Happy New Year! Today marks the start of the Gregorian year 2016, but I promised you that I'd start the new year with a new calendar.
But first, I made another promise earlier in December. I said that in January, I would start watching and discussing my birthday gift -- the Great Courses DVD. Well, today is January 1st, and so let me fulfill that promise.
Lecture 1 of David Kung's Mind Bending Math is called "Everything in This Lecture is False." Recall that Dave Kung's series of lectures are all about paradoxes -- and we see the first paradox of the series right in the title of the first lecture. For if this were true, then everything in this lecture -- including the title -- must be false. But, as Kung points out, the for the statement "everything in this lecture is false" to be false only means that something in this lecture must be true. And something was true -- right after he says that, he declares that 1 + 1 = 2, which is surely true.
Kung describes a number of paradoxes throughout this lecture. They include:
-- The Liar Paradox: "This sentence is false."
-- The Sign Paradox: "Notice: Do Not Read This Sign Under Penalty of Law."
-- Godel's Paradox: "This statement is not provable."
-- The Traveler's Paradox (also known as the Missing Dollar Paradox)
-- The Barber's Paradox
I've heard of many of these paradoxes well before watching Kung's lecture. For example, I remember back when I was a senior in high school, and the school was discarding a few library books. I took one of the books -- it was about set theory, yet it mentioned the Barber's Paradox. The book told the story about a barber who shaved all the men of the village who didn't shave themselves. This sounds quite reasonable, until we ask, does this barber shave himself?
-- If he shaves himself, then he is a self-shaver -- yet he doesn't shave self-shavers.
-- If he doesn't shave himself, then someone must shave him -- who can only be the barber himself.
So we see that the barber shaves himself if and only if he doesn't shave himself -- which is why this is a paradox. I remember soon after reading the book that I was to get a special haircut just in time for graduation, and I was tempted to ask the barber whether he shaved himself or not!
But there was one paradox that I'd never heard of before watching Kung's lecture. This is Curry's paradox, named for the American logician Haskell Curry. Here's how it works -- we will give the name Curry to the following conditional statement:
-- If Curry is true, then 1 + 1 = 1.
Now Kung says that it's impossible for Curry to be false. He quickly explains why Curry isn't false, but this may be confusing to someone who isn't familiar with logic. So instead, let me get to an example that I've posted here on the blog.
Let's go back to our discussion of the Parallel Postulate. We know that the Parallel Postulate distinguishes between Euclidean and non-Euclidean geometry. In hyperbolic geometry, the Parallel Postulate is definitely false. But here's the thing -- I claimed that the Parallel Postulate is actually true in spherical geometry!
To see why, let's use our blog's Fifth Postulate, which I'll write as follows:
-- If l | | m, then any line perpendicular to l is perpendicular to m.
(This is the Perpendicular to Parallels Theorem from the U of Chicago text.) Now recall what it means for a conditional to be false -- it means that there's a counterexample, namely an instance when the hypothesis is true, yet the conclusion is false. For "if a pencil is in my right hand, then it's yellow" to be false means that some pencil that is in my right hand that isn't yellow. So if a conditional is false, then the hypothesis must automatically be true.
Now let's look at the postulate above. If it were false in spherical geometry, then its hypothesis, that l | | m, must be true. Yet in spherical geometry, there are no parallel lines -- so the hypothesis can never be true. Since the hypothesis isn't true, the whole conditional isn't false -- and so we conclude that the whole conditional is in fact true.
Of course, there are other equivalents to the Parallel Postulate, but every one of them we've seen can be written as a conditional. The only way for a conditional to be false is for it to have a true hypothesis and a false conclusion, yet this doesn't happen with any version of the postulate. And so it's inescapable -- the Parallel Postulate must be true in spherical geometry! (This is the same logic that allows us to conclude that all unicorns are white -- the only way for it to be false is for non-white unicorns to exist -- that is, for unicorns to exist!)
And so for Curry to be false, its hypothesis must be true. But its hypothesis is "Curry is true." In other words, we have:
-- If Curry is false, then Curry is true.
