1. Rapoport Problem of the Day
2. Implementing the Usher Calendar in the 2020's
3. Implementing the Usher Calendar in the 2030's
4. Implementing the Usher Calendar in the 2040's
5. Implementing the Usher Calendar in the 2050's
6. Implementing the Usher Calendar in the 2060's
7. Loose Ends with the Usher Calendar
8. Loose Ends with Leap Day
9. Loose Ends with the Eleven Calendar
10. Conclusion: Another Quadrennial Task
7. Loose Ends with the Usher Calendar
8. Loose Ends with Leap Day
9. Loose Ends with the Eleven Calendar
10. Conclusion: Another Quadrennial Task
Rapoport Problem of the Day
Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:
Find x.
[All givens are in the diagram. B lies on Circle A, with tangent
From the Radius-Tangent Theorem of Lesson 13-5,
(x - 9)^2 + (x - 8)^2 = x^2
x^2 - 18x + 81 + x^2 - 16x + 64 = x^2
2x^2 - 34x + 145 = x^2
x^2 - 34x + 145 = 0
(x - 5)(x - 29) = 0
x = 5 or x = 29
Yet if the hypotenuse of the right angle is 5, the resulting legs become -3 and -4 -- and of course, leg lengths can't be negative. Therefore the desired hypotenuse length is 29.
And of course, today's date is the 29th -- that is, February 29th, a very significant day on the Gregorian Calendar. It is the first time that February has had 29 days since 2016, and it won't have 29 days again until 2024.
Ordinarily, I don't post on Saturdays unless it's a school vacation. But four years ago on the blog, I made a special calendar-related post for Leap Day. And so I might as well do the same today.
Four years ago today, I described a proposed Calendar Reform by Andrew Usher. The links included in that calendar proposal post are now dead, and I have no idea where Usher is nowadays. Therefore this blog is the only active website where the Usher Calendar is fully described.
Don't worry -- my baby is still the Eleven Calendar. But I know that as much as I like the calendar, it will never truly be implemented. My eleven-day week will never replace the seven-day week.
On the other hand, the Usher Calendar -- unlike most other proposed Calendar Reforms -- actually can become a reality. The trick here is that the Usher Calendar can be implemented quite gradually, starting simply by changing the dates of certain holidays to match their Usher definitions.
If you want simply to read what exactly the Usher Calendar is, then you can refer back to my Leap Day 2016 post to learn more. In today's post, I'll do something different -- I'll describe a timeline of a gradual implementation of the calendar. By reading the timeline, you'll learn not only what the Usher Calendar is, but why Usher considered it to be superior to the Gregorian Calendar. And unlike sudden implementations of calendars all at once (such as the Eleven Calendar -- "Why are we changing from seven to eleven days per week?" -- or even the World Calendar -- "Why are there blank days?"), this implementation is so gradual that most people won't even realize that it's a new calendar at first. So there's nothing for them to object to.
Our gradual implementation begins by redefining, not a calendar or even a holiday, but something that's in the news as we speak -- the length of the National Football League season.
Implementing the Usher Calendar in the 2020's
2021 -- year of transition
2022 -- 17-game NFL season, Super Bowl LVI on February 13th (2nd Sunday in Feb.)
Unlike the other major American sports, the NFL has the most rigid schedule. Baseball, basketball, and hockey can be played on any day of the week, but most pro football games are on Sunday. There are some games played on Mondays and Thursdays, as well as certain Saturday games in December (as well as January in the playoffs). Otherwise, Sunday is football day.
The current football season consists of four weeks of preseason games, 17 weeks of regular season games (16 games plus a bye week), and five weeks of postseason games (Wild-Card Round, Divisional Round, Conference Championships, Pro Bowl, Super Bowl). The first day of the regular season is on the Thursday after Labor Day (usually when the Super Bowl champs receive their rings), with most teams playing on the following Sunday.
It's convenient to number the weeks of the regular season as 1-17. Here I'll do the same, except that I'll number the weeks in Roman numerals, Week I up to Week XVII. Using Roman numerals reminds us that these are "football weeks" (since the Super Bowls also use Roman numerals), since Andrew Usher himself also uses a week-numbering system (that I'll give later in this post). We can extend this week counting into the postseason, so that the Wild-Card Round is Week XVIII, Divisional Round is Week XIX, Conference Champs are Week XX, Pro Bowl is Week XXI, Super Bowl is Week XXII.
Indeed, Usher's weeks run from Sunday to Saturday. But for football weeks, we want Thursday, the following Sunday, and the next day Monday all to be in the same week. This is because a team will never play a game on Thursday and then on Sunday three days later (as these are in the same football week), but they can play on Sunday and then again on Thursday four days later (as these are in different football weeks). Thus for the purposes of this post, we'll assume that football weeks run from Wednesday to Tuesday.
Once we know the date of Labor Day in a certain season, all other dates can be determined. Suppose Labor Day is on its earliest possible date, September 1st. Then Week I of the season starts on Wednesday the 3rd, opening day is on Thursday the 4th, and most teams play on Sunday the 7th. We then count weeks to find that the Sunday of Week XVII is December 28th, Wild-Card Sunday is January 4th, Divisional Sunday is January 11th, Conference Champs are January 18th, Pro Bowl is January 25th, Super Bowl is February 1st.
But if Labor Day is on its latest possible date, September 7th, then the day of the Super Bowl works out to be February 7th. Thus "Sunday of Week XXII" and "first Sunday in February" are equivalent as they both refer to the same date -- the day of the Super Bowl. Given the Labor Day date, we find that the Super Bowl is always on the same calendar date, but five months later in February.
Anyway, there's now a proposal to add an extra regular season game. If a 17th week is added to the regular season, then the schedule becomes:
Weeks I-XVIII -- Regular Season (17 games + a bye week)
Week XIX -- Wild-Cards
Week XX -- Divisionals
Week XXI -- Conference
Week XXII -- Pro Bowl
Week XXIII -- Super Bowl
And since the Sunday of Week XXII is the first Sunday in February, the Sunday of Week XXIII must be the second Sunday in February. The Big Game would now played in the February 8th-14th range.
