Mathematics and the Gregorian Calendar - …danf/talks/Calendar.pdfMathematics and the Gregorian Calendar Daniel Frohardt Department of Mathematics Wayne State University February

Mathematicsand the

GregorianCalendar

DanielFrohardt

Department ofMathematicsWayne State

University

CalendarCalculations

The GregorianCalendar

Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Mathematics and the Gregorian Calendar

Daniel FrohardtDepartment of Mathematics

Wayne State University

February 21, 2012

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Give Us Our Eleven Days

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Doomsday Rule

Doomsday is a function from Years to Days of the week.

The Doomsday assigned to a given year is the day of the weekon which the last day of February falls in that year.

For example, the Doomsday for 2012 is Wednesday.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Doomsday Rule

Doomsday is a function from Years to Days of the week.

The Doomsday assigned to a given year is the day of the weekon which the last day of February falls in that year.

For example, the Doomsday for 2012 is Wednesday.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Finding a day for dates this month

If we have identified a Doomsday in a given month we caneasily calculate the day of the week for any date in that month.We know that February 29 is a Wednesday, so the 22nd, 15th,8th, and 1st days of the month are also Wednesdays.If we want to find the day of the month for Valentine’s Day,February 14th, 2012, we note that it is the day before aDoomsday, hence a Tuesday.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Learning the Doomsday Method

We need to identify a Doomsday within each month so that wecan reckon the day of the week for any day in that month bycomparison. To do this smoothly it is best to learn *all* of theDoomsdays in a year. This will come naturally with practice ifyou follow the procedure of reckoning backwards and forwardsusing Doomsdays until you are within a week of the day you aregiven to work with.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Finding a Doomsday in each month (or nearly so)

1 Jan 5 May 9 Sep2 Feb 6 Jun 10 Oct3 Mar 7 Jul 11 Nov4 Apr 8 Aug 12 Dec

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks


1 Jan 5 May 9 Sep2 Feb 28 [29*] 6 Jun 6 10 Oct 103 Mar 7 Jul 11 Nov4 Apr 4 8 Aug 8 12 Dec 12

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks


1 Jan 31 [32*] 5 May 9 Sep2 Feb 28 [29*] 6 Jun 6 10 Oct 103 Mar 7 7 Jul 11 Nov4 Apr 4 8 Aug 8 12 Dec 12

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A special, ad hoc mnemonic

My Nine to Five job at the Seven Eleven

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks


1 Jan 31 [32*] 5 May 9 9 Sep 52 Feb 28 [29*] 6 Jun 6 10 Oct 103 Mar 7 7 Jul 11 11 Nov 74 Apr 4 8 Aug 8 12 Dec 12

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Moving to different years

For dates between 2000 and 2099 it is convenient to start fromthe Doomsday for that century. This is defined to be theDoomsday for the year 2000.

Fact to memorize: February 29, 2000 was on Tuesday.

Conway’s mnemonic: Y2K — “TwosDay”

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Moving to different years

For dates between 2000 and 2099 it is convenient to start fromthe Doomsday for that century. This is defined to be theDoomsday for the year 2000.

Fact to memorize: February 29, 2000 was on Tuesday.Conway’s mnemonic: Y2K — “TwosDay”

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

July 4, 2026

The two-hundredth anniversary of the signing of theDeclaration of Independence will be on the Doomsday that year(as will every July 4th, because 7/11 is a Doomsday and 11-7=4).To find the Doomsday for 2026, we write 26 as a multiple of 12plus a remainder

26 = 2 ∗ 12 + 2

Each set of twelve consecutive years, 2001–2012 and2013–2024, contains exactly three leap years, so the Doomsdaywill move up a net of one day during each of those intervals,and another in each of the years 2025 and 2026. TheDoomsday for 2026 is therefore 2 + 2 days after Tuesday, orSaturday.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

December 7, 2041

December 7 is two days after the Doomsday, December 5 (5 =12 - 7).To find the Doomsday for 2041, we write 41 as a multiple of 12plus a remainder

41 = 3 ∗ 12 + 5

The 5 additional days contain [5/4] = 2 leap years, so theDoomsday for 2045 is 3 + 5 + 1, congruent to 2 mod 7, daysafter Tuesday: Doomsday(2041) = Thursday.December 7, 2041 thus falls 2 days after Thursday, which is aSaturday.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Other Centuries

During the century from March 1, 1900 to February 29, 2000there were 25 leap days. So the Doomsday advanced100 + 25 = 125 days between 1900 and 2000. Since 125 is 1less than a multiple of 7, this implies that the Doomsday(1900)= Wednesday, one day later than Doomsday(2000).

Going back to 1800, we note that there were only 24 leap daysbetween March 1, 1800 and February 28, 1900, the last day ofFebruary in that year. So Doomsday(1800) = Friday, two dayslater than Doomsday(1900).

Similar reasoning shows that Doomsday(2100) = Sunday, twodays earlier than Doomsday(2000).