So we conclude that Curry must be true. I'll let Kung take it from here:
"But Curry says that if Curry is true, then something else is true. And because we now know that Curry is true, then then conclusion must be true. The conclusion of Curry is that 1 + 1 = 1." QED
And there we have it -- a proof that 1 + 1 = 1! And this is why we call it Curry's paradox. As it turns out, we don't have to worry about our students accidentally proving 1 + 1 = 1 in our Geometry classes, since Curry is not a geometrical statement. On the other hand, Godel's statement "This statement is not provable" can be written as a mathematical statement -- but you have to be as smart as Godel (the Austrian mathematical Kurt Godel) to figure out how. The conclusion is that there exists statements that are true, yet not provable in mathematics.
In each of Kung's lectures, he presents a "Quick Conundrum." In today's Quick Conundrum, Kung shows that if you take a tiny scrap of paper, place it inside the neck of a bottle lying on its side, and then blow hard enough into the bottle, the paper will blow out of -- not into -- the bottle. He says this is because the air blowing into the bottle pushes the existing air out of the bottle -- and in the same way, self-referential statements such as "This sentence is false" push back on themselves.
Kung wraps up the lecture with some classic Knights and Knaves problems, which he attributes to the American mathematician Raymond Smullyan (who apparently is still alive at 96 years of age). Here is a link to some of these problems so you can try them out for themselves:
And now it's time for me to announce my original calendar reform proposal. So before I attempt to create an original calendar, I'd better check to see what calendar reform proposals are out there! An excellent resource out there is the Calendar Wiki:
This links directly to the Calendar Reform page. When we scroll down, we see a list of calendars organized by the length of the week. There are proposals with weeks ranging from three to ten days. I will provide links to the first calendar listed under each week length:
3 days per week: Liberalia Triday Calendar
4 days per week: a few ancient African calendars
5 days per week: Annus Novus Decimal Calendar
6 days per week: Solar Calendar with common Muslim/Christian weekend (or Raenbo calendar)
7 days per week: Gregorian Calendar (no need for a link)
8 days per week: 8-days a week simple calendar
9 days per week: Kalentris
10 days per week: Archetypes Calendar
Last year I mentioned the Raenbo calendar. Here's a link to Raenbo:
Finally, there are two calendars that incorporate all of the above calendars:
The idea between these two calendars is the same -- we begin with the 365-day year and designate one day per year as a blank day. The 364 remaining days can be divided into 91-day quarters, which in turn can be divided into 7- or 13-day weeks. Then if we label an additional blank day per quarter as a blank day, then there are five blank days and 360 days remaining. The number 360 has many factors, including 3, 4, 5, 6, 8, 9, 10, 12, 15, and so on. So all of these can be possible week-lengths -- and they are all listed at the links to the Abysmal and Shift Calendars.
So it will be difficult for me to create an original calendar, since almost any calendar I can think of is basically a version of the Abysmal or Shift Calendars. We notice that there are calendars listed having weeks from three up to ten-days -- but there is no eleven-day calendar. So if I want to create an original calendar, a good one will be one in which every week has 11 days. For lack of a better name, let me hereby announce:
The Eleven Calendar -- An Original Calendar Reform
Actually, I did find one passing reference to a calendar that involves the number 11. I found it at the following website:
This website is called Dozens Online -- which at this point seems strange. What would a website about dozens -- which seems to refer to the number twelve -- have anything to do with a calendar based on elevens?
Well, Dozens Online isn't merely concerned with the number 12, but the number base 12. Indeed, the posters on the Dozens Online message board are dozenalists -- adherents of the idea that humans should use base 12 rather than base 10. Technically speaking, base 12 should be called "duodecimal," just as base 10 is called "decimal." But dozenalists like to call their chosen number base "dozenal" in order to remind people that we already use base 12 every time we talk about dozens. Even a dozen dozen already has a special name in English -- the gross.
Dozenalism is even more radical than Calendar Reform -- Calendar Reformers only wish to change how we count dates, but dozenalists want to change how we count, period. But because of this, dozenalists are often sympathetic to Calendar Reform, and in fact, many of the posters are themselves Calendar Reformers. Timothy Travis -- the creator of the Raenbo Calendar -- is obviously both a dozenalist and a Calendar Reformer, and indeed Travis has posted at Dozens Online a few times.