This proposed agreement has yet to become official. (The owners are eager to add an extra game, but the players aren't so much.) But assuming that it becomes official, the proposal will go into effect starting in fall 2021. Then the first later Super Bowl would be February 13th, 2022. I'm hoping that the proposal becomes official, since it's the first small step towards implementing the Usher Calendar.
2027 -- year of transition, Super Bowl LXI on Sunday, February 14th
At first, the new 17-game season goes smoothly. But then in the year 2027, something strange happens with the calendar. The second Sunday in February 2027 is the 14th, which causes the Super Bowl to coincide with two holidays.
The first, of course, is Valentine's Day. One huge problem here with the Big Game on Valentine's Day is that it challenges traditional gender roles -- in a (heterosexual) relationship, the man will typically want to watch the Super Bowl, while the woman will want her husband do treat her to something romantic for V-Day. This forces the man to make a choice.
My prediction is that most men will choose football. This may lead to a spike in divorces shortly after the game, but it may also cause record TV ratings for the game.
And that's because of the second holiday -- Presidents' Day. This holiday is currently defined as the third Monday in February. Most of the time, the day after the second Sunday of the month isn't the third Monday, except in the case when Sunday and Monday are the 14th and 15th. Thus 2027 becomes an special year when the Super Bowl coincides with both Valentine's and Presidents' Day.
Many people have clamored for the Monday after the Super Bowl to be a holiday so that they won't have to work the day after the Big Game. In 2027, the 17-game season times it perfectly so that the Super Bowl (in Week XXIII) is on Presidents' Day Weekend. This is why I believe that the good luck of having the game on Prez Day weekend outweighs the bad luck of having it on V-Day, much to the dismay of many women.
2028 -- 18-game NFL season, Super Bowl LXII on Sunday, February 20th (3rd Sun. in Feb.)
And indeed, the NFL enjoys the Presidents' Day Super Bowl so much that they wish to change the calendar again to make it permanent. In most years, the day before the third Monday in February is the third Sunday, which would be football Week XXIV. In order to make the Super Bowl land on Week XXIV, we add an 18th day to the regular season:
Weeks I-XIX -- Regular Season (18 games + a bye week)
Week XX -- Wild-Cards
Week XXI -- Divisionals
Week XXII -- Conference
Week XXIII -- Pro Bowl
Week XXIV -- Super Bowl
Notice that it's also possible to reach Prez Day weekend if the 19-week regular season consisted of 17 games and two bye weeks (which is more palatable for the players). But for the purposes of this blog post, we assume that it's 18 games and one bye week. This makes it easy to remember:
16-game season = Super Bowl on 1st Sun. in Feb.
17-game season = Super Bowl on 2nd Sun. in Feb.
18-game season = Super Bowl on 3rd Sun. in Feb.
In any case, with the new extended season, the Super Bowl now lands on Presidents' Day weekend in most years.
Implementing the Usher Calendar in the 2030's
For the next decade -- up until the year 2037 -- the 18-week schedule goes smoothly. The Super Bowl always lands on Presidents' Day weekend, and fans enjoy having the day off after the Big Game.
But unfortunately, the year 2038 is just like 2027. It's a year when Prez Day is on the 15th, and so the day before is the 14th, which is both Valentine's Day and the second Sunday in February (that is, the Sunday of Week XXIII rather than XXIV).
We could revert to a 17-game schedule for 2038 (and years like it) only -- but of course, the owners won't like it. (And if we reached Prez Day by adding a second bye week rather than an 18th game, then the players won't like losing the bye week in 2038).
Or we could simply start the season a week earlier -- on Labor Day weekend in such years. Actually, the NFL used to begin regularly on Labor Day weekend, but fans would rather enjoy their last weekend of summer outdoors or out of town rather than watch football. This is why the league waits until the following weekend to start the season.
The following link discusses the problems with an 18-game schedule in seasons like 2037-38, when Labor Day isn't until September 7th:
https://www.sports-central.org/sports/2020/01/27/enjoy_the_week_off_while_you_still_can.php
Added to this, of course, is that with a second bye week for each team — something the owners have more or less agreed to, and for which the 1993 season provides an oh-so-convenient precedent — an 18-game schedule would give the owners the Presidents' Day weekend Super Bowl they desire. One irregularity arises, and that is when Labor Day falls on September 7; in that case, the owners would have to bite the bullet and begin the regular season on that weekend, otherwise the Super Bowl would be played on February 21 — one week too late. The second bye also puts the owners in a position to give every team that plays a Thursday game an automatic bye the week before — something that will go over big with the NFLPA, which needs to sign off on any lengthening of the schedule.
(OK, so cut-and-paste from this website doesn't work -- highlight it to see. Also, notice that this proposal has both 18 games and two bye weeks, but keeps Super Bowl on Prez Day weekend by eliminating the Pro Bowl week after the Conference Champs. It suggests either waiting until the following Week XXV/4th Sun. in Feb. to play the Pro Bowl, or just the day before the Super Bowl -- Sat. of Prez. Day weekend.)
But this still doesn't solve the Valentine's Day problem. Angry women will still remember what their ex-husbands did to them 11 years earlier, and all heck will break loose if there's a repeat in 2038.
So our goal is for football to avoid both Labor Day and Valentine's Day, but to remain on Presidents' Day weekend in 2038. The solution is staring us in the face -- redefine Presidents' Day so that it's 24 weeks after Labor Day and avoids Valentine's Day weekend:
2037 -- year of transition
2038 -- Super Bowl LXXII on Sunday, February 21st, before Prez Day, Monday, February 22nd
And voila -- that's one of the holidays that Andrew Usher redefines in his calendar! Instead of falling in the February 15th-21st range, Prez Day is now in the 16th-22nd range, thus allowing Super Bowl Sunday one day earlier to be in the 15th-21st range.
