation, or a resetting based on astronomical observation.

This can happen at either solstice.  Once the solstice is reached, begin counting the two weeks.  This will keep the calendar aligned with the actual procession of the seasons.

Earlier, I mentioned that we gain significantly more daylight from March 1 – March 7 than we do from June 1 – June 7.

Based on this formula, the gain in length of day from March 1 – March 7 is about 21 minutes whereas the the gain in length of day from June 1 – June 7 is about 6 minutes.

This formula I’ve used

y=12+3.5sin((2*PI/365)*(x-80))

is an idealized representation based on a shortest day of 8.5 hours and a longest day of 15.5 hours.  The actual lengths of day vary with latitude and, in Portland, Oregon, are closer to 8 hrs. 40 min. for the shortest day and 15 hrs. 40 min. for the longest.

## Language and Counting

There are two topics that I teach which allow me to discuss counting.  When we cover the order of operations (or what I like to call the hierarchy of computation, since exponentiation is not an operation), I begin by saying that the most basic form of mathemtics is counting.  By repeated counting, we arrive at a consideration of addition.  By repeated addition (despite some vigorous dissent) we arrive at multiplication and so on.

In teaching about the set of complex numbers, I begin by saying that the simplest form of mathematics is counting, so the most basic set of numbers is the set of counting numbers.  But a problem arises if we write an equation using the counting numbers – we can write an equation whose solution is not a counting number.

x+5=1

This then requires an extension of the Counting Numbers to the Integers.

When I discuss counting, I mention that almost all cultures count.  From there, different cultures develop different mathematics depending on their different needs.  Ethnomathematics is a very interesting field.  Researchers in ethnomathematics study not only how different cultures use math, but also how different professions use math.  Professor Tod Shockey at University of Maine is a specialist in ethnomathematics and wrote his dissertation on the use of math in the medical sciences.

I mentioned that ALMOST all cultures count.  There are some cultures that do not have words for numbers above 3 or 4.  Why not?  Probably because they don’t need them.

One culture which seems not to count at all is the Piraha people of the Amazon basin.  Here is a link to a pretty good article on this.  Two researchers have argued about whether or not the Piraha count and whether or not they have words for colors.  One researcher who thought that the Piraha had words for numbers was contradicted by another who said that they just have words for smaller and bigger and that these were being mistaken for words for one and two.

What makes this particularly interesting is that this disagreement between the linguists also arose over whether or not the Piraha have words for colors.  In describing a color, they will say that it is the same color as something in their everyday life.  This seems to get at a fundamental property of the Piraha language – it does not have abstractions.  This is an insight into the way that this culture views the world, just as any language is a snapshot of the collective mind of the culture that uses it.