The **Triangular Earth Calendar**, abbreviated as **TEC**, is a proposed new calendar. Its main features are:

- Each year corresponds exactly to a year in the Gregorian calendar.
- Within a year, the calendar uses multiples of 2 or 3 to group the days.
- One can draw the calendar as a
*triangle*.

The calendar divides each year into 10 months. Each month has 6 weeks; each week has 6 days. This provides a total of 360 days. To match the Gregorian calendar, which approximates the tropical year of the Earth, there is an extra week at the end of the year. This week contains only 5 days for most years, but it contains 6 days in a leap year. TEC shares its leap years with the Gregorian calendar.

TEC also provides a *decimal* mechanism to specify the time.

## Contents

## Identifying days[]

Week | ||||||
---|---|---|---|---|---|---|

1 | 00 | 01 | 02 | 03 | 04 | 05 |

2 | 06 | 07 | 08 | 09 | 10 | 11 |

3 | 12 | 13 | 14 | 15 | 16 | 17 |

4 | 18 | 19 | 20 | 21 | 22 | 23 |

5 | 24 | 25 | 26 | 27 | 28 | 29 |

6 | 30 | 31 | 32 | 33 | 34 | 35 |

Week | ||||||
---|---|---|---|---|---|---|

1 | 00 | 01 | 02 | 03 | 04 | 05 |

Shown above is the calendar for each year. Months and days have numbers starting from zero. Thus the first day of the year is `0.0:`, the second is `0.1:`, and the last day of the first month is `0.35:`. Day 20 of month 6 (the seventh month) is `6.20:`. (The dot separates month from day while the colon is always at the end.)

The use of the 0 digit is intentional so that TEC dates are representative of time elapsed: `6.20:` is 20 days after `6.0`, which is 6 months after `0.0:`. In contrast, 20 June in the Gregorian calendar is only 19 days after 1 June, because there is no 0 June. Further, 1 June (`1/6` or `6/1`) is only five months after 1 January (`1/1`), the first day of the year.

### Month A[]

The final week of the year is designated by the letter A, but may also be designated 10. The intention is to allow the use of a base 36 numbering system for abbreviation where necessary. This means that days can be designated 0 through 9 and then A through Z (in lieu of 10 through 35), where such abbreviation is necessary. For common use, the intention is to use 10 through 35 for the days.

### Years[]

Year zero corresponds to Gregorian year 2001. This causes the TEC years 0 to 999 to match the same millenium as the third millenium of the Gregorian calendar. That millenium is 2001 to 3000, because there is no year 0 in the Gregorian calendar.

Thus TEC `0.0.0:` refers to the same day as Gregorian 1 January 2001. TEC `3.6.35:` refers to year 3, month 6 (the seventh month), day 35 (the last day), which is sometime in Gregorian year 2004.

To convert years between the two systems, add 2001 to the TEC year to obtain the Gregorian year.

TEC | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

Gregorian | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 |

### Time[]

Though TEC uses multiples of 6 for weeks and months, it uses a decimal system to represent time. This system is called the *TEC Decimal Day*.

Midnight is `:00000` in TEC. Noon is `:50000`. One can write in short form `:0` for midnight and `:5` for noon.

The time zone is not specified unless you use the amphora **@** to indicate Date Prime. This refers to the time at the western edge of the International Date Line, which is the same as UTC +14. For example, `@:50000` is noon in Date Prime. In the Americas, local noon will occur earlier than noon in Date Prime, while in Japan, which is much farther east, local noon will occur more than one-half day before it does in Date Prime.

## Writing date and time[]

The format used to represent a date is denoted with four symbols. For local dates/times, the symbol used to separate individual date denotations is the period "**.**", and the symbol used to separate the date denotations and time is the colon "**:**". All dates in TEC must use the colon.

An example of this would be in Year.Month.Day:Time format; `0.0.0:00000` refers to year 0, month 0, day 0, time `:00000`, which is the same as midnight at the start of 1 January 2001.

The other two symbols are the amphora "**@**", and the comma "**,**". A leading amphora implies Date Prime, which is the same date/time as the western edge of the International Date Line, or UTC +14. The comma refers to complex dates for scientific and computer use, such as every first day of the week; see examples below.