Of course, now Valentine's Day becomes the weekend of the Pro Bowl, not the Super Bowl. But the Pro Bowl is much less popular than the Super Bowl, and so I expect more men to make their wives happy on V-Day rather than watch the Pro Bowl.
(Note: The proposal at the link above, combined with the new Usher Prez Day, places the Conference Champs on V-Day instead of the Pro Bowl. This restores the angry wife problem, since men would definitely skip dates with their wives to watch the Conference Champs.)
And as a bonus, the new February 16th-22nd range for Presidents' Day now includes the 22nd -- George Washington's actual birthday. (Officially, "Presidents' Day" isn't the name of a holiday -- the only federal holiday in February has always been called "Washington's Birthday.")
By the way, the fact that 2038 would be the year of the first material change in the holiday calendar reminds us of the Year 2038 problem, an impending Y2K-like bug on computers:
https://computer.howstuffworks.com/question75.htm
According to the link, Y2038 will strike on January 19th (Tuesday of week XIX, two days after the regular season ends). So we can strive to fix both the Y2038 bug and the Prez Day calendar bug at the same time.
Implementing the Usher Calendar in the 2040's
Once the NFL expands its season to 18 games (and its playoffs to 14 teams, as is also proposed), I suspect that this might embolden college football to expand its playoffs to eight teams. It will now be possible for an eight-team playoff to end before the end of the 18-week NFL regular season (so that the two playoffs no longer clash).
Here's a link to a possible eight-team playoff proposal:
https://www.sbnation.com/a/college-football-commissioner/playoff-expansion
Have the quarterfinals a week after conference championship games (so around December 8-12), then announce bowl pairings on the following Sunday. Teams that lost in the quarterfinals would have the option of accepting bowl bids, thereby diluting bowl games as little as possible.
Let's see what that looks like using Roman numeral weeks. The date range shows the earliest and latest possible date of the Saturday in that range, since we expect college football games to be played on Saturdays:
Week XIV -- Quarterfinals (December 6th-12th)
Weeks XV-XVI -- two weeks off between rounds (December 13th-19th/20th-26th)
Week XVII -- Semifinals (December 27th-January 2nd)
Week XVIII -- one week off between rounds (January 3rd-9th)
Week XIX -- Finals (January 10th-16th)
It might be possible to juggle the semifinals to that it falls on January 1st regardless of the day of the week (as is currently done if the Rose and Sugar Bowls are hosting them).
And sure enough, the quarterfinal range (December 6th-12th) matches the range given at the link (which is Dec. 8th-12th). According to the link, this is one week after conference championships:
Week XIII -- Conference Championships (November 29th-December 5th)
But this is a problem, because the last Sat. in Nov. is also the Saturday after Thanksgiving. We wouldn't want to have conference champs on Thanksgiving weekend, especially since many colleges play their rivals that weekend.
The problem occurs because Thanksgiving is usually the Thursday of Week XII in most years, but it's the Thursday of Week XIII in certain years. Our solution will be to redefine Thanksgiving so that it's always the Thursday of Week XII.
2040 -- year of transition
2041 -- College Rivalry on Saturday, November 23rd, after Turkey Day, Thursday, Nov. 21st
And -- you guessed it -- this new definition of Thanksgiving matches Usher's definition. Notice that it will occur exactly a century after "Franksgiving" when President FDR redefined Thanksgiving -- the old range of November 24th-30th was moved up by two days, to the 22nd-28th. Then Usher Turkey Day (essentially Franksgiving II) will move the range up by two days again, to the 20th-26th.
So the resulting end of the college season will look like this:
Week XII -- Rivalry Weekend (Saturday, November 22nd-28th)
Week XIII -- Conference Championships (November 29th-December 5th)
Week XIV -- Quarterfinals (December 6th-12th)
Weeks XV-XVI -- two weeks off between rounds (December 13th-19th/20th-26th)
Week XVII -- Semifinals (December 27th-January 2nd)
Week XVIII -- one week off between rounds (January 3rd-9th)
Week XIX -- Finals (January 10th-16th)
And once Thanksgiving is moved, other holidays can follow. Here are some more holidays that can be attached to specific dates in the NFL season:
Columbus Day -- Monday of Week V (October 6th-12th)
Halloween -- Saturday of Week IX (October 25th-31st)
Veteran's Day -- Saturday of Week XI (November 8th-14th)
MLK Day -- Monday of Week XIX (January 12th-18th)
Some of these suggestions have been made outside of the Usher calendar proposal. For example, here's a link to a debate about making Halloween the last Saturday in October:
https://www.timeforkids.com/g34/debate-halloween-saturday-2/
The transition year can be in 2043, when October 31st is on a Saturday anyway:
2043 -- year of transition
2044 -- Halloween on Saturday, October 29th (last Saturday in October)
Meanwhile, the proposed date for MLK Day means that both college and pro football have their respective championship games on three-day weekends.
But so far, these proposals simply change American holidays to fit the American football season. We can't really call Usher's Calendar Reform a new calendar if only one country is adopting it. Also, so far we've only changed holidays during the football half of the year (September-February) and ignored holidays during the non-football half (March-August).
Usher also proposed new dates for Christmas and Easter. If the two big Christian holidays were to change, then it would certainly affect all Christian nations. But how likely is the Church to change the dates of its two main holidays?
Certainly, Christmas on December 25th has been firmly established. The holiday simply falls on the same date every year, and very few people would want to change it. On the other hand, Easter is notoriously difficult to calculate. Indeed, many schools no longer tie their spring breaks to Egg Day, preferring instead to keep it during a fixed week. Therefore, it's much more likely for movement to fix Easter to a specific week.
The current Easter range on the Gregorian Calendar is March 22nd-April 25th, a five-week range. So Usher decided to take the middle week as his Easter range, April 5th-11th. It also fits the rest of his calendar, as there are seven weeks between Super Bowl Sunday and Easter Sunday.