But actually the Raenbo calendar isn't popular at Dozens Online. Instead, the forum posters voted for what they wanted in a new calendar. The two options with the most votes (at seven each) are "Preserve the Abrahamic 7-day week" and "Have the number of days per months following a regular pattern (e.g. alternating 26 and 27)." (As this is a dozenal forum, all the numbers are already converted to dozenal, so 26 and 27 refer to decimal 30 and 31.) Receiving five votes each are the three options "Use a leap-week calendar to ensure that dates fall on the same weekday every year," "Have an astronomical event (e.g. an equinox or solstice) on the same date every year," and "Make that event new year's day."
It was realized that there already exists a Calendar Reform that satisfies the first three of these -- it preserves a seven-day week, has a regular month-length pattern (although not as simple as alternating 26 and 27), and uses Leap Weeks. This is the Symmetry010 Calendar:
Hanke-Henry Permanent Calendar
By now you're saying -- all these links to calendars, none of which have anything to do with 11. So where's this Eleven Calendar that I keep discussing.
Well, the Eleven Calendar is based on the final post in the following thread:
But before we reach that post, we see that the thread begins by describing yet another calendar -- one which is based on the numbers 12 (yes, a dozen!) and 360. This calendar has a 12-day week. Another poster in the thread mentions what he calls the Dozenal Solstice Calendar, which uses a 6-day week and is therefore yet another clone of the Raenbo Calendar (even more so, since both the Raenbo and Solstice Calendars use dozenal.)
Now we finally reach the post that I've been wanting to discuss. The author is an Australian who calls herself the "Dozens Disciple" -- Wendy Krieger:
The week has an importance in that it regulates activities that do not happen daily. For example, one might have a market day or a holy day each week.
In the roman calendar, the weeks were tied to the quarter months, eg 'Ides' and 'Kalens'. In more recent times, the week is tied to the day, so that the particular days repeat every seven or ten or whatever times.
The shortest cycle for an exact calendar is 33 days, giving dozenal 265;2XXXX.
Since 263 is dec 33 by 11, you could bave E; days of 29; days each with no more than 2 to 3 intercalendar days. A 29 year cycle gives then a calendar round.
Notice that Wendy Krieger alternates between decimal and dozenal in her post. She uses the letters X and E to represent the digits ten and eleven. (These are similar to the digits used in the Schoolhouse Rock episode "Little Twelvetoes," which actually introduces a dozenal system.) To make it easier to understand Krieger's post, let's convert all the numbers to decimal:
The shortest cycle for an exact calendar is 33 years, giving 365.24242 as the average year-length.
Since 363 is 33 times 11, you could have 11 months of 33 days each with no more than 2 to 3 intercalary days. A 33 year cycle gives then a calendar round.
So there we have it -- a calendar with exactly 11 months. Krieger saw that the closest multiple of 11 to the number of days in a year is not 360 or 364, but 363. And notice that 363 divided by 11 is 33 -- which just happens to be itself a multiple of eleven! So we can divide each 33-day month into three weeks, each with 11 days.
Therefore, we now have a calendar that incorporates the number 11 twice -- there are 11 days per week and 11 months per year. So this is a Calendar Reform that is truly worthy of the new name, the Eleven Calendar.
By now you may be asking, how can this be considered an original calendar of mine -- wouldn't Krieger be the true creator of this calendar? Actually, I don't consider Krieger's creation to be complete -- she only mentions how it's possible to have a 33x11 calendar. And besides -- she doesn't mention dividing the 33-day months into 11-day weeks (despite starting her post by telling about the historical importance of the week). So I consider myself the originator of the Eleven Calendar.
Our week of eleven days is a rather strange period of time. Notice that 11 days is longer than our standard seven-day week, yet is shorter than a fortnight, which is two standard weeks. The term "fortnight" isn't often used by Americans. It is more commonly used in countries like Great Britain -- indeed, British secondary pupils often have schedules based on the fortnight, where they attend classes at a different time the first week from the second week.