Date and time | Written form for TEC |
---|---|

Noon | :50000 |

Day 15 | 15: |

Month 4, day 7 | 4.7: |

Month A, day 2 | A.2: |

Year 5, month 0, day 24 | 5.0.24: |

Year 11, month 7, day 16, Date Prime | @11.7.16: |

Day 32 at Noon | 32:5 or 32:50000 |

Day 4 of every week (complex date) | 4,10,16,22,28,34: |

In year 3, every third month starting with 0, day 1 of every other week starting with the first week, at midnight and noon | 3.0,3,6,9.1,13,25:0,5 |

Dates before 0.0.0:00000 are currently represented as negative numbers, though a per millennia format might be under study.

## Unique Properties[]

TEC has many unique properties. It breaks down into many mathematical models. One week can be divided into whole days by 2, 3, and 6. They can be divided by 4 as well, with half days. One month can similarly be divided into weeks. One month can be divided into whole days by 2, 3, 4, 6, 9, 12, and 18. The 60 week year, not counting the last week, can be divided evenly by 2, 3, 4, 6, 12, 15, and 30.

Triangular Earth Calendar is named so because of its triangular properties. Unlike most calendars which are viewed only on a square grid, TEC is viewed in its most natural form, the triangle. The following are representations of TEC in triangular form:

Here is the view of a single week.

1 2 3 4 5 6

That is the same for weeks in a month. Here is the view of days in a month (using single numbers only).

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6

Here is the same view, but with days and weeks in a month. Each number is a day, each small triangle is a week, the entire is a month.

1 2 3 4 5 6 1 1 2 3 2 3 4 5 6 4 5 6 1 1 1 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6

Here is the view of the Triangular Earth Calendar by the months. Each number is a day, the final week, plus leap day, is in the blank areas.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 2 2 1 3 3 2 2 2 3 3 3 2 2 2 2 3 3 3 3 2 2 2 2 2 3 3 3 3 3 2 2 2 2 2 2 3 3 3 3 3 3 2 2 2 2 2 2 2 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 5 6 4 4 2 5 5 3 6 6 4 4 4 5 5 5 6 6 6 4 4 4 4 5 5 5 5 6 6 6 6 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 8 9 0 7 7 4 8 8 6 9 9 5 0 0 7 7 7 8 8 8 9 9 9 0 0 0 7 7 7 7 8 8 8 8 9 9 9 9 0 0 0 0 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 0 0 0 0 0 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 0 0 0 0 0 0 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9 0 0 0 0 0 0 0 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 0 0 0 0 0 0 0 0

Here is a total view, each number a day, each smallest triangle a week, the next largest a month, the entire a year, the final week and leap day are displayed as previously.

1 2 3 4 5 6 1 1 2 3 2 3 4 5 6 4 5 6 1 1 1 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 1 1 2 3 1 2 3 4 5 6 4 5 6 1 1 1 1 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 2 3 2 2 3 3 2 3 4 5 6 4 5 6 4 5 6 1 1 1 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 1 1 1 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 1 2 3 4 2 3 6 2 3 5 2 3 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 1 1 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6

## Equations of the TEC[]

As you can see,TEC is triangular, and expands somewhat fractally, extending to years forming larger decade sized triangles, and those decade sized triangles forming century sized triangles, and so on.

The mathematical properties of this calendar are then not based on using a base 6 number alone, but on the fact that it relies on Triangular Numbers and all of the properties of them. TEC at different levels also shares properties with squares, tetrahedrons, cubes, and is similar to Sierpinski's Triangle.

This occurs because TEC is based on Triangular Numbers. Triangular Numbers occur in the third downward diagonal row of Pascal's Triangle. Triangular numbers are formed by adding consecutive whole numbers. 1+2+3+4...+n = Triangular Number. The notation for this can be represented as n(!+). As such, TEC can be expressed in a pseudo-mathematical expression of triangular days multiplied by triangular weeks multiplied by triangular months plus the remainder of days on a normal or leap year. This looks like the following:

4(!+) * 3(!+) * 3(!+) + (5 || 6)

It can also be expressed without using weeks:

4(!+) * 8(!+) + (5 || 6)

## Creation of TEC[]

The following is a shortened account of how the Triangular Earth Calendar was formed.

First, discard what **cannot** be changed, unless the earth sped up around the sun fast enough to remove the 5.25, which is the extra days over 360. It’s actually 5.24… and further decimals, but for the purposes of understanding the calendar and its variables, we’ll simply refer to this number by using a variable, the Greek letter Omega **Ω**, meaning last, as we put it at the end of everything.

