Draft

Decalendar

Introducing Decalendar, a solar calendar which measures time in years and days without the need for months or weeks.

Author

Martin Laptev

Published

2026+097

Modified

2026+110

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '32px'}}}%%
flowchart LR
   A[Dec]-->B[date]-->C[time]-->D[snap]-->E[span]
   click A "/dec"
   click B "/dec/date"
   click C "/dec/time"
   click D "/dec/snap"
   click E "/dec/span"

Diagram 1

Decalendar

My website serves as a demonstration of both the Quarto publishing📤system and the Dec measurement📐system. I use several clever hacks to get Quarto to display all of the dates on my website in the Dec year+day format. Knowing the basics of the Dec calendar🗓️(Decalendar) will help you to understand the filter and include articles in the Quarto section of my site.

Among its many features, Quarto offers support for the Observable data visualization system. Observable is my top choice for interactive graphics. We can interact with the two Observable calendar plots below⬇️using the adjacent Observable inputs. The scrubber🧽input is a great place to start because it cycles🔄through every value of the range🎚️inputs beneath it.

0 Day of year (doy)

To activate the scrubber input, press the “Play”▶️button above⬆️the range inputs. Upon activation, the box around the selected day in each plot will move back and forth between the first “day of year” (doy), d0, and the last doy, which is either d364 or d365. To insert or remove d365, use the “Year length” radio📻input to set the number of days in the year.

The insertion of d365 shifts 306 dates, d to d, in the Gregorian calendar by 1 day, but does not change the order of any Dec dates, because d365 is the last day of any Dec leap year and is always followed by d0 of the subsequent Dec year (y+1). The “Year length” radio input also changes the value of the negative “Day of year” range input by 1 day.

Similarly, the “Coordinated Universal Time (UTC) offset” radio input shifts the Gregorian calendar date selected by the “Month” and “Day of month” range inputs by 1 day. The “UTC offset” radio input will also shift the Decalendar plot cell colors🎨by 1 day if the “Color scheme” radio input is set to the “Month” discrete scheme instead of the “Day” continuous scheme.

From the perspective of Dec, month color labels🏷️are only useful if we want to compare the Dec and Gregorian calendars. In contrast, Dec day color labels can help us sort days into groups of 100 called hectodays (h) and groups of 10 named xún (x). Dec defines meterological seasons in terms of h and uses x instead of Gregorian calendar months and weeks.

The “Plot layout” radio input rotates the calendar plots by a quarter turn, interchanging the horizontal (↔︎) and vertical (↕) axes. The axis labels demonstrate that x and “days of xún” (dox) are analogous to weeks and “days of week” (dow). If we multiply an x axis label by ten and add it to a dox axis label, we get a “positive integer doy” (pid) cell value: × 10 + = .

\[\text{pid} = \text{x} \ast 10 + \text{dox} \tag{1}\] \[\text{dyl} = \text{pid} - \text{nid} \tag{2}\]

There are two range inputs labelled “Day of year” because every doy can be expressed as either a positive or a negative integer. The pid is the number of days that have passed in the year, the absolute value of the “negative integer doy” (nid) is the number of days left in the year, and their difference is the “Decalendar year length” (dyl), which can be 365 or 366.

First dow of the Gregorian calendar year

The distinction between pid and nid can be explained in terms of computer programming. If we think of a year as an array and each day as an array element, dyl is the number of elements in the array, pid is a positive index, and nid is a negative index. Array indexes can be used to obtain specific array elements individually via indexing or in groups via array slicing.

The year+day Dec date format is short for year+day/dyl. Dec truncates dates because the dyl is not needed to specify a date, remains constant for 366, 1095, or 2920 days, has only 2 possible values: 365 or 366, and can be determined by passing Year y to Equation 3 below. Nevertheless, we can use the dyl to convert between different kinds of Dec dates.

\[\text{dyl}=\begin{cases} 366&{\begin{align}\text{if } (\text{y} + 1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} \ \ \ \ 4=0\\ \href{https://en.wikipedia.org/wiki/Logical_conjunction}{\land}(\text{y}+1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 100\neq0\\ \href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}( \text{y} + 1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 400=0\end{align}}\\\\ 365&{\text{otherwise.}}\end{cases} \tag{3}\]