Even though the word "fortnight" literally means "14 nights," some Calendar Reformers use the word to refer to slightly shorter periods of time. In the Abysmal House Calendar, each house is divided into four 13-day fortnights, and the original poster in the Krieger thread first wanted to call her dozenal (12-day weeks) fortnights. On the other hand, the ten-day weeks of the Archetypes Calendar mentioned above are still known as weeks (though the author tweaks this, and shortens "ten-day weeks" to "tweeks").
So 11-day periods seem to be in between these -- it's too long to call it a "week," yet too short to call it a "fortnight." As it turns out, I have a dental appointment on January 12th, which is exactly 11 days from today. I'm likely to think of the appointment as "two weeks away," since there are two weekends between today and the appointment. But if it were this upcoming Monday (that is, January 4th) and I had an appointment 11 days later, on the 15th, I'd probably think of the appointment as "next week" since I'd have only one weekend left to enjoy before the appointment.
So 11 days is this strange "week-and-a-half" long period. Is there any other significance to this period of 11 days -- one that would make us want to base a calendar on it? Well, as it turns out, the average Taco Bell customer visits the fast food restaurant once every 11 days:
Of course, this is just an average -- I doubt that any particular customer counts out 11 days before deciding to go to Taco Bell. Most likely, many people eat there once a week and others eat there once a fortnight, so that the average works out to be once every 11 days.
Still, this gives me the idea of associating each of the 11 days of the Eleven Calendar week with some particular food item -- so one day could be Taco Bell day. I wouldn't actually call the day Taco Bell Day (even though we already have a day of the week devoted to tacos -- Taco Tuesday), since if I were eating at Taco Bell, I wouldn't want everyone else in the world eating there with me. But if we number the days 1-11, then I might choose to go to Taco Bell every Day 2 (that is, Taco 2's Day).
Now one food that I definitely need to limit is sweets. (That's why I have that dental appointment coming up in 11 days!) I need to reduce or eliminate how much candy I eat, and I could stand to limit other sweets such as cookies as well.
Now I want to time it so that I can purchase and eat sweets on the days on which they are most likely to be sold at a discount -- days like November 1st and December 26th. Notice that those two dates are exactly 55 days apart -- and 55 is a multiple of 11. So they would fall on the same day of the week in the Eleven Calendar. On each day of the calendar, I'm limited to a particular junk food -- but of course I can eat healthy food like fruits and vegetables on any day.
This version of the calendar begins on March 1st. This is convenient because now the 2-3 blank days can be February 27th, 28th, and 29th.
Simplest Version of the Eleven Calendar
(Maximizes Compatibility with Gregorian Calendar)
The 363 days are divided into 33 months, as follows:
Month 1: March 1st - April 2nd
Month 2: April 3rd - May 5th
Month 3: May 6th - June 7th
Month 4: June 8th - July 10th
Month 5: July 11th - August 12th
Month 6: August 13th - September 14th
Month 7: September 15th - October 17th
Month 8: October 18th - November 19th
Month 9: November 20th - December 22nd
Month 10: December 23rd - January 24th
Month 11: January 25th - February 26th
Blank Days: February 27th, 28th, 29th
Each month is divided into 11-day weeks, with days numbered 1 to 11.
Even though I haven't named the days, it may be desirable to name the months. This is easy -- just as the 13-month calendar can just add a new month name like "Smarch," an 11-month calendar can simply drop one of the days.
One easy way to name the months is just to name each 33-day month after the Gregorian month in which it begins. So Month 1 will be called "March," Month 2 will be called "April," and so on. This means that Month 8 will be called "October," just as an eight-sided polygon is an octagon. Many previous Calendar Reformers have sought to restore the names of the months from September to December back to their original Latin meanings.
So Month 11 will be called January, which means that February is dropped. It's often said that February is the least popular month of the year, especially in the North where the coldest days of the year fall in February.
There's also that holiday in February -- Valentine's Day. It's often said that only one group of people actually like Valentine's Day -- attached females, as these are the recipients of V-Day gifts. Attached males dislike the holiday as tend to be the givers of V-Day gifts, and singles dislike the holiday as they aren't involved in any gifts at all. Actually, I'd say that there's another group of people who do like the holiday -- young children. For them, Valentine's Day represents a break from the usual curriculum and one of the last days of fun at school before the Long March.
