Omer Counting in Different Bases

My father-in-law showed me a ספר that discusses whether or not you can fulfill your obligation to count the עומר using other base systems besides decimal. This a good case where the question is far more interesting than the answer. Surely, one should not do that. However, it was a very interesting concept I had never thought of before. So, I added a widget on the blog's sidebar which will display the day of the Omer in various relevant bases.

On the insistence of reader Pi (that's his shorter name), I have added base 7 as well as a pick-your-own-base section.

Here is the excerpt from the ספר.

Happy π Day

We wish you all a happy Pi Day, today being March 14th which, in the US anyway, is expressed as 3-14. Pi day was first observed in the year 1593. Ok, I'm just making that up (and rounding.) Just to give this some semblance of a Torah flavour, here is our post on Pi in the Torah 

In European countries where the day is written before the month, Pi Day is observed on April 31. For information on that, you would have needed to contact me this past Sunday morning at 2:30 am.

והמבין יבין.

Here are 10 ways to celebrate Pi Day, including this young chap who memorized 2,552 digits (eat your heart out, Brodsky.)

10,000 Kikars

Someone just sent me a brilliant piece explaining a very difficult תוספות regarding the 10,000 ככר of silver which המן gave for the extermination of the Jews. Check it out!

טומאה under a beam

In the משנה יומית, we recently encountered a משנה I had long been awaiting. Just as a background, טומאה which is inside of a room will generally make anything else in the room טמא. In order for the טומאה to spread, however, it needs to be inside a space which is at least a cubic טפח. So, the משנה אהלות י"ב:ז discusses how big a cylindrical beam would have to be in order to be certain that there is a cubic טפח underneath it. The משנה gives the figure of a 24 טפח circumference. Let us explore the math behind this:

The key to figuring this out is actually simple geometric fact mentioned in the .גמרא סוכה ח

כל אמתא בריבועא אמתא ותרי חומשא באלכסונא

An isosceles, right-angle triangle whose two sides are a טפח will have a diagonal of 1.4 טפחים. Or, more simply, if you draw a line from one corner of a 1x1 טפח square to the other, the length will be 1.4 טפחים. This is actually a simple (and rounded) version of the Pythagorean Theorem. The square of the hypotenuse is equal to the sum of the squares of the other two sides. 1²+1²=2 so the hypotenuse should be √ 2 which is more like 1.414. Let's be as precise as we can for this.

Above you will find a rough sketch of the problem we are trying to solve. We are going to solve for the radius of the beam. So in order for the space to be a cubic טפח, the diagonal has to be √ 2 טפח. If we draw a line from the centre of the beam to the corner of the square which surrounds the beam we can form a triangle which I have highlighted in faint green. It is a right-angle triangle where the two sides are of length x and the hypotenuse would be x+√ 2. But using our rule from סוכה (and Pythagoras) we know that the length of the hypotenuse can also be expressed as √ 2x. Therefore, we have

√ 2x	= √ 2 + x
√ 2x - x	= √ 2
(√ 2 - 1)x	= √ 2
x	= √ 2 / (√ 2 - 1)
x	≈ 3.41
Double that to get the diameter and multiply by π to get the circumference
c	≈ 21.45 טפחים

Now we know that the תלמוד does not necessarily use 100% precise figures. The aforementioned גמרא in סוכה clearly uses a round figure of 3 for π. But how would the משנה have come to a figure of 24 for this calculation?

רע"ב explains: A circumference of 24 means a diameter of 8 (when using the talmudic π of 3) If the sides of the large square above are 8, the diagonal, from one corner of that square to the other, which includes the diameter and two cubes, would be 8 * 1 ²/₅. Basically, that's ¹⁶/₅ extra, ⁸/₅ diagonal on either side. But for a טפח x טפח cube, we were looking for a diagonal of just ⁷/_{5. רע"ב explains that משנה was not concerned with the minuscule margin of the extra ¹/5. The troubling issue is that if you go through all those calculations with a circumference of 21, a diameter of 7, you get exactly ⁷/5 on the dot! I saw explained according to one source that since a beam would likely be sunken into the ground slightly, the משנה did not want to give the exact measurement which would end up being overly stringent.}

To be continued...

Frequency of חזק's and סיומי דפ יומי

This coming שבת, the סיום דף יומי מסכת גיטין coincides with the חזק of finishing ספר שמות. See this fascinating article by R' Dovid Heber of the Star-K. In it, he shows that since the beginning of דף יומי there have in fact been the same number of סיומים as חזק's and goes through the numbers to show exactly why that is. עיין שם,

I decided to take this matter a little further and investigate how many times this has actually happened or will happen, assuming the exact structure of דף יומי remains the same. This is what I found, going back to the beginning in 1923 all the way to the year 2050:

Beitzah 40	1/16/2044	16 Teves	5804	Vayechi
Avodah Zarah 76	7/7/2040	26 Tammuz	5800	Matos Masei
Moed Katan 29	7/14/2029	2 Av	5789	Matos Masei
Gitin 90	3/12/2016	2 Adar II	5776	Pekudei
Shabbos 157	3/9/2013	27 Adar	5773	Vayakhel Pekudei
Kesubos 112	12/22/2007	13 Teves	5768	Vayechi
Yevamos 122	12/24/1977	14 Teves	5738	Vayechi
Temurah 34	7/7/1945	26 Tammuz	5705	Matos Masei

It is interesting to note that it does not occur on בחוקתי at all in this time, nor does it ever land on שמחת תורה. In fact, looking even further ahead, the only time it will ever land on שמחת תורה between now and the year 2200 will be the following on שמחת תורה in ארץ ישראל:

Horiyos 14

Thu Oct 26, 2062

22 Tishrei, 5823

For anyone curious as to how I was able to figure all this out, I wrote a very short and simple Java application and made use of the brilliant Zmanim API by KosherJava.