Dec categorizes each date as a countdown or countup date, depending on whether the date counts up the days since Year y or counts down the days until Year y+1. The current year+day UTC date, +, informs us that Year began days ago, whereas its countdown equivalent, -, lets us know that Year will begin in days.

\[\text{y}+\dfrac{\text{pid}}{\text{dyl}} = \text{y} + 1 + \dfrac{\text{nid}}{\text{dyl}} \tag{4}\]

Both pid and nid can be useful. If we wanted to add 285 days to the doy selected below, for example to predict when a pregnant🤰woman will give birth to a baby👩‍🍼(Jukic et al. 2013+215), we should add 285 to the pid if it is less than 80 in a common year or less than 81 in a leap year, but otherwise we should add 1 to the year and add 285 to the nid: + 285 = .

First dow of the Gregorian calendar year

The radio input beneath the plots selects the dow for d, the first day of the Gregorian calendar year. Changing the d dow shifts every Gregorian calendar date by one to six days without affecting Decalendar. A leap year that begins on the last dow, Dow 6, has an extra “week of year” (woy), but its first and last woy, Weeks 0 and 53, have only one day.

Although weeks determine the shape of the Gregorian calendar plot, its cell values are “days of month” (dom). We can uniquely identify🪪a specific day in any year with a pid, instead of a month and a dom. Except for d365 in leap years, every year has the same x, h, and months, but not the same weeks, because the d dow varies from year to year.

1 Day of xún (dox)

In contrast to a week, an x can be split evenly into either five pairs of days or two equal halves called “pentadays of xún” (pox). Likewise, a common year can be divided evenly into 73 “pentadays” (p). The last p of a leap year, p73, consists of the final day of the leap year, d365, and the first four days of the subsequent year: Days 0 to 3.

In the context of a common year, p73 is synonymous with p0 of the subsequent year. To obtain the current p, we double the current x and add the current pox, which is 1 if the current dox is greater than 4 and 0 otherwise: = × 2 + [ > 4]. If we divide a dox by 5, the remainder will be its corresponding “day of pentaday” (dop): mod 5 = .

\[\text{pox} = \left[\text{dox} > 4\right] \tag{5}\] \[\text{p = x} \ast 2 + \text{pox} \tag{6}\] \[\text{dop = dox} \href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 5 \tag{7}\]

In Diagram 2 below, each row is a pox and each square node is a dox. Diagram 2 visualizes Schedule L, a Dec schedule that plans for exactly 219 work days per year, which is close to the 208 to 210 work days per year provisioned by a four-day workweek. Schedule L designates Dox 1, 2, 3, 6, 7, and 8 as work days and Dox 0, 4, 5, and 9 as rest days.

Schedule L (Dox 0 to 9)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  B~~~M[ ]:::empty----N[ ]:::empty
  W---M
  N---O[ ]:::empty
  Q[ ]:::empty---L
  W[ ]:::empty-->A[0]-->B[1]-->C[2]-->D[3]-->E[4]---L[ ]:::empty
  Y[ ]:::empty-->F[5]-->G[6]-->H[7]-->I[8]-->J[9]---O[ ]:::empty
  B~~~P[ ]:::empty---Q
  Y---P
  subgraph workdays[work]
  B
  C
  G
  H
  D
  I
  end
  subgraph restdays[ ]
  A
  F
  E
  J
  workdays
  end
  classDef empty width:0px;

Diagram 2

Dec identifies groups of days between Dop 0 and 4 and betwixt Dox 0 and 9 as “pentaday interquintile ranges” (pir) and “xún interdecile ranges” (xir), respectively. The acronyms pir and xir are derived from the terms quintile, decile, and interquartile range. If we follow Schedule L, a pir is to a workweek as a p is to a week and as a x is to a fortnight.

The pair of days between two pir is called a “liminal interconnecting margin” (lim). The last lim of a common year, Lim 73, comprises Days 364 and 0 and is synonymous with Lim 0 of the subsequent year. In a leap year, Lim 73 consists of Days 364 and 365 and overlaps with Lim 74, which is composed of Days 365 and 0 and is equivalent to Lim 0 of the succeeding year.

Except for Lim 74, every even-numbered lim is the border that separates two xir. With the exception of Lim 73, every odd-numbered lim is flanked by the two pir within each xir. Diagram 3 below shows the last 5 doy of a common year and the first 5 doy of the following year, which include the last day of Lim 72, Pir 72, Lim 0, Pir 0, and the first day of Lim 1.