The proposed year of transition is 2045, when all three Easters (Gregorian, Orthodox, and Usher) fall on the same date, April 9th:
2045 -- year of transition
2046 -- Easter on Sunday, April 8th (Sunday in April 5th-11th range)
The new Easter date will unite all Christian countries. And the Western and Orthodox churches can finally celebrate its holiest day on the same date worldwide. (Note: I don't try to place Jewish, Islamic, Hindu, or Chinese holidays on the calendar as these are based on lunar calendars, whereas most Christians observe a solar calendar outside of Easter, which we're now fixing.)
5. Implementing the Usher Calendar in the 2050's
In the Usher Calendar, several Christian holidays fall near American secular holidays. In particular, the day after Presidents' Day is Mardi Gras while the day before Memorial Day (keeping its original definition as Monday, May 25th-31st) is Pentecost.
And so the Big March now suddenly lines up with Lent. Indeed, I like the idea of changing the current four-day Presidents' Day weekend (in my new district) to Saturday-Tuesday (as opposed to Friday-Monday) so that it includes Mardi Gras. The big party atmosphere (whether it's Mardi Gras, Super Bowl, or Prez Day) all ends on Ash Wednesday, the first day of the Big March and Lent.
Schools can now revert tying Easter to spring break -- but should it be the week before Easter (Holy Week) or the week after Easter (Bright Week)? Since Holy Week is still a part of Lent, we should probably make Easter/Bright Week into spring break. (Many schools will probably include Good Friday as part of spring break.)
The idea here is that other nations, after adopting the new Easter, will want to start tying their own secular holidays to the new Easter as well, thus establishing the Usher Calendar there. Indeed, this all reminds me of an attempt in Great Britain, back in 1928, to define Easter as the Sunday in the range of April 9th-15th:
https://whorunsbritain.blogs.lincoln.ac.uk/2017/04/17/what-happened-to-the-easter-act/
The April 9th-15th range is strange -- why the 9th-15th rather than the 8th-14th (which would simply be "the second Sunday in April")? I suspect April 9th-15th was chosen because this is exactly 22 days before another British holiday -- Labour Day, the first Monday in May. (I intentionally spell "Labour" the British way, in order to distinguish this from the American "Labor" Day in September.) In other words, the goal of the Easter Act was to fix the holy day to 22 days before Labour Day.
Notice that 22 days after Usher Easter is now April 27th-May 3rd. The British can define this to be the new Labour Day (near International Labour Day on May 1st). If it's preferable for the holiday to be always in May, then we can change it to the first Friday in May, 26 days after Usher Easter.
Some Canadian holidays fall near American holidays anyway, so these readily convert to the new Usher Calendar. Canadian Thanksgiving remains on the same day as Columbus Day (range October 6th-12th), and Family Day stays on the same day as Presidents' Day (range February 16th-22nd). And Victoria Day remains a week before Memorial Day (no change).
We wouldn't want to refer to the American NFL calendar to define international holidays. So instead, we will number the weeks just as Usher does.
Usher begins by defining his weeks as running from Sunday to Saturday. He decides to take the week containing the Christian holiday "Assumption of Mary" (August 15th) as Week 53, and then we count backwards from there. Most years contain 52 weeks numbered 2-53, but occasionally there is a year containing 53 weeks numbered 1-53. (That's why I used Roman numerals earlier, to distinguish the football weeks from Usher weeks.)
Thus the Usher Calendar is really a leap week calendar -- one that contains an extra week (Week 1) rather than an extra day (today, February 29th). But most laypeople won't see the week numbers -- instead, they check the calendar to see the dates (day numbers) of various holidays, even though these are defined as certain days of certain numbered weeks. For example, American Labor Day is now defined as Monday of Week 4, but instead most people see it as September 1st, or 7th, or whatever.
If August 15th is a Saturday, then the following week is Week 1, Leap Week. Thus it's easy to tell whether a year has a Leap Week -- check whether August 16th-22nd is Sunday-Saturday. If so, then it's Leap Week.
During the decade of the 2050's, it's possible that the Usher Christmas may be adopted -- once again, I mention it last as it's the least likely to be changed. Usher defines his Christmas as Sunday of Week 20, which is in the range December 21st-27th. Perhaps 2050 itself can be a year of transition, since December 25th is already on Sunday that year.
2050 -- year of transition
2051 -- Christmas on Sunday, December 24th (Sunday in December 21st-27th range)
This definition of Christmas also fixes other related holidays -- for example, the first Sunday of Advent is now equivalent to the Sunday after Thanksgiving. Epiphany, after the twelve days of Christmas, is now the Friday in the January 2nd-8th range.
Implementing the Usher Calendar in the 2060's
On this blog, I don't use the phrase "leap year," since there are two types of leap units -- Leap Days and Leap Weeks. For example, 2054 contains a Leap Week (Week 1), but not a Leap Day (February 29th), while 2052 and 2056 contain Leap Days but not Leap Weeks.
But I do use the phrase Double Leap Year to refer to a year with both a Leap Week and Leap Day. So 2048 works out to be a Double Leap Year. Actually, 2020 would also be a Double Leap Year if the Usher Calendar were already in effect (but it isn't). We can see that 2020 contains a Leap Day (February 29th), and August 15th is coming up on a Saturday (thus making August 16th-22nd into Week 1, the Leap Week).
There are two types of Double Leap Years. One of them, coming up in 2048, is harmless. But there's another type of Double Leap Year that causes problems for the Usher Calendar, and it might occur in 2060 unless we avoid it.
