Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

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Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Karl Palmen

Dear Calendar People

 

On the May 16 this year in reply to the issue of continued fractions I said:

 

I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

 

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.

 

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

 

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

 

I found one measure of structural complexity, which involves Ford circles:

 

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

 

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

 

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in  pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

 

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11).  Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

 

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

 

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840  for the leap month years of lunisolar calendars

 

 

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule  (with yerms mixed) that applies to this measure of complexity.

 

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

 

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

 

 

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

 

½ [1]+

1/3 [1]-

3/8 [2]+

4/11 [2]-

11/30 [3]

 

½ [1]-

8/15 [2]+

9/15 [2]-

26/49 [3]

 

So not every iteration increases complexity.

 

 

Karl

 

16(01(13

 

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Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Karl Palmen

Dear Calendar People

 

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

 

COMPLEXITY 0

0/1 0.0 ‘c

 

COMPLEXITY 1

1/8 0.125     ‘cccccccL

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL

1/5 0.2       ‘ccccL’

1/4 0.25      ‘cccL’  Julian Calendar

 

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

 

COMPLEXITY 1

1/3 0.333333… ‘ccL’

 

COMPLEXITY 2

3/8 0.375     ‘ccLccLcL’ Octaeteris

2/5 0.4       ‘ccLcL

3/7 0.428571… ‘ccLcLcL

 

COMPLEXITY 1

1/2 0.5        ‘cL’         

 

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

 

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

 

COMPLEXITY 3

5/12  0.416666… ‘ccLcLccLcLcL

 

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

 

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

https://en.wikipedia.org/wiki/Pell_number

 

COMPLEXITY 4

12/29  0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL

 

COMPLEXITY 5

29/70 0.4142857…  ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL

 

 

Karl

 

16(01(17

 

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 14 October 2016 13:04
To: [hidden email]
Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Calendar People

 

On the May 16 this year in reply to the issue of continued fractions I said:

 

I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

 

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.

 

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

 

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

 

I found one measure of structural complexity, which involves Ford circles:

 

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

 

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

 

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in  pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

 

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11).  Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

 

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

 

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840  for the leap month years of lunisolar calendars

 

 

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule  (with yerms mixed) that applies to this measure of complexity.

 

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

 

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

 

 

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

 

½ [1]+

1/3 [1]-

3/8 [2]+

4/11 [2]-

11/30 [3]

 

½ [1]-

8/15 [2]+

9/15 [2]-

26/49 [3]

 

So not every iteration increases complexity.

 

 

Karl

 

16(01(13

 

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Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Walter J Ziobro

Dear Karl and Calendar List

What would be the complexity level of my 13 month leap month calendar in which every month,,, including the leap month always has 28 days?

I'm guessing complexity level 2

Walter Ziobro

Sent from AOL Mobile Mail




On Tuesday, October 18, 2016 Karl Palmen <[hidden email]> wrote:

Dear Calendar People

 

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

 

COMPLEXITY 0

0/1 0.0 ‘c

 

COMPLEXITY 1

1/8 0.125     ‘cccccccL

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL

1/5 0.2       ‘ccccL’

1/4 0.25      ‘cccL’  Julian Calendar

 

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

 

COMPLEXITY 1

1/3 0.333333… ‘ccL’

 

COMPLEXITY 2

3/8 0.375     ‘ccLccLcL’ Octaeteris

2/5 0.4       ‘ccLcL

3/7 0.428571… ‘ccLcLcL

 

COMPLEXITY 1

1/2 0.5        ‘cL’         

 

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

 

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

 

COMPLEXITY 3

5/12  0.416666… ‘ccLcLccLcLcL

 

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

 

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

https://en.wikipedia.org/wiki/Pell_number

 

COMPLEXITY 4

12/29  0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL

 

COMPLEXITY 5

29/70 0.4142857…  ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL

 

 

Karl

 

16(01(17

 

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 14 October 2016 13:04
To: CALNDR-L@...
Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Calendar People

 

On the May 16 this year in reply to the issue of continued fractions I said:

 

I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

 

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.