Schedule L (p72 and p0)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  W~~~~O[ ]:::empty
  W[ ]:::empty-->A[360]-->B[361]-->C[362]-->D[363]-->E[364]----L[ ]:::empty
  Y[ ]:::empty-->F[0]-->G[1]-->H[2]-->I[3]-->J[4]-->K[ ]:::empty
  Q[ ]:::empty---L
  Y---Q
  subgraph workdays[work]
  B
  C
  G
  H
  D
  I
  end
  subgraph restdays[ ]
  A
  F
  E
  J
  workdays
  end
  classDef empty width:0px;

Diagram 3

The diagrams above illustrate that the transition from a common year preserves the alternating pattern of two-day lim and three-day pir. After 4 or 8 years, this pattern is interrupted by Lim 73 and 74 at the end of a leap year. In Diagram 4 below, this interruption manifests as an extra doy per row which puts Day 364 alongside Day 365 and Day 4 beside Day 5.

Schedule L (d360 to d365 and d0 to d5)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  W~~~~O[ ]:::empty
  W[ ]:::empty-->A[360]-->B[361]-->C[362]-->D[363]-->E[364]-->L[365]----S[ ]:::empty
  Y[ ]:::empty-->F[0]-->G[1]-->H[2]-->I[3]-->J[4]-->K[5]-->T[ ]:::empty
  Q[ ]:::empty---S
  Y---Q
  subgraph workdays[work]
  B
  C
  G
  H
  D
  I
  end
  subgraph restdays[ ]
  A
  F
  E
  J
  K
  L
  workdays
  end
  classDef empty width:0px;

Diagram 4

According to Schedule L, pir only contain workdays and lim are solely made up of rest days. When we follow Schedule L, a lim is the Dec analog of a weekend. To make lim appear like weekends we can start from Dox 1 instead of Dox 0 as in the Diagram 5 below, which displays its lim as a two-by-two square grid on the right like Lim 73 and 1 in Diagram 4 above.

Schedule L (Dox 1 to 0)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  B~~~M[ ]:::empty----N[ ]:::empty
  W---M
  N---O[ ]:::empty
  Q[ ]:::empty---L
  W[ ]:::empty-->A[1]-->B[2]-->C[3]-->D[4]-->E[5]---L[ ]:::empty
  Y[ ]:::empty-->F[6]-->G[7]-->H[8]-->I[9]-->J[0]---O[ ]:::empty
  B~~~P[ ]:::empty---Q
  Y---P
  subgraph workdays[work]
  A
  B
  C
  F
  G
  H
  end
  subgraph restdays[ ]
  D
  E
  I
  J
  workdays
  end
  classDef empty width:0px;

Diagram 5

The order of dox in Diagram 5 is different than all of the previous diagrams but all of the diagrams above show Schedule L because the categorization of dox as work or rest days remains unchanged. If we left rotate (↺) the dox categories of Schedule L by 1 day, we get the Schedule X Dec schedule: L ↺ 1 = X. Schedule X groups rest days at the end of each p.

Schedule X (Dox 0 to 9)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  B~~~M[ ]:::empty----N[ ]:::empty
  W---M
  N---O[ ]:::empty
  Q[ ]:::empty---L
  W[ ]:::empty-->A[0]-->B[1]-->C[2]-->D[3]-->E[4]---L[ ]:::empty
  Y[ ]:::empty-->F[5]-->G[6]-->H[7]-->I[8]-->J[9]---O[ ]:::empty
  B~~~P[ ]:::empty---Q
  Y---P
  subgraph workdays[work]
  A
  B
  C
  F
  G
  H
  end
  subgraph restdays[ ]
  D
  E
  I
  J
  workdays
  end
  classDef empty width:0px;

Diagram 6

If we follow Schedule X, there will be 4 consecutive work days during any transition from a leap year. To limit the number of consecutive work days to 3, we could right rotate (↻) Schedule L and obtain Schedule F: L ↻ 1 = F. Unlike Schedule X, Schedule F handles yearly transitions just as gracefully as Schedule L and provisions the exact same number of work days per year.

