# The Symmetry454 Calendar

Created by Dr. Irv Bromberg, University of Toronto, Canada

Don’t be Anti-Symmetric!

# Call for Calendar Reform

The Gregorian calendar is deficient in the following ways:

• Each consecutive Gregorian year starts on a different weekday. This is the most serious deficiency of the Gregorian calendar. If it were impossible to correct this deficiency by adopting a perpetual (perennial) calendar reform then we wouldn’t care about the rest of this list. This deficiency causes the dates of holidays and events to shift. Those that fall on a fixed day number in a month fall on a different weekday. Those that fall on the nthoccurrence of a specified weekday in a month fall on a different day number. These shifts compel organizations such as governments, businesses, industries, and academic institutions to consume vast amounts of time, energy, and paper rescheduling annually recurring events just because of the changing weekday-date relationships. Even monthly recurring events are cumbersome to schedule. Easter can land on any date from March 22 through April 25, and numerous ecclesistical days counted before and after Easter wander likewise.
• The lengths of the Gregorian months follow an illogical, irregular pattern: 31, 28 or 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 days.
• When dividing the Gregorian calendar year into 4 groups of 3 months the resulting quarters have unequal day counts and unequal ratios of workdays to weekend days.
• Appending the leap day at the end of February causes unfortunate calendrical complexity, because it increments the ordinal day number of every date following February in leap years. This problem could have been avoided by appending the leap day to the end of the year.
• The Gregorian calendar mean year is currently almost 12 seconds too long relative to its intended target, the mean northward equinoctial year (also imprecisely known as the spring or vernal equinox year).
• Given a Gregorian date, moderately complicated arithmetic is required to determine the weekday.
• The weekday occurrences of the leapdays in each full 400-year Gregorian leap cycle are unequal:
Monday=15, Tuesday=13, Wednesday=15, Thursday=13, Friday=14, Saturday=14, and Sunday=13.
• Many people are superstitious about bad luck occurring on Friday the 13th.

A perpetual calendar reform is of prime importance, starting every year on the same weekday while conserving the traditional 7-day sabbatical cycle. The following recent precedents support conserving the sabbatical cycle as an absolute requirement:

• The French Republican calendar was implemented during the French Revolution until Napoléon abolished it just over 12 years later, and again for only 18 days in 1871. Its perpetual calendar year began at the autumn equinox and had twelve 30-day months with three 10-day weeks per month, plus 5 national holidays at the end of the year (6 in leap years). Religious objections to its disruption of the 7-day sabbatical cycle and workers’ objections to 10-day weeks ultimately led to its downfall.
• The 13 Month calendar was a proposed perpetual calendar reform having thirteen 28-day months plus an extra day at the end of the year outside the 7-day sabbatical cycle. A month called Sol was inserted between June and July. Its leap day was inserted before Sol and was also outside the sabbatical cycle. Every month started on Sunday, so every month had a Friday the 13th. The arithmetic for calendrical calculations was never published by any of the individuals or organizations who promoted it. The League of Nations rejected it in 1937 because of religious objections to its disruption of the 7-day sabbatical cycle, and because it wasn’t divisible into 4 equal quarters containing whole months.
• In 1929 USSR switched to a 5-day week with work schedules rotated to maintain continuous productivity, but that failed because family and friends had different days off work, and it was difficult to maintain equipment that was in continuous use. By 1932 they switched to a 6-day week with a common day off. In 1940 they reverted to the 7-day week, to increase productivity! (1 day off in 7 days is more productive than 1 in 6)
• The World calendar was a copyrighted proposed perpetual calendar reform having 4 equal quarters of 31+30+30 days each, plus Worldsday at the end of the year, outside the 7-day sabbatical cycle. In leap years Leapyear Day was inserted after the end of the second quarter and was also outside the sabbatical cycle. Every quarter started on Sunday, so the first month of each quarter contained a Friday the 13th. The arithmetic for calendrical calculations was never published by any of the individuals or organizations who promoted it. The USA vetoed it at the United Nations in 1955 because of religious objections to its disruption of the 7-day sabbatical cycle.
• The International Organization for Standardization formally specified the 7-day week as the international standard in ISO 8601.
Any calendar reform proposal must be openly free for use (no copyright) and must also include royalty-free open-source public domain documentation of arithmetic for calendrical calculations:
• to encourage standardized error-free implementation in computer systems and electronic devices
• to assist with conversions of pre-reform date records.

Minimal change isn’t important for calendar reform. Although it’s possible to make the Gregorian calendar perpetual without correcting any other deficiencies, if any change is to be made then we may as well correct every deficiency, to obtain an optimal outcome and enjoy all the benefits.

The Symmetry454 or its sister Symmetry010 calendar reform will solve all of the Gregorian calendar deficiencies.