In that year, Presidents' Day falls on its earliest possible date, Monday February 16th. Thus Usher Easter should also be on its earliest possible date -- Sunday, April 5th, seven weeks later. But due to Leap Day, April 5th, 2060 is on Monday rather than Sunday. But Easter can't be on Sunday, April 4th, as this is out of range (April 5th-11th). Thus Easter must be on its latest possible date, April 11th -- so now there are suddenly eight weeks between Presidents' Day and Easter. This leads to problems, especially if we define both Presidents' Day and Mardi Gras to be school-closing holidays -- instead of a four-day weekend, we have off-days on a Monday and a Tuesday that are a week apart!
Easter falls on April 6th, 2059 and April 10th, 2061. These are 105 weeks apart, and so there must be one 52-week year and one 53-week year between them. But if we give 2059 a Leap Week, then Labor Day that year is on September 8th, one week too late. And if we wait until 2060 for the Leap Week, then Memorial Day that year is on May 24th, one week too early. There is no year for Leap Week that will make all holidays in both 2059 and 2060 work out correctly.
And it's easy to see why. Leap Week is supposed to occur only when August 15th is on Saturday. But due to the 2060 Leap Day, August 15th leaps directly from Friday in 2059 to Sunday in 2060. Since August 15th is never on Saturday, the Leap Week never works out.
Thus there are two types of Double Leap Years -- ones where August 15th leaps to Saturday, like 2048 (and 2020 if the Usher calendar had already started), and one where it leaps over Saturday, as it would in 2060. The former is acceptable, but the latter can't occur.
And so Usher had to come up with a new rule for Leap Days. If August 15th is on a Friday, then there cannot be a February 29th six months later. This is equivalent to saying that February 29th can fall on any day except Sunday. It's also equivalent to saying that if a year starts (on January 1st) on Thursday then February can't have 29 days.
On the Usher calendar, 2060 doesn't have a Leap Day -- instead, 2061 has the extra day. We must come up with a new leap rule other than the Gregorian rule. And this is where I come in -- I actually communicated with Usher and helped him come up with a leap rule for his new calendar.
2060 -- 28 days in February, start of new Usher Leap Rules
2061 -- 29 days in February, first new Usher Leap Day
We discovered that if we simply stated that Leap Days occur every four years, except that if the fourth year starts on Thursday than we wait for the fifth year (which would start on Friday), then this produces a 17-year cycle. (If a year with a Leap Day starts on Friday, then every fourth year after that starts on Wednesday, then Monday, then Saturday, then back to Thursday again -- 16 years after the first Leap Day year. So we postpone Leap Day one year to the 17th year starting on Friday again.)
But four Leap Days in 17 years are much too few to match the tropical year. (After all, the original reason for Leap Days is because the tropical year isn't a whole number of years.) In order to match the tropical year, Usher allows for seven such 17-year cycles, followed by a five-year cycle with two Leap Days (and one Leap Week). This produces a 124-year cycle with thirty Leap Days, along with 22 Leap Weeks. This can be cut in half to a 62-year cycle -- fifteen Leap Days, eleven Leap Weeks.
So which years in each cycle contain Leap Days? Well, notice that 2046 -- the year in which Usher Easter begins -- is a multiple of 62, so we can start the count from that year. Then since 2048, 2052, 2056, and 2061 have Leap Days, we state that the 2nd, 6th, 10th, and 15th years of each 17-year cycle contain Leap Days:
First 17-year cycle: 2048, 2052, 2056, 2061
Second 17-year cycle: 2065, 2069, 2073, 2078
Third 17-year cycle: 2082, 2086, 2090, 2095
Last 11-year cycle: 2099, 2103, 2107
The last cycle contains eleven years so that they fit in a larger 62-year cycle. Thus, to determine whether a year contains a Leap Day, we divide the year by 62 and then divide the remainder by 17 -- if the second remainder is 2, 6, 10, or 15, then it contains a Leap Day. We also find that the Leap Weeks occur in years whose second remainder is 2, 8, or 14. And so years with second remainder of 2 are Double Leap Years.
Notice that this isn't exactly the leap rule that Usher and I came up with -- the original 124-cycles were never divided into two 62-cycles. But I post it here because 2046 is a multiple of 62, but not a multiple of 124. Since the Usher Calendar didn't exist in 1984, we'd have to wait until 2108, the next multiple of 124, for full implementation of the calendar. But I'd rather start it in 2046.
Thus completes the definition of the Usher Calendar.
Loose Ends with the Usher Calendar
This is what I wrote four years ago today about the Usher Calendar:
About ten years ago, the American poster Andrew Usher also proposed a calendar with a new Leap Day rule. A long discussion of this calendar has been archived at Mathforum (the same place where some of the old John Conway discussions are archived):
[2020 update: link removed as it's now dead]
Here is how Usher states his Leap Day rule:
3. That the leap year rule be changed to have a leap year occur every
fourth save that it be delayed when the leap year would start on a
Thursday
At first glance, the Usher Calendar appears to be just another version of the Dee Calendar. Leap years usually occur four years apart -- occasionally five years apart. The only difference is that instead of the simple 33-year cycle of the Dee Calendar, the Usher leap year depends on the day of the week.
Here's the answer -- the Usher Calendar is actually a Leap Week Calendar in disguise! To see what's going on here, let's go back to another Leap Week Calendar -- the Icelandic Calendar:
http://www2.math.uu.se/~svante/papers/calendars/iceland.pdf
sumarauki (leap week): A leap week inserted after aukanætur just before miðsumar. 72 In the Gregorian version, it begins on 22 July, when that day is a Sunday, or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.
But this doesn't explain that last part of the Icelandic Leap Week rule: or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.
This phenomenon occurs whenever we try to convert from a Leap Week Calendar (like the Icelandic) to a Leap Day Calendar (like the Gregorian). I give it a special name -- a Double Leap Year. Loosely speaking, a Double Leap Year is a year with both February 29th and a Leap Week. Here the year 1995-1996 (measured from the first day of the Icelandic "summer" in April 1995 to April 1996) contains both a Leap Week (in July 1995) and a February 29th (in 1996).