 

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

 

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

 

I found one measure of structural complexity, which involves Ford circles:

 

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

 

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

 

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in  pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

 

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11).  Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

 

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

 

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840  for the leap month years of lunisolar calendars

 

 

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule  (with yerms mixed) that applies to this measure of complexity.

 

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

 

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

 

 

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

 

½ [1]+

1/3 [1]-

3/8 [2]+

4/11 [2]-

11/30 [3]

 

½ [1]-

8/15 [2]+

9/15 [2]-

26/49 [3]

 

So not every iteration increases complexity.

 

 

Karl

 

16(01(13

 

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Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Karl Palmen

Dear Walter and Calendar People

 

The complexity of such a calendar would depend on the leap year rule for the leap months and would apply only if the leap years were spread as smoothly as possible.

 

I have forgotten what his leap year rule was, but I guess it is 13 leap years in 293 years. The intervals between the leap years alternate between 22 & 23 years, except for one pair of consecutive 23s. This I reckon to be complexity 3.

 

Complexity 2 is when all intervals between years of the minority type (leap years) are equal with ONE exception which is ONE year different.  Examples are the 33-year cycle of 8 leap years or the 45-year cycle of 2 leap years for a 28-day month calendar.

 

I think the reason Walter thinks it is complexity 2,  may be that he has been looking at the sequence of intervals between the leap years rather than the sequence common years and leap years. The intervals form two types that are spread as smoothly as possible and so have a complexity and this complexity is 1 less (if the leap years are a minority). This process can be repeated again and again this we have just one interval and in so doing we measure the complexity.

 

13/293 ccc…cLc… etc.  complexity 3

7/13   23:22:23:22:23:22:23:22:23:22:23:22:23 complexity 2

Taking intervals between the 22s which are the minority type

1/6  3:2:2:2:2:2 complexity 1

Then we have only 1 interval between the 3s (of 6).

 

 

The complexity can be measured by means of repeated mixing in which each mix has one of the one type and one or more of another type of equal complexity.

I show this mixing for the 28-day month calendar:

 

0/1: All years have 13 months, no leap years: Complexity 0.

1/1: All years have 14 months, every year is a leap year: Complexity 0.

 

1/23: Mix 22 of 0/1 with 1 of 1/1 and we get a 23-year cycle with one leap year:  complexity 1.

1/22: Mix 21 of 0/1 with 1 of 1/1 and we get a 22-year cycle with one leap year: complexity 1

 

2/45: Mix 1 of 1/23 with 1 of 1/22 and we get 45-year cycle with two leap years: complexity 2

3/68: Mix 2 of 1/23 with 1 of 1/22 and we get 68-year cycle with three leap years: complexity 2

 

13/293: Mix 5 of 2/45 with 1 of 3/68 and we get 293-year cycle with thirteen leap years: complexity 3.

 

 

For leap weeks the corresponding cycle is 52/293, which has complexity 4.

 

0/1: All years have 52 weeks, no leap years: Complexity 0.

1/1: All years have 53 weeks, every year is a leap year: Complexity 0.

 

1/6: Mix 5 of 0/1 with 1 of 1/1 and we get 6-year cycle with one leap year: complexity 1

1/5: Mix 4 of 0/1 with 1 of 1/1 and we get 5-year cycle with one leap year: complexity 1

 

3/17: Mix 2 of 1/6 with 1 of 1/5 and we get 17-year cycle with three leap years: complexity 2

2/11: Mix 1 of 1/6 with 1 of 1/5 and we get 11-year cycle with two leap years: complexity 2

 

11/62: Mix 3 of 3/17 with 1 of 1/11 and we get 62-year cycle with 11 leap years: complexity 3

8/45:  Mix 2 of 3/17 with 1 of 1/11 and we get 45-year cycle with 8 leap years: complexity 3

 

52/293: Mix 4 of 11/62 with 1 of 8/45 and we get 293-year cycle with 52 leap years: complexity 4

 

The Gregorian leap week cycle 71/400 would require one more step,, because it is 5 of 11/62 and 2 of 8/45 and so has complexity 5 as shown in my original note. Walter’s  400-leap week cycle based on his modified 33-year cycle is an example of this.