In this context we can state that:

1 year = 360 + Ω.

Most mathematicians agree that the 3-base system is the most beautiful. This system is called ternary. Binary = 0’s and 1’s, Ternary = 0’s, 1’s, and 2’s (or a “balanced” system is -1’s, 0’s, and 1’s).

However, it is not quite so easy for humans to learn. One might say that humans like most things in even numbers. So how do you get the benefits of 3 and 2 at the same time? If one assumes that the shared benefits between two things results in the Lowest Common Denominator of those two things, then one can find the benefits of a binary (2 number system) and ternary (3 number system) by multiplying them.

3 * 2 = 6

That’s easy enough. That was the basic idea that started Triangular Earth Calendar. But the author was unsure it would work. It seemed that you could make 6 days to a week and 6 weeks to a month, but it left a curious 10 month remainder. The factors of 10 are 2 and 5, so that it apparently has no properties of 3. But is there an advantage to 10, 6, 6, other than the fact that modern day culter works well with the number 10 (as opposed to the Babylonians who apparently used Base 60, for instance). To discover the number 10 and its relationship to 2, 3 and 6, the author returned to the properties about 6:

- 6 is divisible by 3 and by 2.
- All even numbers divisible by 3, are also divisible by 6.
- 6 has a honeycomb property with respect to geometric shapes.
- Hexagrams are made up of 6 triangles.

This leads to the idea of triangles. The triangle is the most basic 2 dimensional polygon, that can be combined with itself to make all other shapes that can **replicate** themselves.

- 3 equilateral triangles of the same size put together make a larger equilateral triangle. That equilateral triangle can thus replicate itself in the same fashion.
- 2 right triangles of the same size put together with their hypotenuses touching evenly will create a rectangle or a square, both of which can replicate themselves by combining them in groups of 4.
- 6 specifically angled isosceles triangles of specific dimensions together will form a hexagon (a 6-sided object of equal proportions). When combined, these form a
**honeycomb-like**shapes.

Hexagons **cannot** replicate larger versions of themselves, but they have what can be termed as **strong** properties like that of an equilateral triangle. In terms of this strength, the **strongest** structure is a unit circle. Spheres, which may be thought of in this context as the 3D equivalent of a circle, can equally distribute pressure and has extremely low friction. Even certain ovals carry some of these properties. This is why eggs are not cubes, but spherical-like shapes… very strong, no friction when depositing them. Circles are also the most **economic** shape.

How does the circle relate to the hexagon? A circle surrounded by circles of similar size create a hexagon-like shape. A hexagon could then be determined as the representation of the equal distances between the outermost points of the most compact form of 7 circles, which makes the hexagon the least-sided polygon to economically contain 7 circles. Triangles would be the least-sided polygon to economically contain a single circle. Hexagons and triangles, being related as previously stated, could be assumed to share some properties with circles, which is the least-sided 2D shape of all, if one does not consider a line or a fragment to be a shape. Humans have marvelled at the construction done by bees, which apparently know nothing of circles, but seem to have an instinctual knowledge of hexagons. From this example, it may seem that circles, triangles, and hexagons have a lot in common in terms of economy and strength.

In terms of a building construction, strength may be the most desired quality, with economy being a benefit. In terms of a calendar, economy is the only desired quality. In this regard, a base-6 calendar makes the most sense, and helps to confirm the validity of starting with a base-6 calendar. However, this did not answer the question of 10.

Looking back at the properties of 6, there is yet something else held in common with hexagons, triangles, and circles. Back to the Triangular Number Formula, we see that increasing n(!+) from 0 results in the number of sides of all of these objects.

1(!+) = 1 = Sides to a Circle 2(!+) = 1 + 2 = 3 = Sides to a Triangle 3(!+) = 1 + 2 + 3 = 6 = Sides to a Hexagon

If this notion is taken a step further, we see the following:

4(!+) = 1 + 2 + 3 + 4 = 10

By simple arithmetic, the author recognizes that 10 is a Triangular number. This is tested simply:

1 2 3 4 5 6 7 8 9 0

And so, the number 10, like 6, can form an equilateral triangle, which can in turn replicate itself. This was found to be of benefit for a full triangular year, but further that successive groups of 10 (decades, centuries, millennia) would replicate the basic year structure flawlessly and economically. It did not require any connection via factoring to the number 3 to enjoy the benefits of the 3-sided object, the triangle. The properties of 4(!+) are still under investigation with relation to other objects, including tetrahedrons.