Other nations, particularly in Asia, already have a solution to the Valentine's Day problem without any sort of Calendar Reform. Such countries have three Valentine's Days. The first is on the usual February 14th date, when attached males (!) receive gifts. One month later, March 14th (yes, that's Pi Day!) is White Day, when attached females receive gifts, and one month after that, April 14th, is Black Day, their Singles Appreciation Day.
But this country has only one Valentine's Day. And as long as only two groups really enjoy the day (along with the weather), people are going to name February as the least popular month, and so this Calendar Reform can eliminate it. Notice that as an adult male, I'm a member of one of the groups that dislikes Valentine's Day, but of course I personally have no problem with sales on February 15th.
Since today is New Year's Day, I ought to make it my New Year's Resolution to follow the Eleven Calendar and eat only the foods that I say I will on each day. Unfortunately, I don't have yet enough self-discipline to follow this calendar -- and besides, I haven't decided which foods I'll assign to each of the 11 days yet. Still, I'm proposing the Eleven Calendar today in order to discuss its properties as a calendar, without worrying about food.
And besides, I can't really celebrate New Year's Day on March 1st or skip Valentine's Day. I still live in a world that follows the Gregorian Calendar. So far in this post, I've only defined how I would tie the Eleven Calendar to the Gregorian Calendar. But a true Calendar Reform would eliminate the Gregorian Calendar completely and establish a new calendar from scratch.
Also, notice that there's one part of Wendy Krieger's post that I haven't implemented yet:
The shortest cycle for an exact calendar is 33 years, giving 365.24242 as the average year-length.
Recall that the difference between the Julian and Gregorian Calendars is that the former adds February 29th every four years, but the Gregorian Calendar skips the date in years like 1700, 1800, 1900, then 2100, 2200, 2300, then 2500 and so on. This decreases the average year-length from 365.25 for the Julian calendar to 365.2425 for the Gregorian.
Notice that every time we have a Gregorian exceptional year, we have eight years between Leap Days, such as from 1896 to 1904, or 2096 to 2104. But this doesn't make sense -- this is just like saying, "Most of the time four cookies satisfy my appetite, but today I am a bit hungrier, so I'll just eat eight cookies instead of four." If four cookies aren't enough, we should just eat five cookies, not eight -- and so if our Leap Day is so hungry that four years aren't enough, let's just feed it five years.
This is what Krieger is doing here. In this post, she points out that there's a 33-year cycle such that eight of the years have Leap Days. The eighth Leap Day occurs 28 years after the first -- and then we wait five years before the next Leap Day, which is the first of the new 33-year cycle. This drops the average year-length from the 365.25 of the Julian to 365.24242 -- which is even more accurate than the Gregorian year, since the length of the tropical year is about 365.24219 days. And it takes only 33 years for the cycle to repeat, not 400 as in the Gregorian.
Moreover, our Eleven Calendar already incorporates the number 33 -- there are 33 days in a month and 33 weeks in a year. So it's natural to use a 33-year cycle for this calendar as well.
As soon as we introduce the 33-year cycle, the calendar is no longer compatible with the Gregorian Calendar with its 400-year cycles. So this is truly a brand-new calendar.
Instead of starting this calendar on March 1st, I decided that the first day of the new calendar will be February 5th, 2014 -- the date on which Krieger first mentioned the 33-year cycle. And incidentally, I was 33 years old back on that date, so there's yet another connection to 33. February 5th may seem like strange date for New Year's Day, but notice that it's fairly close to Chinese New Year, which falls on February 8th this year. (The Archetypes Calendar is also based on Chinese New Year.)
I decided that I would create two versions of the Eleven Calendar. One of them will be a blank day calendar where we add 2-3 extra days to the basic 363-day year, and the other will be a Leap Week calendar (just like the Symmetry010 or Hanke-Henry calendars) with 11 extra days in certain years.