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Seasonal examples

March 20, 2012 = 20 on Wednesday = 1 from WednesdayTUESDAY

June 20, 2012 = 14 on WednesdayWEDNESDAY

September 22, 2012 = 17 on Wednesday = 3 on WednesdaySATURDAY

December 21, 2012 = 9 on Wednesday = 2 on WednesdayFRIDAY

March 20, 2013 = 20 on Thursday = 1 from ThursdayWEDNESDAY

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

The seasons have different lengths

Spring starts on Tuesday and ends on Wednesday.Spring is 7 ∗ 13 + 1 = 92 days long.

Summer starts on Wednesday and ends on Saturday.Summer is 7 ∗ 13 + 3 = 94 days long.

Autumn starts on Saturday and ends on Friday.Autumn is 7 ∗ 13− 1 = 90 days long.

Winter starts on Wednesday and ends on Saturday.Winter is 7 ∗ 13− 2 = 89 days long.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A question

What causes the seasons to have different lengths?

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

How eccentric is the Earth’s orbit?

The distance from the Earth to the Sun varies between

93− 1.5 million miles and 93 + 1.5 million miles.

So when the Earth is closest to the Sun it is a little more than1.5% closer than its average distance.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

How eccentric is the Earth’s orbit?

The distance from the Earth to the Sun varies between

93− 1.5 million miles and 93 + 1.5 million miles.

So when the Earth is closest to the Sun it is a little more than1.5% closer than its average distance.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Kepler’s second law of planetary motion (1609)

The line segment joining the Sun to an orbiting planet sweepsout equal areas during equal intervals of time.

2/21/12 7:15 AM

Page 1 of 1file:///Users/daniel-frohardt/Kepler_laws_diagram.svg

f2f2

f3f3

a1a1

a2a2

f1 (sun)f1 (sun)

A2A2A1A1

planet 2planet 2 planet 1planet 1

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Needed facts

The Earth is closest to the Sun in early Winter, typicallyaround January 3.

The area of a sector of a circle of radius r subtending anangle θ is 1

2 r2θ.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

“Back of the envelope” consequence

rW = Winter distance to Sun

rS = Summer distance to Sun

θW = Daily angular progress in Winter

θS = Daily angular progress in Summer

r2W θW = r2

SθS , rW ≈ (1− .03) ∗ rS

=⇒ θW ≈ (1 + 2 ∗ .03) ∗ θS

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Explanation for variation in lengths of seasons

The Earth is moving about 6% faster early in the winter thanearly in the summer, so it moves through that part of its orbitmore quickly, making winter shorter. The difference in thelengths of the seasons is a bit less, because the Earth slowsdown in later Winter and speeds up in late Summer.

The table below shows that Summer is slightly more than 5%longer than Winter.

Summer 93.65 days Autumn 89.85 daysWinter 88.99 days Spring 92.75 days

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Explanation for variation in lengths of seasons

The Earth is moving about 6% faster early in the winter thanearly in the summer, so it moves through that part of its orbitmore quickly, making winter shorter. The difference in thelengths of the seasons is a bit less, because the Earth slowsdown in later Winter and speeds up in late Summer.The table below shows that Summer is slightly more than 5%longer than Winter.

Summer 93.65 days Autumn 89.85 daysWinter 88.99 days Spring 92.75 days

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Types of day

A solar day is the length of time it takes for the Sun to“return” the same longitude.

A sidereal day is the length of time for the “fixed stars” toreturn to their locations.

The standard 24 hour day is actually the mean solar day, thelength of a solar day averaged over the course of a year.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Another consequence of the Earth’s eccentricity

In the Winter, solar days are about 7 seconds longer than 24hours.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Explanation

Because of the way the Earth revolves around the Sun, asidereal day is shorter than a solar day.

It takes approximately 23 hours and 56 minutes for the Earthto spin around relative to bodies outside the solar system.

The remaining 4 minutes per day (=24 hours/365+ days) isthe length of time it takes for the Earth to spin enough tomake up for its progress in solar orbit.

During the Winter, when the Earth is moving 3% more rapidlyin its orbit, it will take an extra .12 minutes (about 7 seconds)of spinning to complete the solar day.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

High Noon

High Noon is the timewhen the sun is at itszenith, midway betweensunrise and sunset.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

High Noon


Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

High Noon


Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Drift in high noon

The cumulative effect of all those 24 hour plus 7 seconds daysin the winter will be to push high noon later in the day. We’lluse a basic property of the sine curve to estimate this drift.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A general property of sinusoidal curves

Peak amplitude = Maximum slope * Period / 2π

��

��

��

��

��

��

Period/4

π/2∗Peak amplitude

..................

..................

..................

..................

....................

....................

......................

.......................

...........................

................................

...............................................

...............................................................................................................................................................................................................................................................................................................................................................................