And if you're even more curious, here's my code:

import java.util.Calendar;

import java.util.Date;

import net.sourceforge.zmanim.hebrewcalendar.HebrewDateFormatter;

import net.sourceforge.zmanim.hebrewcalendar.JewishCalendar;

public class Daf {

      public static void main(String[] args) {

            int daf;

            HebrewDateFormatter hdf = new HebrewDateFormatter();

            Calendar day = Calendar.getInstance();

            day.set(Calendar.YEAR,2050);

            //System.out.println(day);

            boolean siyum=false;

            // While loop to expose all siyumim that happen on Shabbos. I have to go in manually afterwards to identify the chazaks.

            // I could have possibly used getParshaIndex() and played with that to expose just the chazaks but that would have been clunkier

            while (day.get(Calendar.YEAR) > 1922){

                  JewishCalendar jewCal = new JewishCalendar(day);           

                  try{

                      daf = jewCal.getDafYomiBavli().getDaf();

                  } catch (IllegalArgumentException iae){

                      break;

                  }

                  if (siyum){

                        if (day.get(Calendar.DAY_OF_WEEK) == Calendar.SATURDAY){

                       

                        System.out.println(jewCal.getDafYomiBavli().getMasechtaTransliterated() + " " +

                                    daf + ", " + day.getTime() + "," +  jewCal + "," + jewCal.getParshaIndex());

                        }

                        siyum=false;

                  }

                  if (daf == 2) siyum = true; //If today is the beginning of the masechta, yesterday was the siyum of the previous

                  day.add(Calendar.DAY_OF_YEAR, -1);

            }

            day.set(Calendar.YEAR,2200);

            //Separate loop to identify Simchas Torah since it isn't usually on Shabbos

            while (day.get(Calendar.YEAR) > 1922){

                  JewishCalendar jewCal = new JewishCalendar(day);           

                  try{

                      daf = jewCal.getDafYomiBavli().getDaf();

                  } catch (IllegalArgumentException iae){

                      break;

                  }

                  if (siyum){

                        // If the 22nd or 23rd day of Tishrei

                        if (jewCal.getJewishMonth() == 7 && (jewCal.getJewishDayOfMonth() == 22 || jewCal.getJewishDayOfMonth() == 23)){

                        System.out.println(jewCal.getDafYomiBavli().getMasechtaTransliterated() + " " +

                                    daf + ", " + day.getTime() + "," +  jewCal + "," + hdf.formatParsha(jewCal));

                        }

                        siyum=false;

                  }

                  if (daf == 2) siyum = true;

                  day.add(Calendar.DAY_OF_YEAR, -1);

            }           

      }

}

Pi in the Torah

This week, the הפטרה is read from מלכים א פרק ז. There is a difference of opinion as to which part of the פרק is read. The ספרדים and some אשכנזי shuls will read the part that contains a reference to the figure of π. פסוק כג states that the pool was 10 אמות in diameter and 30 אמות in circumference. The obvious difficulty is that π is more than 3. Even if the פסוק were to round to the nearest whole number, it would be 31.

There is a famous approach to this problem which is said over in the name of the גר"א although I am not sure there is any written evidence of it. The word used for the circumference is וקוה. However, it is read וקו. If you drop the leading ו which is simply a וי"ו החיבור, you are left with קו and קוה. The respective גמטריות of those words are 111 and 106. If you divide 111/106 and multiply that by the apparent figure of π used - 3 - then the result is 3.1415094! That is correct to 3 decimal places!

I once heard an interesting addendum to this. There needs to be some meaning behind the figure of 30 אמות. The thickness of the walls was 1/4 אמה (I'm not sure where we know that from.) Therefore, the inner diameter was 9 1/2 אמות. If you multiply that by π you come awfully close to 30. I thought heard this in the name of the מלבי"ם but I looked it up and he definitely did not say that.

BlackBerry® App

I am happy to announce that the Al Pi Cheshbon BlackBerry® App is now available, compatible with all devices, including the new BB10 devices.

Apps for other WeeklyShtikle blogs are available as well.

A Pythagorean Question

A math-oriented cross-post from the Daily Leaf. I guess we should be expecting more of those over the course of מסכת עירובין.

Jewish Pi Day

Today is Jewish Pi Day: 3rd month (Sivan), day 14. Shall we eat pi(e) at 1:59 shaot zmaniyot into the day?

Most Possible Seudos

As this year comes to a close, here is a great little factoid I heard from the Rav's Friday night דרשה:
    This year, being a leap year, and since no יום טוב ever fell out on שבת we have a unique situation where we have the maximum number of סעודות you could possibly have in one year:

54 שבתות x 3 סעודות	=	162 סעודות
12 ימים טובים x 2 סעודות	=	24 סעודות
	=	186 סעודות

This hasn't happened since 1984!