Schedule F (Dox 0 to 9)

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '29px'}}}%%
flowchart LR
  M[ ]:::empty----N[ ]:::empty
  W---M
  N---O[ ]:::empty
  Q[ ]:::empty---L
  W[ ]:::empty-->A[0]-->B[1]-->C[2]-->D[3]-->E[4]---L[ ]:::empty
  Y[ ]:::empty-->F[5]-->G[6]-->H[7]-->I[8]-->J[9]---O[ ]:::empty
  P[ ]:::empty---Q
  Y---P
  subgraph workdays[work]
  C
  D
  E
  H
  I
  J
  end
  subgraph restdays[ ]
  A
  B
  F
  G
  workdays
  end
  classDef empty width:0px;

Diagram 7

Each of the 32 base Dec schedules can be expressed as a 5-bit binary (base2) sequence. Of these 32 binary sequences, 8 are palindromes. If a Dec schedule can be represented by a 5-bit palindrome, we can identify its work and rest days by the last digit of not only the pid but also either the subsequent nid (nid) in common years or the nid after next (nid) in leap years.

To get the nid, we sum the nid and one: nid = nid + 1. Similarly, the nid is the sum of the nid and two: nid = nid + 2. Table 1 below displays the pid, nid, and nid of the first and final 11 days of a common year. We can convert any nid into a “mixed integer doy” (mid) and then use the last digit of the mid to discern between the work and rest days of any of the 32 base Dec schedules in common years.

The horizontal line above all but the last digit of the mid in Table 1 is called a vinculum. In Dec, a vinculum negates whatever is beneath it. Negating all of the digits of an integer flips its sign. Negative integers can be denoted by vinculum instead of a minus sign. For example, d1 and d19 are two equivalent ways to write Day -1, the last day of the Decalendar year.

pid nid nid mid
0 -365 -364 375
1 -364 -363 376
2 -363 -362 377
3 -362 -361 378
4 -361 -360 379
5 -360 -359 360
6 -359 -358 361
7 -358 -357 362
8 -357 -356 363
9 -356 -355 364
10 -355 -354 365
… … … …
354 -11 -10 29
355 -10 -9 10
356 -9 -8 11
357 -8 -7 12
358 -7 -6 13
359 -6 -5 14
360 -5 -4 15
361 -4 -3 16
362 -3 -2 17
363 -2 -1 18
364 -1 -0 19
pid nid nid mid pid nid nid mid
0 -365 -364 375 354 -11 -10 29
1 -364 -363 376 355 -10 -9 10
2 -363 -362 377 356 -9 -8 11
3 -362 -361 378 357 -8 -7 12
4 -361 -360 379 358 -7 -6 13
5 -360 -359 360 359 -6 -5 14
6 -359 -358 361 360 -5 -4 15
7 -358 -357 362 361 -4 -3 16
8 -357 -356 363 362 -3 -2 17
9 -356 -355 364 363 -2 -1 18
10 -355 -354 365 364 -1 -0 19
Table 1

The correlation between a digit and the absolute value (magnitude) of its mixed integer is positive for positive digits and negative for negative digits. Each negative digit in the mid column of Table 1 pulls the magnitude of its mid towards one, meanwhile each positive digit in that column moves the magnitude of its mid in the opposite direction along the number line.

In a common year, the last digits of pid and nid run antiparallel to each other like complementary strands of deoxyribonucleic acid🧬, but instead of adenine to thymine and cytosine to guanine, the pattern is 0 to 4, 1 to 3, 2 to 2, 3 to 1, 4 to 0, and so on. The final digits of pid and nid follow the same pattern in leap years: 0 to 4, 1 to 3, 2 to 2, 3 to 1, 4 to 0, and so on.