The author concluded that Base 6 alone was not the basis of the TEC, but that it was based on Triangular Numbers. After creating the basic symmetrical structures that appear above, it was given the name Triangular Earth Calendar for its uncanny ability to form triangles at all levels and its triangle-like ability to economically represent the calendar, even in the cubic 6x6 form featured above, in the smallest container. Upon further study, the author found that being triangular afforded further benefits when dividing the calendar into ever smaller units by keeping its symmetry beyond the scope of the traditional units.

## Date Shift[]

Triangular Earth Calendar ends date shift. Date shift is the fact that the days of the week are not the same from year to year. TEC removes date shift, which has repercussions on notable dates. Birthdays, anniverseries, holidays, and other historical dates are on the same day every year. The Gregorian calendar not only causes date shift because of a non-terminating year week, but the fact that the leap day is placed near the beginning of the year. This causes all days after the leap day on a leap year to shift 1 day later. TEC keeps the 129th day of the year on the 129th day. The only date that changes is New Years Eve Day, because the leap day is placed at the very end of the year. The last week (pseudo-month) of the TEC calendar is a terminating week. This means that on non-leap years, when it does not create a full week, it does not place the first day of the year on the last day of the week. The terminating week is a representation of TEC in triangular form, where the placement of the ... is partially symmetrical, and is not part of a regular month/week structure. This is why the TEC promotes the use of the triangular structure over the cubic structure of the printed calendar, though both are acceptable for common use.

Further, it is seen that the special qualities of the terminating week are not outweighed by its benefits. An example of this is that to set the day on a traditional electronic calendar, it must perform complex operations on the year that the day of the year is on to determine which day of the week it is. Also, days of the month slip from month to month. This occurs to the Gregorian calendar because of several factors, including 7-day weeks, non-symmetrical rigid months, and an ill-placed leap day. Thus, a terminating week solves two of those problems by moving the leap day to the end of the calendar year and as a non-rigid month. The 6-day weeks solve the final problem by creating month-to-month symmetry.

## Why[]

The Triangular Earth Calendar is meant to solve many problems with the modern world. The financial world uses different Fiscal calendars so that each month is comparatively equal. By using TEC businesses can work on the same calendar as everyone else. Workers are now subjected to new schedules to fit into a 24/7 world. TEC is meant to make it a 6x6 world.

Workers currently on 12-hour, 2/2/3 schedules may be required to work 3 days in a row every other weekend to stay synchronized with a 7-day week, which causes inconsistent days off. This can be disruptive to families. TEC allows workers to work the same days of the week, month after month, year after year, without disrupting their lives. Companies may also create never-before-possible schedules, such as 3 on, 3 off.

Adoption of this calendar would also alter the definition of a full-time job. Ideally, a full work week would be 36 hours. While this may seem like a reduction, the worker would be working for 60 weeks, not 52. The same work is done, but with less disruption to the schedule.

The final week may be proposed as a "Week of light" as other calendars have tried to implement. This week could possibly be an international week of rest, reflection, and peace.

## Shifts and Schedules[]

This section does not completely list every possible permutation of schedules, but is meant to outline a few of the more notable possibilities.

### Full Time Examples[]

#### Single Week Schedules[]

Full Time hours conform to a full 1.5 days of work per week, or a 1:50000 work week, equivalent to a 36 hour work week over 6 days.

##### 3 Consecutive Days[]

**0 1 2** 3 4 5

0 **1 2 3** 4 5

0 1 **2 3 4** 5

0 1 2 **3 4 5**

**0** 1 2 3 **4 5**

**0 1** 2 3 4 **5**

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

##### 3 Evenly Spaced Days[]

**0** 1 **2** 3 **4** 5

0 **1** 2 **3** 4 **5**

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

##### 4 Consecutive Days[]

**0 1 2 3** 4 5

0 **1 2 3 4** 5

0 1 **2 3 4 5**

**0** 1 2 **3 4 5**

**0 1** 2 3 **4 5**

**0 1 2** 3 4 **5**

- Each day consists of 3/8 day of work, or :37500. Equivalent to 9 hours per day.