We can now see why Usher came up with his strange Leap Day rule -- the whole point of it is to avoid the Double Leap Year problem! Let's look at his Leap Week rule:
4. That the perpetual calendar can be made, by considering the first
day of the year of weeks to occur on the Sunday after the Assumption,
and if this is the first possible calendar day, it is called week 1,
and otherwise week 2, and every year runs through week 53. And this
calendar ensures that everything can be fixed to a day of a certain
week
Now the Assumption of Mary is a holiday celebrated in some Christian churches -- most notably the Orthodox churches. It falls on August 15th. So Usher's Leap Week Calendar actually begins on the following Sunday, which falls in the August 16th-22nd range. Usher writes that if this Sunday is the first possible calendar day -- that is, August 16th, it is Week 1, otherwise it is Week 2. We see that Week 1 is actually the Leap Week, since Usher numbers his weeks from 1-53 in years with a Leap Week and 2-53 in years without the extra week. We notice that August 16th, 2015 fell on a Sunday, so 2015 would contain a Leap Week in the Usher Calendar. The next Usher Leap Week is in 2020.
Notice that January 1st, 2004, fell on a Thursday. So Usher avoids the Double Leap Year problem simply by forbidding February 29th in years that begin on Thursday. Then whenever January 1st falls on a Thursday, the following August 15th must fall on Saturday, and so Leap Week must occur.
Let's compare the Leap Week Rules for the Icelandic and Usher Calendars:
Sumarauki (leap week) begins on 22 July, when that day is a Sunday, or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.
Usher Week 1 (leap week) begins on 16 August, when that day is a Sunday.
Usher doesn't need a special Double Leap Year case. He simply prevents the special Double Leap Year case from ever occurring by making sure that February 29th can never fall on Sunday (that is, a year with Thursday, January 1st can never contain February 29th).
We can see why Usher does this. Any Leap Week Calendar would have the following box checked:
( ) the solar year cannot be evenly divided into seven-day weeksand possibly several other boxes as well. But the visible part of the Usher Calendar only changes which weeks have a February 29th -- and I already stated that the box that corresponds to Leap Days doesn't count since the Gregorian Calendar already has them. There might be less resistance to the Usher Calendar than to any other Leap Week Calendar since it makes no radical changes to the structure of the calendar, unlike calendars which have a visible Leap Week.
The most obvious differences between the Usher and Gregorian Calendars are the holidays. Usher places each holiday on a particular day in a numbered week (1-53) in the Leap Week Calendar.
So August 23rd is both the latest that Week 2 can begin and the earliest that Week 2 can end. That is, August 23rd is part of Week 2 no matter what. We can even define Week 2 to be the (Sunday to Saturday) week containing August 23rd. Likewise August 30th, exactly seven days after the 23rd, must be part of Week 3 no matter what, and September 6th must be part of Week 4 no matter what.
Here is a complete list of these anchor days that must belong to a particular week:
Week 2 -- Aug. 23rd
Week 3 -- Aug. 30th
Week 4 -- Sept. 6th
Week 5 -- Sept. 13th
Week 6 -- Sept. 20th
Week 7 -- Sept. 27th
Week 8 -- Oct. 4th
Week 9 -- Oct. 11th
Week 10 -- Oct. 18th
Week 11 -- Oct. 25th
Week 12 -- Nov. 1st
Week 13 -- Nov. 8th
Week 14 -- Nov. 15th
Week 15 -- Nov. 22nd
Week 16 -- Nov. 29th
Week 17 -- Dec. 6th
Week 18 -- Dec. 13th
Week 19 -- Dec. 20th
Week 20 -- Dec. 27th
Week 21 -- Jan. 3rd
Week 22 -- Jan. 10th
Week 23 -- Jan. 17th
Week 24 -- Jan. 24th
Week 25 -- Jan. 31st
Week 26 -- Feb. 7th
Week 27 -- Feb. 14th
Week 28 -- Feb. 21st
Week 29 -- Feb. 28th
Week 30 -- Mar. 7th
Week 31 -- Mar. 14th
Week 32 -- Mar. 21st
Week 33 -- Mar. 28th
Week 34 -- Apr. 4th
Week 35 -- Apr. 11th
Week 36 -- Apr. 18th
Week 37 -- Apr. 25th
Week 38 -- May 2nd
Week 39 -- May 9th
Week 40 -- May 16th
Week 41 -- May 23rd
Week 42 -- May 30th
Week 43 -- June 6th
Week 44 -- June 13th
Week 45 -- June 20th
Week 46 -- June 27th
Week 47 -- July 4th
Week 48 -- July 11th
Week 49 -- July 18th
Week 50 -- July 25th
Week 51 -- Aug. 1st
Week 52 -- Aug. 8th
Week 53 -- Aug. 15th
I like to refer to these anchor days as "Doomsday." The important thing to note here is that the Usher Doomsday matches the Conway Doomsday from March to August (or January to August, depending on how you interpret his "March 0th" rule). After the Week 53 Doomsday on Assumption Day itself, Usher Doomsday advances one day, so it's always the day after Conway Doomsday.
Returning to 2020, here's a link to another Leap Week Calendar Reform:
https://www.hermetic.ch/cal_stud/palmen/wkmth.htm
This calendar eliminates individual day numbers, which the author (Karl Palmen) says are "the trickiest part of the date to remember." Instead, he divides the Gregorian year into twelve months, each with four or five weeks (named for Greek letters Alpha, Beta, Gamma, Delta, Epsilon).
It turns out that if Gregorian August 15th is on a Saturday, then the Palmen year has 53 weeks. And notice that the months listed above with five Doomsdays are May, August, November, and January -- and three of these months also have an Epsilon week on the Palmen Calendar.
Thus Usher dates can be mapped readily onto the Palmen Calendar and back. Of course, there are still a few differences between the two calendars. Usher weeks go Sunday-Saturday, while Palmen weeks go Monday-Sunday. And Palmen places his Leap Week at the end of the year. If we wished to place it closer to Usher Leap week, we can have an Epsilon week at the end of July rather than December.