 

Note that these mixes are not determined by continued fraction convergents. Sometimes a convergent may be skipped. Instead they are determined by Ford circles on which this complexity is defined.

 

 

Karl

 

16(01(18

 

Yerm Calendar page updated to show yerm 16 at

http://www.hermetic.ch/cal_stud/palmen/yerm1.htm#moon

 

 

From: Walter J Ziobro [mailto:[hidden email]]
Sent: 19 October 2016 04:17
To: [hidden email]; Palmen, Karl (STFC,RAL,ISIS)
Subject: Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Karl and Calendar List

What would be the complexity level of my 13 month leap month calendar in which every month,,, including the leap month always has 28 days?

I'm guessing complexity level 2

Walter Ziobro

Sent from AOL Mobile Mail

 


On Tuesday, October 18, 2016 Karl Palmen <[hidden email]> wrote:

Dear Calendar People

 

I list all examples, with mean year 0 to ½ , up to 8 years long, in the order of their mean year.

I’ve started each example cycle, so that the leap years occur as late as possible and so year Y of a C-year cycle with L leap years is I leap year if and only if

Remainder (Y*L) divided by C is less than L.

I’ve also coloured red any leap year preceded by an odd number of common years along with those common years to show the structure.

 

COMPLEXITY 0

0/1 0.0 ‘c

 

COMPLEXITY 1

1/8 0.125     ‘cccccccL

1/7 0.142857… ‘ccccccL’

1/6 0.166666… ‘cccccL

1/5 0.2       ‘ccccL’

1/4 0.25      ‘cccL’  Julian Calendar

 

COMPLEXITY 2

2/7 0.285714… ‘cccLccL’

 

COMPLEXITY 1

1/3 0.333333… ‘ccL’

 

COMPLEXITY 2

3/8 0.375     ‘ccLccLcL’ Octaeteris

2/5 0.4       ‘ccLcL

3/7 0.428571… ‘ccLcLcL

 

COMPLEXITY 1

1/2 0.5        ‘cL’         

 

All cycles whose mean year lies between those of any two cycles on consecutive lines (of same complexity) are of greater complexity. There are an infinite number of cycles of complexity 2 with mean year between a cycle of complexity 1 and an adjacent cycle of complexity 2 and they converge to the cycle of complexity 1.

 

I reckon the shortest example of complexity 3 has 12 years of which either 5 or 7 are leap

 

COMPLEXITY 3

5/12  0.416666… ‘ccLcLccLcLcL

 

It is the mediant of 2/5 and 3/7 and with leap years placed as late as possible is also the concatenation of these two in the order of their mean year.

This cycle would apply to a 12 month year of 365 days with the 30 & 31 day month spread as smoothly as possible (and as shown here, with the 31-day months as late as possible).

It also corresponds to the white and black notes of a piano keyboard (and as shown here, starting from B).

 

I reckon the shortest cycles of each level of complexity are

1/2, 2/5, 5/12, 12/29, 29/70, … converging to the fractional part of the square root of two.

The integer number sequence they contain is the Pell numbers in which P(n+1) = 2*P(n) + P(n-1).

https://en.wikipedia.org/wiki/Pell_number

 

COMPLEXITY 4

12/29  0.413793… ‘ccLcLccLcLccLcLcLccLcLccLcLcL

 

COMPLEXITY 5

29/70 0.4142857…  ‘ccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLccLcLcLccLcLccLcLcLccLcLccLcLcL

 

 

Karl

 

16(01(17

 

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 14 October 2016 13:04
To: CALNDR-[hidden email]
Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Calendar People

 

On the May 16 this year in reply to the issue of continued fractions I said:

 

I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

 

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.

 

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

 

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

 

I found one measure of structural complexity, which involves Ford circles:

 

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

 

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

 

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in  pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

 

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11).  Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

 

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

 

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840  for the leap month years of lunisolar calendars

 

 

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule  (with yerms mixed) that applies to this measure of complexity.

