e and lunisolar ratio

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e and lunisolar ratio

Irv Bromberg
Dear Kalendarists:

The traditional Hebrew calendar has 7 leap years in 19 solar years. The fraction 19/7 is a pretty good approximation to e, accurate to within an error of less than 0.004.

My Rectified Hebrew calendar has 130 leap years in 353 solar years. The fraction 353/130 is a better approximation to e, accurate to within an error of less than 0.003.

What is the significance of this good approximation?
(This isn't a puzzle, I'm just asking for ideas.)

-- Irv Bromberg, University of Toronto, Canada
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Re: e and lunisolar ratio

Amos Shapir-2
Hi Irv and calendar people,

This is just a coincidence that the ratio of the length of the mean tropical year y and the mean lunar month m, then y/m-12 is close to 1/e.

If it were exactly 1/e, for the current length of the mean lunar month, the tropical year would have to be 365.23076 days -- way too short for any tropical year.
(Incidentally, it's very close to 3 leap days in 13 years).

FWIW, since the mean lunar month is increasing and the mean tropical year is decreasing, this ratio will be reached some time in the future (I'd leave it as an exercise to the reader to find out exactly when).

Have fun,

Virus-free. www.avg.com

On Tue, Dec 26, 2017 at 7:11 PM, Irv Bromberg <[hidden email]> wrote:
Dear Kalendarists:

The traditional Hebrew calendar has 7 leap years in 19 solar years. The fraction 19/7 is a pretty good approximation to e, accurate to within an error of less than 0.004.

My Rectified Hebrew calendar has 130 leap years in 353 solar years. The fraction 353/130 is a better approximation to e, accurate to within an error of less than 0.003.

What is the significance of this good approximation?
(This isn't a puzzle, I'm just asking for ideas.)

-- Irv Bromberg, University of Toronto, Canada



--
Amos Shapir
 
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Re: e and lunisolar ratio

Victor Engel
This coincidence reminds me of the one I discovered some time ago relating imperial and metric systems to solar and lunar cycles. If you take a circle and measure it's diameter in dm so that the diameter in dm matches the length of a lunation in days, then the circumference is close to the length of the year in inches.


Victor

On Wed, Dec 27, 2017 at 2:33 AM, Amos Shapir <[hidden email]> wrote:
Hi Irv and calendar people,

This is just a coincidence that the ratio of the length of the mean tropical year y and the mean lunar month m, then y/m-12 is close to 1/e.

If it were exactly 1/e, for the current length of the mean lunar month, the tropical year would have to be 365.23076 days -- way too short for any tropical year.
(Incidentally, it's very close to 3 leap days in 13 years).

FWIW, since the mean lunar month is increasing and the mean tropical year is decreasing, this ratio will be reached some time in the future (I'd leave it as an exercise to the reader to find out exactly when).

Have fun,

Virus-free. www.avg.com

On Tue, Dec 26, 2017 at 7:11 PM, Irv Bromberg <[hidden email]> wrote:
Dear Kalendarists:

The traditional Hebrew calendar has 7 leap years in 19 solar years. The fraction 19/7 is a pretty good approximation to e, accurate to within an error of less than 0.004.

My Rectified Hebrew calendar has 130 leap years in 353 solar years. The fraction 353/130 is a better approximation to e, accurate to within an error of less than 0.003.

What is the significance of this good approximation?
(This isn't a puzzle, I'm just asking for ideas.)

-- Irv Bromberg, University of Toronto, Canada



--
Amos Shapir
 

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Re: e and lunisolar ratio

Karl Palmen

Dear Victor and Calendar People

 

If the diameter were 29.53059 dm, the circumference would be (Pi/0.254)*29.53059 = 365.24836… inches.

 

I look at the idea of a leap year rule with leap years spaced as smoothly as possible for a year whose fractional part is 1/e:

1/e = 0.36787944117144232159552377016147…

7/19 = 0.368421….

4/11 = 0.363636…

So it can be approximated by a number of Metonic cycles one of which is truncated to 11 years, similar to the 334-year cycle of 123 leap months, but considerably shorter.

39/106 = 0.3679245…

32/87 = 0.3678161…

These approximations have structural complexity 4, equal to the complexity of the 334-year cycle of 123 leap months.

Mixing them together equally gives:

71/193 = 0.3678756…

This is made up of 11 Metonic cycles 2 of which are truncated and has structural complexity of 5, equal to the complexity of the 1040-year cycle of 383 leap months.

 

Karl

 

16(17(16

 

From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Victor Engel
Sent: 27 December 2017 19:10
To: [hidden email]
Subject: Re: e and lunisolar ratio

 

This coincidence reminds me of the one I discovered some time ago relating imperial and metric systems to solar and lunar cycles. If you take a circle and measure it's diameter in dm so that the diameter in dm matches the length of a lunation in days, then the circumference is close to the length of the year in inches.

 

 

Victor

 

On Wed, Dec 27, 2017 at 2:33 AM, Amos Shapir <[hidden email]> wrote:

Hi Irv and calendar people,

This is just a coincidence that the ratio of the length of the mean tropical year y and the mean lunar month m, then y/m-12 is close to 1/e.

If it were exactly 1/e, for the current length of the mean lunar month, the tropical year would have to be 365.23076 days -- way too short for any tropical year.

(Incidentally, it's very close to 3 leap days in 13 years).

FWIW, since the mean lunar month is increasing and the mean tropical year is decreasing, this ratio will be reached some time in the future (I'd leave it as an exercise to the reader to find out exactly when).

Have fun,