The last digits of pid and mid are misaligned by 4 days in leap years and by 5 days in common years. Dec maintains a constant 5-day misalignment by replacing the mid with the next mid (mid) in leap years. The accents above nid and mid both advance the apparent doy by one day. Table 2 below shows the pid, nid, nid, and mid of the first and last 11 days of a leap year.

pid nid nid mid
0 -366 -364 375
1 -365 -363 376
2 -364 -362 377
3 -363 -361 378
4 -362 -360 379
5 -361 -369 360
6 -360 -358 361
7 -359 -357 362
8 -358 -356 363
9 -357 -355 364
10 -356 -354 365
… … … …
355 -11 -9 10
356 -10 -8 11
357 -9 -7 12
358 -8 -6 13
359 -7 -5 14
360 -6 -4 15
361 -5 -3 16
362 -4 -2 17
363 -3 -1 18
364 -2 -0 19
365 -1 -0 00
pid nid nid mid pid nid nid mid
0 -366 -364 375 354 -11 -9 10
1 -365 -363 376 355 -10 -8 11
2 -364 -362 377 356 -9 -7 12
3 -363 -361 378 357 -8 -6 13
4 -362 -360 379 358 -7 -5 14
5 -361 -369 360 359 -6 -4 15
6 -360 -358 361 360 -5 -3 16
7 -359 -357 362 361 -4 -2 17
8 -358 -356 363 362 -3 -1 18
9 -357 -355 364 363 -2 -0 19
10 -356 -354 365 364 -1 -0 00
Table 2

The Schedule L rule for categorization of work and rest days can be summarized as [doy mod 5 ∈ {1,2,3}], where ∈ means “is an element of” and {1,2,3} is Set L, a set which contains all of the dop that are work days according to Schedule L. The doy in the Schedule L rule can be the mid or nid in common years, the mid or nid in leap years, or the pid in all years.

Base32

The Schedule L pattern of rest and work days can be expressed in binary (base2) as 01110, decimal (base10) as 14, or Dec duotrigesimal (base32) as L. Each of the 32 Dec base schedules can be represented by a single letter of the Dec base32 (b32) alphabet. The b32 letters are listed in Table 3 below alongside their base10 (b10) and base2 (b2) values.

A 0 00000 N 16 10000
Á 1 00001 O 17 10001
B 2 00010 Ó 18 10010
C 3 00011 P 19 10011
D 4 00100 Q 20 10100
E 5 00101 R 21 10101
É 6 00110 S 22 10110
F 7 00111 T 23 10111
G 8 01000 U 24 11000
H 9 01001 Ú 25 11001
I 10 01010 V 26 11010
Í 11 01011 W 27 11011
J 12 01100 X 28 11100
K 13 01101 Y 29 11101
L 14 01110 Ý 30 11110
M 15 01111 Z 31 11111
A 0 00000 G 8 01000 N 16 10000 U 24 11000
Á 1 00001 H 9 01001 O 17 10001 Ú 25 11001
B 2 00010 I 10 01010 Ó 18 10010 V 26 11010
C 3 00011 Í 11 01011 P 19 10011 W 27 11011
D 4 00100 J 12 01100 Q 20 10100 X 28 11100
E 5 00101 K 13 01101 R 21 10101 Y 29 11101
É 6 00110 L 14 01110 S 22 10110 Ý 30 11110
F 7 00111 M 15 01111 T 23 10111 Z 31 11111
Table 3

Table 3 above shows that the b32 alphabet includes the 26 letters of the English alphabet and combines the 6 vowels, a, e, i, o, u, and y, with acute accents ( ́) to create 6 additional letters, á, é, í, ó, ú, and ý, for a total of 32 letters. The 6 additional accented letters are included immediately after their unaccented antecedents as per the order of the English alphabet.

If we need more work days than those provided by Schedule L, we can switch to the Schedule LM Dec schedule by following Schedule L on even numbered p and Schedule M to odd numbered p. Schedule LM has 1 more work day than Schedule L and provisions 255 work days per year without modifying the yearly transition shown in the Diagrams 3 and 4 above.

Unlike weekly schedules, Dec schedules like Schedules L and LM provide a consistent🎯number of work days every year. While Days 364, 365, and 0 can be work or rest days in the Gregorian calendar️, these days are always rest days if we follow Schedules L or LM. Therefore, Schedules L and LM do not require any holidays to smooth the transition between years.

There are 11 United States (US) Federal holidays. US Federal holidays that fall on a Gregorian calendar️ rest day, Dow 0 or Dow 6, are observed on the nearest Gregorian calendar️ work day: Dow 1 or Dow 5. Instead of applying this rule to Schedule L and moving holidays from Dox 0 to 1, 4 to 3, 5 to 6, or 9 to 8, we can switch between Dec schedules as needed.