 

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

 

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

 

 

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

 

½ [1]+

1/3 [1]-

3/8 [2]+

4/11 [2]-

11/30 [3]

 

½ [1]-

8/15 [2]+

9/15 [2]-

26/49 [3]

 

So not every iteration increases complexity.

 

 

Karl

 

16(01(13

 

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Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Karl Palmen
In reply to this post by Karl Palmen

Dear Calendar People

 

A summary of examples of the complexity of cycles whose leap years are spread as smoothly as possible.

 

Definition: The complexity of a cycle is how far its Ford circle is from either Ford circle of 0/1 or 1/1 in steps of touching Ford Circles.

See https://en.wikipedia.org/wiki/Ford_circle for Ford Circle.

The Ford circle of a cycle is the Ford circle of the fractional part of its mean year, which will serve as notation of the example cycles below.

 

For example, the 293-year leap week cycle 52/293 has its Ford circle touch 11/62, which in turn touches 3/17, which in turn touches 1/6, which in turn touches 0/1. There are 4 touchings, therefore 4 steps and also this is the minimum and so 52/293 has complexity 4. Note that the Ford circles of a/b & c/d touch if and only if ad & bc differ by 1.

 

A cycle has complexity 1 if it has only one leap year or one common year.

A cycle has complexity 2 if its number of years is one different from a proper multiple of the number of leap years or the number of common years.

 

Solar leap day calendars

Complexity 1: ¼ Julian Calendar

Complexity 2: 8/33 33-year cycle

Complexity 3: 31/128 & 71/400

 

Leap Week Calendars

Complexity 1: 1/6 & 1/5

Complexity 2: 3/17 & 2/11

Complexity 3: 14/79, 11/62, 8/45 & 5/28

Complexity 4: 69/389, 74/417, 52/293 & 41/231

Complexity 5: 159/896 & 71/400

 

Lunar Calendars by month

Complexity 1: ½

Complexity 2: 9/17 & 8/15 yerm

Complexity 3: 26/49 3-yerm cycle

Complexity 4: 399/752, 425/801, 451/850, 477/899, … and all other one-era cycles

Complexity 5: 876/1651 & 928/1749 some multi-era cycles

Complexity 6: 13752/25920 Hebrew Yerm Calendar

 

Lunar Calendars by 12-month year

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 11/30 Tabular Islamic

Complexity 4: 29/79 Suggested improvement of Tabular Islamic

 

Lunisolar Calendar leap month cycles

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 7/19 Metonic cycle

Complexity 4: 123/334 & 130/353 and other cycles made from 7/19s one of which is truncated to 4/11.

Complexity 5: 239/649, 383/1040 & 267/725

Complexity 6: 622/1689 was used in an unsmooth form in 2nd Goddess Lunar calendar

Complexity 7: 2519/6840 Meyer-Palmen cycle

 

28-day month Calendar leap month cycles

Complexity 3: 13/293

 

30-day month Calendar leap month cycles

Complexity 3: 7/40 Julian calendar mean year

Complexity 4: 18/103

 

Karl

 

16(02(20

 

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Re: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

Karl Palmen
In reply to this post by Karl Palmen

Dear Calendar People

 

In my original note I showed examples grouped according to their complexity. Here I show them according to cycle type. This may better show the relationship between cycles of different complexity. I gain write each cycle down as the fractional part of its mean year in units of intercalation. The complexity applies only to cycles whose leap years are spread as smoothly as possible.

I use a comma to separate mixer cycles (Ford circles touch) and semicolon for non-mixers.