##### 4 2x2 Evenly Spaced Days[]

**0 1** 2 **3 4** 5

0 **1 2** 3 **4 5**

**0** 1 **2 3** 4 **5**

- Each day consists of 3/8 day of work, or :37500. Equivalent to 9 hours per day.

##### 6 Consecutive Days[]

**0 1 2 3 4 5**

- Each day consists of 1/4 day of work, or :25000. Equivalent to 6 hours per day.

#### 2 Week Schedules[]

##### 1 on 1 off[]

**0 1 2 3 4 5** | 0 1 2 3 4 5

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

##### Alternating Halves[]

**0 1 2** 3 4 5 | 0 1 2 **3 4 5**

0 **1 2 3** 4 5 | **0** 1 2 3 **4 5**

0 1 **2 3 4** 5 | **0 1** 2 3 4 **5**

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

##### 2/4 Alternating[]

**0 1** 2 3 4 5 | 0 1 **2 3 4 5**

**0 1** 2 3 4 5 | **0 1 2 3** 4 5

0 1 **2 3** 4 5 | **0 1** 2 3 **4 5**

0 **1** 2 3 **4** 5 | **0** 1 **2 3** 4 **5**

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

#### 6 Week Schedules[]

##### 3 on 3 off[]

Month View - Single Digits

**0 1 2 3 4 5**

**6 7 8 9 0 1**

**2 3 4 5 6 7**

8 9 0 1 2 3

4 5 6 7 8 9

0 1 2 3 4 5

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day.

##### Triangular Progressions[]

Month View - Single Digits

Left to Right

**0** 1 2 3 4 5

**6 7** 8 9 0 1

**2 3 4** 5 6 7

**8 9 0 1** 2 3

**4 5 6 7 8** 9

**0 1 2 3 4 5**

Center with Last Week of Month Off

0 1 **2 3** 4 5

6 **7 8 9 0** 1

**2 3 4 5 6 7**

8 **9 0 1 2** 3

4 5 **6 7** 8 9

0 1 2 3 4 5

- Each day consists of 1/2 day of work, or :50000. Equivalent to 12 hours per day

### Part Time[]

Not Finished

## Computer Code[]

**PHP**

<?php /* Working code is licenced under the GPL (http://www.gnu.org/licenses/gpl.txt). Reproduced as text here under the GFDL (http://www.gnu.org/licenses/fdl.txt). Original Author - Copyright (C) 2004 DeWayne Lehman. This script comes with ABSOLUTELY NO WARRANTY. This script demostrates how to output Date Prime via PHP regardless of which time zone PHP is running in. Working demostration at http://i8-d.com. */ putenv("TZ=Pacific/Kiritimati"); $year = date(Y) - 2001; $month = 0; $day = date(z); $time = round((date(G)/24 + date(i)/1440 + date(s)/86400)*100000); $time = sprintf("%05s",$time); while ($day > 35) { $day = $day - 36; $month = $month +1; } echo "<b>@".$year.".".$month.".".$day.":".$time."</b>"; ?>

## Copyright[]

This calendar and related works done by its author are copyrighted by DeWayne Lehman. He grants full, unrestricted use to anyone. The copyright is simply to prevent any third party from copyrighting TEC for their own and restricting unhindered access. This copyright owner grants GFDL. This copyright extends full, unrestricted use to documents not covered by GFDL and related works not located on this site.

## References[]

The Chinese may have been the first to have devised a symmetrical 10-month calendar, though they lacked essential information needed to complete it.

More detailed graphics, including geometrical breakdowns of TEC and its mathematical advantages will be forthcoming.

Triangular Numbers - MathWorld

All You Ever Wanted to Know About Pascal's Triangle and more

#### References[]

- Adapted 12 Dec 2005 from the Wikibooks module, "Triangular Earth Calendar" http://en.wikibooks.org/wiki/Triangular_Earth_Calendar, used under the GNU Free Documentation License
- That was adapted from http://www.i8-d.com/calendar.html, "Copyright 2004, DeWayne Lehman. Free for the world to use, no strings at all attached, except that nobody else can ever copyright and keep it from the world."

This page uses content from Wikinfo. The original article was at Triangular Earth Calendar. The list of authors can be seen in the page history.