The last US Federal holiday of the Gregorian calendar year is Christmas🎄. Even though it occurs on d299, which is the last day of Hectoday 2 (h2), Christmas is likely to be celebrated on d300, the first day of Hectoday 3 (h3), by people who do not use Dec and live in a UTC time zone with a negative offset. The Dec analog of the holiday season is Hectoday -1 (h1).

2 Day of hectoday (doh)

In addition to the four astronomical seasons, there are four Dec seasons which were chosen from two overlapping sets of four consecutive h to match daily global mean temperature patterns. The two sets called are “positive integer hectodays” (pih) and “negative integer hectodays” (nih). Equation 8 organizes doy into one of the four pih: h0, h1, h2, or h3.

\[\text{pih} = \left\lfloor\dfrac{\text{pid}}{100}\right\rfloor \tag{8}\] \[\text{nih} = \left\lfloor\dfrac{\text{nid}}{100}\right\rfloor \tag{9}\]

Similarly, Equation 9 groups doy into one of the four nih: h1, h2, h3, or h4. In either case, the pid or nid of a doy is divided by 100 and then floored to discard the fractional part of the quotient. In the line plot below, the four Dec seasons, h0, h1, h2, and h1, are labelled red, yellow, cyan, and violet, respectively, but h2 is truncated to remove its overlap with h1.

The line plot shows how well the four Dec season match ERA5 daily global mean temperatures. If we think of the method for assigning doy to Dec seasons in Equation 10 as a statistical model, its “goodness of fit” is evidenced by the fact that the hottest doy on average, d149, is near the middle of h1 and the coldest doy on average, d316, is close to the center of h1.

\[\text{season} = \left\lfloor\dfrac{\text{pid} - \text{dyl} * [\text{nid} \ge -100]}{100}\right\rfloor \tag{10}\]

In general, the hottest days are in h1, the coldest days are in h1, temperatures increase with time in h0, and temperatures decrease with time in h2. Therefore, we can refer to h0, h1, h2, and h1 as the rise📈, crest🔥, fall📉, and trough❄️, respectively, of global mean temperatures. Table 4 below shows the Dec season names in the Northern and Southern Hemispheres.

Hemisphere h0 h1 h2 h1
Northern Spring Summer Autumn Winter
Southern Autumn Winter Spring Summer
Table 4

The remainder after dividing a doy is a “day of hectoday” (doh), which essentially is the percent of a pih that has passed if the dividend is a pid, the percent of a nih that remains if the dividend is a nid, the percent of a pih that is left if the dividend is a mid that is equal to a pid, or the percent of a nih that has passed if the dividend is a mid which is equal to a nid.

\[\text{doh} = \text{doy} \href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 100 \tag{11}\]

To highlight an h and doh, we can use a three digit mid with a vinculum over either the first digit or the last two digits. If its vinculum is above only its first digit, the current three-digit mid is equal to the the current nid and signals that h is % done: mid = nih. If its vinculum is over its last two digits, the current three-digit mid is equal to the current pid and indicates that h is % unfinished: mid = pid.

We can also use mid to draw attention to the current x. If its vinculum is above its first two digit, the current mid is equal to the the current nid and signals that h is % done: mid = nih. If its vinculum is over only its last digit, the current mid is equal to the current pid and indicates that h is % unfinished: mid = pid.

The connection between pid and seasonal temperature changes requires pid to reset after d1. If we do not need to visually keep track of years, h, doh, months, dom, x, dox, weeks, dow, p, or dop, we can translate a a Dec year+day date into a single integer called a “day of era” (doe).

Every doy belongs to one pih, one nih, one “positive integer xun” (pix), and one “negative integer xun” (nix). In total, that is two h and two x per doy.

3 Day of era (doe)

Dec refers to midnight on d0 as the “beginning of year” (boy). At the boy, the pid rolls over from 364 or 365 to 0. If the nid did not reset to -365 or -366 at the boy, it would continue from -1 to 0 and thus become a pid. The doe is like a nid that became a pid at the “beginning of era” (boe), midnight on d0 of Year 0 (y0), and never restarted before or after the boe.

Each of the ten Dec time zones has its own boe, doe, boy, and doy. The boe of the Zone 0 Dec time zone is called the Dec epoch. We can convert Julian day numbers (JDN) to Zone 0 doe by subtracting the number of full days in between the start of the Julian period and the Dec epoch, which is 1721119 if the Zone 0 time is later than noon and 1721120 otherwise.