The Symmetry454 calendar is a simple perpetual solar calendar that fully conserves the traditional 7-day week (by using a leap week instead of a leap day), has symmetrical equal quarters each having 4+5+4 weeks, and starts every month on Monday. The Symmetry454 calendar arithmetic is openly documented, royalty-free.

# Overview of the Symmetry454 Calendar

The Symmetry454 calendar is a simple perpetual solar calendar that conserves the traditional 7-day week, has symmetrical equal quarters each having 4+5+4 weeks, and starts every month on Monday.

Holidays and special days, indicated with background yellow shading, are permanently fixed.
Hover over the day number to see the pop-up event description.
All Easter-related ecclesiastical calendar days are shown on the fixed dates that they would have if the proposed perpetually fixed Easter date of Sunday April 7thwere adopted (based on the median astronomical Easter date, and the fact that Julian Sunday April 9, 30 AD was Symmetry454 Sunday April 7, 30 AD).

### First Quarter:

January February March
week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun
1 1 2 3 4 5 6 7 5 1 2 3 4 5 6 7 10 1 2 3 4 5 6 7
2 8 9 10 11 12 13 14 6 8 9 10 11 12 13 14 11 8 9 10 11 12 13 14
3 15 16 17 18 19 20 21 7 15 16 17 18 19 20 21 12 15 16 17 18 19 20 21
4 22 23 24 25 26 27 28 8 22 23 24 25 26 27 28 13 22 23 24 25 26 27 28
9 29 30 31 32 33 34 35

### Second Quarter:

April May June
week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun
14 1 2 3 4 5 6 7 18 1 2 3 4 5 6 7 23 1 2 3 4 5 6 7
15 8 9 10 11 12 13 14 19 8 9 10 11 12 13 14 24 8 9 10 11 12 13 14
16 15 16 17 18 19 20 21 20 15 16 17 18 19 20 21 25 15 16 17 18 19 20 21
17 22 23 24 25 26 27 28 21 22 23 24 25 26 27 28 26 22 23 24 25 26 27 28
22 29 30 31 32 33 34 35

### Third Quarter:

July August September
week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun
27 1 2 3 4 5 6 7 31 1 2 3 4 5 6 7 36 1 2 3 4 5 6 7
28 8 9 10 11 12 13 14 32 8 9 10 11 12 13 14 37 8 9 10 11 12 13 14
29 15 16 17 18 19 20 21 33 15 16 17 18 19 20 21 38 15 16 17 18 19 20 21
30 22 23 24 25 26 27 28 34 22 23 24 25 26 27 28 39 22 23 24 25 26 27 28
35 29 30 31 32 33 34 35

### Fourth Quarter:

October November December
week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun week Mon Tue Wed Thu Fri Sat Sun
40 1 2 3 4 5 6 7 44 1 2 3 4 5 6 7 49 1 2 3 4 5 6 7
41 8 9 10 11 12 13 14 45 8 9 10 11 12 13 14 50 8 9 10 11 12 13 14
42 15 16 17 18 19 20 21 46 15 16 17 18 19 20 21 51 15 16 17 18 19 20 21
43 22 23 24 25 26 27 28 47 22 23 24 25 26 27 28 52 22 23 24 25 26 27 28
48 29 30 31 32 33 34 35 53 29 30 31 32 33 34 35

# Epoch

Provided that years always start on Monday, the Symmetry454 calendar shares the same epoch as the Gregorian calendar, starting on Monday, January 1, 1 AD. This was also the same epoch as that of the Symmetry010 calendar, the ISO calendar, and the Revised Julian calendar. The Symmetry454 year number before the epoch was year zero, to be consistent with the astronomical convention.

# Leap Rule

Non-leap (common) years have 52 weeks = 364 days. Leap years, which occur at intervals of 6 or 5 years, have a leap week appended to December, shown above as week 53 of the calendar year = 371 days.

The Symmetry454 calendar employs a superior symmetrical smoothly-spread leap rule that ensures excellent long-term astronomical accuracy:

The simple fixed arithmetic 52/293 leap rule has 52 leap years that are automatically and inherently symmetrically spread as smoothly as possible within each repeating cycle of 293 years:

It is a leap year only if the remainder of ( 52 × Year + 146 ) / 293 is less than 52

(If a leap year remainder is less than 33 then the next leap year will be 6 years later, otherwise 5 years later.)

Click here to see the leap year list.

With this simple single-step leap rule, leap year intervals occur in groups of either 6 + 6 + 5 = 17 years or 6 + 5 = 11 years,
which symmetrically group into sub-cycles of 17 + 11 + 17 = 45 years or sub-cycles of 17 + 17 + 11 + 17 + 17 = 79 years.
In each full calendar cycle these sub-cycles inherently occur symmetrically in the sequence 45 + 79 + 45 + 79 + 45 = 293 years.