If we do it right, we can use the same cycle for both calendars. So the blank day calendar won't just add 2-3 days to the end of the year, but instead will add a day every few months. Then we can just multiply everything by 11 to obtain the Leap Week calendar -- which means that the Leap Week version will contain a 363-year cycle.
A basic 33-year cycle contains 363 months of 33 days each, for a total of 11,979 days. But we need 33 years to contain 365 days each plus 8 Leap Days, for a total of 12,053 days. The difference is 74, so we need 74 of the 363 months to contain a blank day. The corresponding Leap Week calendar will require 74 Leap Weeks in 363 years.
Now all we need is a rule to determine with months or years have the extra day or week. There are two ways to accomplish this. One rule is based on what Peter Meyer -- the creator of the Archetypes Calendar -- uses for his own Leap Week calendars:
Two Integral-Week Solar Calendars
For example, in his year based on five-day weeks, Meyer writes:
By definition, a year in the Integral Five-Day-Week Calendar is a long year if and only if the year number, mod 165, is exactly divisible by 21.
As it turns out, a similar rule almost works for 11-day weeks:
A year is a long year (i.e., has a Leap Week) if and only if the year number, mod 363, is exactly divisible by five.
This almost works because 363/74 is very close to five. We see that this indeed almost works -- if the years 0, 5, 10, 15, ..., 350, 355, 360 contain Leap Weeks, then there are 73 such weeks -- but we need there to be 74. So we can just throw in an extra week:
A year is a long year (i.e., has a Leap Week) if and only if the year number, mod 363, is either 192 or exactly divisible by five.
I chose 192 because it's near the midpoint of the 363-year cycle. (The exact midpoint is 181, but I added 11 in order to avoid two consecutive long years.)
The other method is to use a symmetrical cycle -- the type of cycle that Dr. Irv Bromberg uses in his Sym010 and other calendars. We need our Leap Weeks, on average, to occur once every 363/74 years, so we begin with this fraction, which I'll call x.
To find the long years, we take the half-multiples of x -- that is, we find 0.5x, 1.5x, 2.5x, and so on until we reach 73.5x. We then round this numbers down to the nearest integer if we are using 0-based numbering, or round them up if we are using 1-based numbering. Assuming 0-based numbering, the resulting long years are:
2, 7, 12, 17, 22, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 183, 188, 193, 198, 203, 208, 213, 218, 223, 228, 233, 237, 242, 247, 252, 257, 262, 267, 272, 277, 282, 286, 291, 296, 301, 306, 311, 316, 321, 326, 331, 336, 340, 345, 350, 355, 360
On the blank day calendar, this tells us which months have an extra day -- again assuming that the first month is 0. As it turns out, this also gives us a symmetrical 33-year cycle when considering which years have 366 days -- starting from 0, this are 2, 6, 10, 14, 18, 22, 26, and 30. Then there are five years from year 30 of the old cycle to year 2 of the next.
If February 5th, 2014, is the start of year 0, then the first year with 366 days is 2016. The pattern tells us that the first, fifth, and tenth months have a blank day -- and if the first month starts on February 5th, then this month contains February 29th. As it turns out, the years from 2016 up to 2040 all have a blank day close to a February 29th, after which the 33-year cycle diverges from the Gregorian.
From now until year 2043, the following dates are the first day of a month in the blank day calendar:
April 12th or 13th
May 15th or 16th
June 17th or 18th
July 20th or 21st
November 30th-December 1st
Thus completes the Eleven Calendar -- at least for now. I'll continue to work on it -- especially the Leap Week and other rules to decide whether I like them or not.
Let me conclude this post with a quote from Dave Kung. At the end of the lecture, he offers the following challenge to the viewers: "Find something you're wrong about. Find some area of your life, your world, and stop ignoring the cognitive conflict." By resolving the paradoxes inherent in our lives, we can become not only better mathematicians, but better people. To me, nowhere is this more important than in the classroom.
And so I declare this to be my actual New Year's Resolution for 2016. I want to find something wrong about the way I'm teaching and managing my classroom and resolve it, so that I can become a better teacher.
This marks the end of this post. Once again, Happy New Year -- and the next post will be on Monday, January 4th, which will be Day 76 of the blog calendar.