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

The effect of orbital eccentricity on high noon

The maximum daily change in the length of a solar day isabout 7 seconds per day. Using the formula from the previousslide, we would expect a maximum offset of about 7 ∗ 365/(2π)seconds, which is about 7.5 minutes.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Today’s sunrise and sunset times

Sunrise: 7:21 AMSunset: 6:12 PM

High noon today is around 12:46

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Today’s sunrise and sunset times

Sunrise: 7:21 AMSunset: 6:12 PM

High noon today is around 12:46

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A comparison to think about

Time of sunrise, February 21: 7:21 AMTime of sunset, February 21: 6:12 PMTime of sunrise, November 13: 7:21 AMTime of sunset, November 13: 5:12 PM

The sun will rise at the same time on November 13 that it didtoday, but the sunset will be an hour earlier. This means thathigh noon will arrive half an hour earlier in the day then.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A comparison to think about

Time of sunrise, February 21: 7:21 AMTime of sunset, February 21: 6:12 PMTime of sunrise, November 13: 7:21 AMTime of sunset, November 13: 5:12 PM

The sun will rise at the same time on November 13 that it didtoday, but the sunset will be an hour earlier. This means thathigh noon will arrive half an hour earlier in the day then.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Uh-oh

The previous slide shows that the variation in the time of highnoon over the course of the year is at least half an hour.

The eccentricity of the Earth’s orbit accounts for only about 15(twice 7.5) minutes of drift.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Eastern Standard Times of sunrise, sunset, solarnoon, Detroit, 2012

5pm6pm7pm8pm

noon1pm

5am6am7am8am ................................................................................................................................................................................................................

.................................................

..............................................

................................................

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............................................

............................................

...............................................

.......................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Drift of high noon, 2012

..................................

.............................................................................................................................................................................................................................................................................................

...............................................

12:30

12:45

12:15

x

Feb 11

x

May 15

x

Jul 23

x

Oct 31

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

The Equation of Time

2/21/12 8:43 AM

Page 1 of 1http://upload.wikimedia.org/wikipedia/commons/7/7a/Equation_of_time.svg

-18

-12

-6

6

12

18

90 180 270 360

Min

utes

Day of year

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Axial tilt

The Earth is tilted approximately 23.5 degrees from the planeof the ecliptic. This means that the path of the Sun over theEarth traces out a great circle in a plane that makes a 23.5degree angle with the plane of the equator.The Sun moves at a constant speed along this great circle, butit does not change longitude at a constant rate of speed.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Arc length on a great circle

Let φ denote latitude from the equator and θ be longitude.Using 1 for the Earth’s radius, arclength ds on the Earth’ssurface satisfies

ds2 = dφ2 + cos2 φ dθ2

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Minimum and maximum values of ds/dθ

Let ε be the axial tilt of the Earth. (ε is about 23.5◦.)

The minimum value of ds is cos ε dθ, which occurs at thesolstices when the Sun is angle ε from the equator.

The maximum value of ds is (tan2 ε+ 1)1/2 dθ = sec ε dθ.This occurs at the equinoxes when the Sun is over the equator.

This shows that

cos ε ≤ ds

dθ≤ sec ε.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks





This shows that

cos ε ≤ ds

dθ≤ sec ε.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks





This shows that

cos ε ≤ ds

dθ≤ sec ε.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks





This shows that

cos ε ≤ ds

dθ≤ sec ε.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

The effect of axial tilt on solar noon

The East-West drift in the Sun’s position is the difference

betweends

dtand 1.

cos ε ≈ .92 and sec ε ≈ 1.09.

−.08 ≤ ds

dθ− 1 ≤ .09.

So the maximum daily drift in high noon due to the axial tilt ofthe Earth is about 8-9% of 4 minutes, approximately 20seconds.

Using the sine curve formula for estimating the maximumcumulative drift, and noting that the axial tilt has period onlyhalf a year, we estimate the maximum drift in solar noon beroughly 20 ∗ 183/(2π) seconds, which is about 10 minutes.

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Components of the Equation of Time

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Some Calendar Questions

1 Has the same date (MM/DD/YYYY) ever fallen on 2different days of the week?

2 On what day of the week is the 13th day of the monthmost likely to fall in the long run under the Gregoriancalenar?

3 Is there a date in Microsoft Excel c© that does not exist?

4 Has there ever been a February 30?

5 Have two consecutive days ever have fallen on the sameday of the week?

6 How many different month lengths have there been in thepast few years?

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

A related question

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Sundial, somewhere in Minnesota

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Te Anau, New Zealand

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Mathematicsand the

GregorianCalendar

DanielFrohardt


University



Doomsday Rule

Lengths ofSeasons

Solar days

Eccentricinfluences

Tilting influences

Concludingremarks

Singapore

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Mathematics and the Gregorian Calendar - …danf/talks/Calendar.pdfMathematics and the Gregorian Calendar Daniel Frohardt Department of Mathematics Wayne State University February