Still, it might be interesting to drop Usher weeks 1-53 and use Palmen's month-Greek instead. In either case, these week names form the invisible Leap Week Calendar -- the visible part of the Usher Calendar uses only day numbers and Leap Days.
Also, I notice that Palmen isn't as optimistic as Usher that the Church will actually agree to change the Christmas and Easter dates. Thus Palmen describes how to keep Christmas on Gregorian December 25th as well as maintain the usual Easter rules.
Oh, and I notice that Steve Hanke is using Leap Day to promote his own Leap Week Calendar, the Hanke-Henry Calendar, again:
https://www.cnn.com/2020/02/29/us/permanent-calendar-leap-year-trnd/index.html
One difference between Hanke-Henry and Usher is that Hanke-Henry Christmas is always on a Monday, while Usher Christmas is always on a Sunday. Both Hanke-Henry and Palmen place Leap Week at the end of the year (although an older version of Hanke-Henry had a summer Leap Week, just like Usher).
Speaking of summer, that's one holiday I didn't convert to the Usher Calendat -- the 4th of July. Once again, we look at the QNTM Calendar Reform checklist:
( ) good luck trying to move the Fourth of JulyOf course, the "Fourth of July" can only occur on July 4th. The question is whether we want to keep the holiday on July 4th, or change it to a fixed day of the week like all other holidays.
If we want to keep both the Leap Week structure of the calendar and make it fall on the "first ______day in July," then this works out to be Wednesday (range July 1st-7th), exactly five weeks and two days after Memorial Day. But a Wednesday holiday isn't desirable. It's probably better to push it back to the following Friday (range July 3rd-9th), since a Monday would either include June dates in its range or fail to include the 4th.
The Usher Calendar will be a strong Calendar Reform once it's fully implemented. And the first step of that implementation may be just around the corner -- the 17-game NFL proposal.
Loose Ends with Leap Day
This is what I wrote four years ago about the history of Leap Day itself:
Let's get back to Leap Days and February 29th. Why is there a Leap Day? On Google today, there is a special Google Doodle with the 29th leaping between the 28th and 1st. If you click on this Doodle, the search query "Leap Day" appears. Well, by now we should know the answer to why there are leap days:
( ) the solar year cannot be evenly divided into solar days
The following link explains the origin of Leap Days and the Julian Calendar. It also explains the Doomsday Algorithm -- a method of determining the day of the week given its date, first devised by one of my favorite mathematicians, John Conway:
http://people.cst.cmich.edu/graha1sw/Pub/Doomsday/Doomsday.html
Julius Caesar introduced the concept of the Leap Year; i.e., the idea of adding an extra day to February in every year divisible by 4. His calendar is called the Julian Calendar, and it was used throughout the Western world until 1582. In 1582, Pope Gregory XIII decreed a modification of the Julian Calendar. He declared that Century years (that is years divisible by 100) are leap years if and only if they are divisible by 400. Thus 1700, 1800, and 1900 are not leap years, but 2000 will be a leap year. The resulting calendar is called the Gregorian Calendar.
The following link is, as of the time I clicked on the Google Doodle, these are the top news results to the query "why is there a leap day?"
http://www.telegraph.co.uk/news/uknews/12177017/Leap-Year-2016-Why-does-February-have-29-days-every-four-years.html
http://www.csmonitor.com/Science/2016/0229/Why-is-there-a-leap-day-Is-there-an-alternative
http://fivethirtyeight.com/features/lots-of-parents-dont-want-their-kids-to-be-born-on-leap-day/
I've incorporated information from all three links into today's post.
Here's a link to an old Mathforum post where Conway explains how to calculate the weekday of the assassination of Julius Caesar, the founder of the Julian calendar:
[2020 update: link removed as it's now dead]
In the original Julian Calendar, 366-day years were called "bissextile" years. This name "bissextile" means something like "double-sixth," referring to the fact that the extra day was not February 29th but the sixth day before March 1st (the Calends of March, not to be confused with the Ides). If we count the Calends itself as the first day, then the repeated sixth day is February 24th. Until 1968, the official Leap Day was February 24th -- not until 1972 was February 29th made the actual Leap Day.
So the Julian Calendar is the simplest possible Leap Day calendar -- we simply add an extra day every four years, so each year would average 365.25 days in length. On your first birthday, you were either 365 or 366 days old, on your second birthday you were either 730 or 731 days old, and on your third birthday, you were either 1,095 or 1,096 days old. But on your fourth birthday, you were exactly 1,461 days old (according to the Julian Calendar). Because of this, I often refer to one's fourth birthday as the first Julian birthday. Then one's eighth birthday becomes the second Julian birthday, one's twelve birthday becomes the third Julian birthday, and so on.
The Telegraph link above mentions that in the 19th century play The Pirates of Penzance, Gilbert and Sullivan propose the idea that a person born on February 29th only has Julian birthdays. Therefore its main character, Frederic, must remain a pirate until his 21st Julian birthday -- when he is actually 84 years old.
Also according to the Telegraph link, a Maryland high school teacher once proposed that Leap Day should be a day off -- why should we work an extra day for free? As much as I wouldn't mind having an extra day off leading up to the Long March, I disagree with this -- especially for teachers. We work the same number of days every year -- 180 plus a few extra PD days -- regardless of whether there is February 29th or not. If our work schedule actually increased a day during years with a Leap Day, he'd have a point.
[2020 update: At a school I subbed at this year, I saw a "Leap Year correction" holiday for classified workers who work all 12 months of the calendar year, from July to June. The Leap Year correction occurred at the beginning of this fiscal year, on Friday, July 5th -- just in time to make the Fourth of July into a four-day weekend.]
I myself was born in 1980. I did once meet a Leaper -- a girl from my school who was born on February 29th, 1980, which makes today her tenth Julian birthday. According to the 538 link above, Leapers are rare -- parents avoid scheduling C-sections on Leap Day because they want their children to celebrate a birthday every year, not just the Julian birthdays.