 

Solar Leap Day:

Complexity 1: ¼

Complexity 2: 8/33

Complexity 3: 15/62, 23/95, 31/128, 39/161, 47/194, 55/227,  63/260, 71/293; 97/400

Complexity 4: 109/450

 

Solar Leap Week:

Complexity 1: 1/6, 1/5

Complexity 2: 3/17, 2/11

Complexity 3: 14/79, 11/62, 8/45, 5/28

Complexity 4: 69/389; 74/417, 63/355, 52/293, 41/231

Complexity 5: 159/896, 71/400

 

Lunisolar Leap Month:

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 7/19

Complexity 4: 116/315, 123/334, 130/353, 137/372, 144/391

Complexity 5: 239/649; 383/1040; 267/725

Complexity 6: 622/1689

Complexity 7: 2519/6840

 

12-Month Lunar Leap Day:

Complexity 1: 1/3, ½

Complexity 2: 4/11, 3/8

Complexity 3: 11/30

Complexity 4: 29/79

 

Lunar:

Complexity 1: ½

Complexity 2: 8/17, 7/15 (yerm)

Complexity 3: 26/49 (3-yerm cycle)

Complexity 4: 451/850 (or any other 1-era cycle)

Complexity 5: 876/1651; 928/1749

Complexity 6: 13753/25920

Yerms hide two levels of the complexity and yerm-eras hide another two levels.

 

Eclipse Season Lunar (5+fraction months = mean eclipse season)

Complexity 1: 6/7, 7/8 (Hepton & Octon)

Complexity 2: 13/15, 20/23 (Tzolkinex & Tritos)

Complexity 3: 33/38, 53/61 (Saros & Inex)

 

Lunisolar Abundance:

Complexity 1: 1/6, 1/5

Complexity 2: 6/31, 7/36

Complexity 3: 13/67, 20/103

Complexity 4 : 97/500; 33/170

 

Months of 1/13 year as in Victor’s 28/293 and 43/450 calendars:

Complexity 1:  1/11, 1/10

Complexity 2:  2/21, 3/31

Complexity 3:  15/157, 13/136

Complexity 4:  43/450; 28/293

 

Karl

 

16(04(03

 

 

From: Palmen, Karl (STFC,RAL,ISIS)
Sent: 14 October 2016 13:04
To: [hidden email]
Subject: Structural Complexity of a Cycle with Leap Years spread as Smoothly as Possible

 

Dear Calendar People

 

On the May 16 this year in reply to the issue of continued fractions I said:

 

I think the best thing about continued fractions applied to calendars is that they reveal the structure of cycles in which the leap years or long months are spread as smoothly as possible. Chapter 4 the conclusion goes into this for the example of the leap day solar calendar, showing intervals of 4 or 5 years between the leap year.”

 

I now think Ford circles are better for this. Each calendar cycle of C years and L leap years can be given a circle of diameter 1/(C^2) if these circles are placed on a horizontal line each touching the line at position L/C, then neighbouring circles will touch without overlapping. See

https://en.wikipedia.org/wiki/Ford_circle for more details about Ford circles.

 

If a calendar cycle has just two types of year, a common year and a leap year and these are spread as smoothly as possible, then the jitter of the calendar is minimum, there are general algorithms to convert such a calendar and such a calendar is a minimum displacement calendar for a constant calendar mean year (Y). One objection to such cycles is that the sequence of leap years can be complicated and this had led to proposals in which the leap years are not spread as smoothly as possible, leading to greater jitter and more difficulty working out a conversion algorithm for the calendar.

 

Here I just look a way of measuring the complexity of a cycle with leap years spread as smoothly as possible. The type of complexity I’m looking at is in the structure of the pattern of the leap years and common years. It is different from other types of complexity, such as the length of the cycle or the complexity of doing a calendar conversion.

 

I found one measure of structural complexity, which involves Ford circles:

 

The simplest of all cycles are those that have no leap years (or every year is a leap year). For example, the Egyptian Calendar. These have complexity 0. The corresponding Ford circles are 0/1 & 1/1 and are shown in brown in the Wikipedia picture.

 

Next in complexity are the cycles with just one leap year or just one common year. For example, the Julian Calendar, which has a leap year once every 4 years. The corresponding Ford circles are those that touch those of complexity 0 and are

… 1/6, 1/5, ¼, 1/3, ½, 2/3, ¾, 4/5, 5/6, …

These cycles have complexity 1.