With 52 leap weeks in the cycle, and 52 weeks in a regular year, the fixed cycle length equals exactly 294 regular years, and the average interval between leap weeks is exactly 294 weeks.

The Symmetry454 calendar mean year ≡ 365+71/293 days ≡ 365 days 5 hours 48 minutes 56+152/293 seconds. This is intentionally slightly shorter than the present era mean northward equinoctial year of about 365 days 5 hours 49 minutes 0 seconds, ensuring essentially drift-free performance for more than 4 future millennia.

Due to the symmetrical arrangement of leap years, the timing of the mean northward equinox moment always falls at the cycle average in the first year of every 293-year cycle. This feature simplifies astronomical performance evaluations. Click here for more information about symmetrical leap cycles.

Using this leap cycle as described, in the present era the mean northward equinox lands near midnight at the start of Symmetry454 date March 17th(reference meridian at Jerusalem, Israel).

The Solar Calendar Leap Rules web page at <http://individual.utoronto.ca/kalendis/leap/> explains why the 52/293 leap rule is preferred and compares it with a wide range of alternative leap rules.

# Tweet

Within the original 140-character limit of Twitter.com, the following Tweet defines the rules of the Symmetry454 calendar:

Common year 52 weeks = 4+5+4 weeks/quarter; append 7 days if remainder of (52×Year+146)/293 < 52; months start Monday; epoch Jan 1, 1 AD

# Benefits of the Symmetry454 Calendar

• The Symmetry454 calendar is perpetual — a permanent copy can be reused every year.
• It conserves the 7-day week.  There are no intercalated or null or leap days outside of the traditional 7-day weekly cycle.
• Its symmetrical structure paves the way to simpler, aesthetically pleasing calendar designs.
• The symmetrical 52/293 leap rule has excellent long-term calendar drift performance relative to the astronomical mean northward equinox.
• Every Symmetry454 year, quarter, month and week starts on Monday and ends on Sunday.
• Every day number within each Symmetry454 month is always on the same weekday in every month.
• Weekday = DayInMonth MOD 7, where Sunday=0, Monday=1, Tuesday=2, etc. No need for Zeller’s Congruence here!
• Monthly meetings on a fixed weekday are always on the same day number in every month, simplifying scheduling, for example the 3rd Thursday is always the 18th day of every month.
• Its symmetrical 13-week quarters are always identical, having the same number of weekdays and weekend days.
• In leap years the 4thquarter has 14 weeks, but this occurs only once per 24 or 20 quarters.
• Every date has permanently fixed week-in-year and day-in-year ordinal numbers, facilitating administrative, academic, commercial and industrial applications, and simplifying calendar arithmetic.
• There is always a whole number of weeks in every year (non-leap year = 52 weeks, leap year = 53 weeks), in every quarter (13 weeks), and in every month (short month = 4 weeks, long month = 5 weeks).
• Every secular holiday, event, anniversary, birthday, and memorial day has a permanently fixed weekday and date, because the calendar is perpetual.
• Holiday and/or special day overlaps are less likely to occur and are easy to predict and avoid.
• Sunday, April 7th is proposed as a perpetually fixed Symmetry454 date for Easter, based on the median date of the Sunday after the day of the astronomical lunar opposition that is on or after the day of the astronomical northward equinox, calculated for the meridian of Jerusalem.
• The first 4 weeks of every Symmetry454 month are identical.
• The coherent structure of the calendar enables simple arithmetic expressions in calculating the following for statistical or business purposes: weekday; day number of year, quarter or month; week number of year, quarter or month; month number of year or quarter.
• Symmetry454 calendar and its arithmetic is in the public domain, allowing royalty-free computer implementation and its free use without copyright restrictions.
• The freeware Kalendis computer program demonstrates the calendar and inter-converts dates, and is freely available at <http://individual.utoronto.ca/kalendis/kalendis.htm>.
• Friday the 13th never happens.

# Symmetry454 Calendar Documentation

The following documentation is in Adobe Acrobat Portable Document Format (PDF).

All Symmetry454 and Symmetry010 calendar layouts show the calendar for this year and every year, because every year is the same!

Click here to access quick reference manual date conversion sheets.

Come back for updates -- check the version numbers and revision dates, as well as the web site news.

Short File Name and PDF Size Description (see also the Kalendis freeware program)
Symmetry454 Summary 426 KB Read This First!

Introduction to the Symmetry454 Calendar — the Executive Summary:
Overview, 3 by 4 month layout, quad stack design, and list of benefits.

Symmetry010 Summary 428 KB For those who feel that the 4+5+4 weeks per quarter of the Symmetry454 Calendar is too radical, the Symmetry010 variant has a more uniform 30+31+30 days per quarter structure.

Introduction to the Symmetry010 Calendar — the Executive Summary:
Overview, 3 by 4 month layout, quad puzzle design, and list of benefits.