Now this is when we must consider religion in more detail. Pope Gregory XIII was the one who reformed the calendar -- but why would Protestants care about what the Pope has to say? For that matter, why would the Eastern Orthodox Church care about what the Pope has to say? Throughout this post, we must remember that there are three main branches of Christianity -- and these three branches observe the calendar differently.
Most Protestant countries switched to the Gregorian calendar by the 18th century (most notably Great Britain in 1752). But some Orthodox (most notably the Russian Orthodox Church) have never made the switch -- they are still using the Julian calendar. Other Orthodox Churches, such as the Greek, observe the Gregorian Christmas and the Julian Easter.
There have been attempts to improve further on the Gregorian Leap Year, since even its average length of 365.2425 days is still slightly longer than the length of an average year. The Revised Julian calendar, accepted in some Orthodox churches, has Leap Days in 2000 and 2400 just like the Gregorian calendar, but then the next century Leap Days are 2900, 3300, 3800, and so on. The gaps alternate between 400 and 500 years. The average year length in this calendar is 365.2422... days.
Loose Ends with the Eleven Calendar
Today, Leap Day, is the third blank day of the Eleven Calendar. In a common year, there are only two blank days, but Leap Day causes there to be three. Tomorrow, March 1st, is considered to be New Year's Day on the Eleven Calendar.
You might notice that I spend lots of time placing various Christian holidays (Mardi Gras, Pentecost, Advent) on the Usher calendar, yet for the Eleven Calendar, we only placed Christmas, Easter, and the Assumption. (I was inspired by Usher to mention the Assumption, but I included it only so that I have three Christian holidays to divide my year into thirds, with three full-week breaks per year.) Oh, and you might notice that I, like Usher, placed Christmas on Sunday in the Eleven Calendar. (But that was because weekends work differently in my calendar -- Sunday is the only day that Christians are guaranteed to have off, so I had to place Christmas there.)
Anyway, let's check out what Lent looks like on the Eleven Calendar. The definition of Lent is that is lasts for 40 days before Easter, not counting Sundays. On the Eleven Calendar, there are eleven days per week, with one of those days being Sunday. Thus there are ten non-Sundays per week -- the 40 non-Sundays are exactly four weeks.
And so the last day before Lent in the Eleven Calendar is the Sunday four weeks before Easter -- so instead of Mardi Gras, it's "Sunday Gras" (Dimanche Gras). Lent itself begins the next day, which would be "Ash Fourday" (or "Clean Fourday").
In fact, there are also Quinquagesima Sunday, Sexagesima Sunday, and so on. These Latin names imply that they occur 50, 60, etc., days before Easter -- but mathematically, those days can't literally both be Sundays (whether we include Sundays in the count or not). In the Eleven Calendar, we can make these days both be Sundays if we count only the ten non-Sundays in each week.
(The only Christians who pay attention to Quinquagesima, Sexagesima, etc., are probably Orthodox Christians for whom these dates determine what they may eat -- Cheesefare, Meatfare, and so on.)
There's also Pentecost Sunday, which is the fiftieth day after Easter (with Easter as the first day). But unlike Lent, Sundays are included in the Pentecost count. We can't make Pentecost fall on Sunday in the Eleven Calendar unless we decide suddenly not to include Sundays in the count. (This might be worth it, since Sunday is the only day that Christians are guaranteed to have off from work.) We also have the Ascension, the fortieth day after Easter. Once again, if we don't count Sundays, then we can make the Ascension land on Sunday as well. (It's actually Thursday in the Gregorian Calendar.)
There's still a Friday in my calendar, so we can keep Good Friday two days before Easter. If we want Palm Sunday to remain on Sunday, then it's four extra days between Palm Sunday and Good Friday.
As for holidays near Christmas, Advent is tricky in the Eleven Calendar. Advent is defined not as 20, 30, etc., days before Christmas, but as four Sundays before the holiday. But in the Eleven Calendar, four Sundays before Christmas is 44 days before the holiday, which might be too long. We might instead have Advent start two Sundays or 22 days before Christmas instead.
If Christmas Day -- that is, the first day of Christmas -- is on Sunday, then 11 days later is the twelfth day of Christmas, also on Sunday. This is January 5th Gregorian -- we ordinarily consider Epiphany to be the next day, January 6th. Then again, we can always cheat and use inclusive counting so that Epiphany also falls on a Sunday. (Yes, so we keep manipulating inclusive/exclusive counting in order to make as many Christian holidays fall on Sunday as possible.)
Conclusion: Another Quadrennial Task
Today I celebrate Leap Day with a $2.29 pizza from 7-Eleven. Tonight I'll watch Hidden Figures -- I've been meaning to watch it ever since Katherine Johnson died. Since the main character is a black woman, I can't decide whether to watch it for Black History Month or Women's History Month. So I'll watch it tonight at 11PM, which will straddle both months.
Leap Day isn't the only day that occurs once every four years. Three days from today is another day that's every four years -- Super Tuesday.
Four years ago, I wrote about the presidential election and how the candidates fared on various issues (especially education, since this is an education blog). But I haven't written as much about the election so far this year.
This year, California is a Super Tuesday state. But this year, all polling places are open for four days starting today, while some are open for a full week. (Oops, by "a full week" I mean eleven days -- here I go thinking in terms of the Eleven Calendar again!) The polling place near my house is open today -- and I finished voting today, on Leap Day itself. (Another state, South Carolina, holds its primary on Leap Day as well.)
It was tough for me to choose a candidate -- indeed, I was literally undecided about my presidential vote until I pulled up to the curb outside the polling place. I don't post endorsements or favor a candidate on the blog. But I will make a special post comparing the candidates on education and other issues.
I'll make my special political post on Monday, the day before Super Tuesday. And so I wish everyone a Happy Leap Day!