 

The cycles of complexity 2 are those whose Ford circles touch a Ford circle of a cycle of complexity 1, but do not already have a lesser complexity. For example, the 33-year cycle of 8 leap years (touches ¼) or the Octaeteris of 8 years with 3 leap years (touches 1/3). Examples previously mentioned on this list have fractions:

8/33 for solar leap day calendars

2/11 & 3/17 for leap week calendars (but not 5/28)

8/15 & 9/17 for months in  pure lunar calendars. These are the yerms in my lunar yerm calendar

3/8 & 4/11 for the leap month years in a lunisolar calendar.

 

You have probably guessed by now that the cycles of complexity 3 are those whose Ford circles touch a Ford of a cycle of complexity 2 and have no lower complexity. For example the 19-year Metonic cycle with 7 leap years (touches 3/8) and the Tabular Islamic calendar (touches 4/11).  Another example is the 28-year cycle of 5 leap weeks (touches 2/11 & 3/17). Examples previously mentioned have fractions:

Walter’s 97/400 for solar leap day calendars (also 71/293 & 31/128)

11/62 for leap week calendars (also 5/28, 8/45 & 14/79)

26/49 for months in for pure lunar calendars (3-yerm cycle)

11/30 for 12-month years in pure lunar calendars (but not 29/79)

7/19 for leap month years in a lunisolar calendar

 

Complexity 4 includes the 334-year cycle, 353-year cycle and others that have once truncation of a Metonic cycle by removing one octaeteris and the 293-year leap week cycle used in the Symmetry454 calendar and also the months of my yerm calendar. Examples previously mentioned have fractions:

52/293 & 69/389 for leap week calendars (also 41/231)

451/850 for months of my yerm calendar or any other one-era cycle

29/79 for 12-month years in a pure lunar calendars

123/334 & 130/353 for the leap month years of a lunisolar calendar.

 

Complexity 5 or more are usually easy to avoid. Examples previously mentioned have fractions:

71/400 & 159/896 for leap week calendars

928/1749 months of a pure lunar calendar (mentioned by Helios) or any other multi-era cycle

239/649, 267/725 & 2519/6840  for the leap month years of lunisolar calendars

 

 

In my Lunar Yerm Calendar and similar the yerms have a complexity of 2 and form a cycle whose complexity is 2 less than the cycle of months and so for example my yerm calendar yerm cycle (17/52) has complexity 2, while the months (451/850) have complexity 4. This is an example of a mixing addition rule  (with yerms mixed) that applies to this measure of complexity.

 

Take any two cycles whose Ford circles touch (such a pair has been referred to as mixer cycles) with equal complexity. Then the complexity of any cycle formed by mixing them is equal to the complexity of either of the two mixers plus the complexity of the mix, which is defined through the Ford circle of the mix proportion.

 

I show another example: The Tabular Islamic calendar 11/30. This is a mix of 1/3 and ½, which both have complexity 1. The mix is 8/11, which has complexity 2. They add up to 3.

cLc|cL|cLc|cLc|cLc|cL|cLc|cLc|cLc|cL|cLc|

Also this cycle is a mix of 3/8 and 4/11, which both than complexity 2. The mix is 1/3, which has complexity 1. They add up to 3.

cLccLcLc|cLccLccLcLc|cLccLccLcLc|

Both steps of mixing can be shown as

cLc|cL|cLc||cLc|cLc|cL|cLc||cLc|cLc|cL|cLc|

 

 

Finally, going back to continued fractions, mentioned at the start of this note, the adjacent convergents of any continued fraction have their Ford circles touching. So each iteration raises the complexity of the cycle by no more than 1. I show with 11/30 & 26/49 (both complexity 3) as examples. I list its convergents and their complexity in [] followed by + or – to indicate whether the convergent is above or below the target.

 

½ [1]+

1/3 [1]-

3/8 [2]+

4/11 [2]-

11/30 [3]

 

½ [1]-

8/15 [2]+

9/15 [2]-

26/49 [3]

 

So not every iteration increases complexity.

 

 

Karl

 

16(01(13

 

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