Dec uses doe for calendrical calculations, such as finding the POSIX zero-based dow of a given date. This year, the dow of Christmas🎄is according to Equation 12: ( + ) mod 7 = . Unlike dow, the dox can be found without much effort, because it is the last digit of the pid or equivalently the remainder after dividing the pid by 10 as per Equation 13.

\[\text{dow} = (\text{doe} + 3) \href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 7 \tag{12}\] \[\text{dox} = \text{pid} \href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 10 \tag{13}\]

Equation 12 is derived from Howard Hinnant’s weekday_from_days algorithm. The Dec epoch dow is 3 = (0 + 3) mod 7. The UNIX epoch dow is 4 = (719468 + 3) mod 7. Depending on how mod is defined, a negative doe could yield a negative dow: -1 = (-60 + 3) mod 7. We can add 7 to a negative dow in the bottom row of Table 5 to turn it into the positive dow above it.

Sun Mon Tue Wed Thu Fri Sat
+ 0 1 2 3 4 5 6
- 7 6 5 4 3 2 1
Table 5

Christmas is an anchored⚓️holiday because it occurs on the same pid every year. In contrast, floating🛟holidays like Thanksgiving🦃are always planned for the same dow and thus can fall on various pid. We can use Equation 14, which is inspired by Howard Hinnant’s weekday_difference algorithm, to find the floating holiday date associated with a given year.

\[\text{dow}_\Delta = (\text{dow}_\text{M} - \text{dow}_\text{S} + 7) \href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 7 \tag{14}\]

In Equation 14, dowM is the minuend, dowS is the subtrahend, and dowΔ is the difference between them that ranges from 0 to 6. To get the pid of the first Dow 4 after d266, which is Thanksgiving in the United States🇺🇸and Brazil🇧🇷, we plug 4 as dowM and the dow of d267 as dowS into Equation 14, = (4 - + 7) mod 7, and then add 267: = + 267.

WarningBad Pun Alert

Deckaday the halls with dows of holly! Fa + la × 8! ’Tis the hectoday to be jolly! Aren’t you thankful that I couldn’t thank of a Thanksgiving pun?

4 Year of era (yoe)

Almost all of the equations above depend on the pid. Most often, the pid is calculated using Equations 15, 16, and 18, which are based on the days_from_civil algorithm created by Howard Hinnant and described in his manuscript entitled chrono-Compatible Low-Level Date Algorithms (2021+185), to convert the two components of a Dec date, namely a “year of era” (yoe) and a pid, into a doe.

\[\text{coe} = \Biggl \lfloor \frac{\begin{cases}\text{yoe}&{\text{if } \text{yoe} \geq 0;}\\\text{yoe}-399&{\text{otherwise.}}\end{cases}}{400} \Biggr \rfloor \tag{15}\]

\[\text{yoc} = \text{yoe} - \text{coe} \ast 400 \tag{16}\]

\[\text{doc} = \text{yoc} \ast 365 + \lfloor\frac{\text{yoc}}{4}\rfloor - \lfloor\frac{\text{yoc}}{100}\rfloor + \text{pid} \tag{17}\]

\[\text{doe} = \text{coe} \ast 146097 + \text{doc} \tag{18}\]

The Dec date equations are the inverse🔁of the Dec doe equations above, based on Howard Hinnant’s civil_from_days algorithm, and useful for obtaining Dec dates from doe, UNIX timestamps, and JDN. Along the way to obtaining a yoe and pid from a doe, the Dec date equations calculate the “cycle of era” (coe), “year of cycle” (yoc), and “day of cycle” (doc).

\[\text{coe} = \Biggl \lfloor \frac{\begin{cases}\text{doe}&{\text{if } \text{doe} \geq 0;}\\\text{doe}-146096&{\text{otherwise.}}\end{cases}}{146097} \Biggr \rfloor \tag{19}\]

\[\text{doc} = \text{doe} - \text{coe} \ast 146097 \tag{20}\]

\[\text{yoc} = \biggl \lfloor \frac{\text{doc} - \lfloor \frac{\text{doc}}{1460} \rfloor + \lfloor \frac{\text{doc}}{36524} \rfloor - \lfloor \frac{\text{doc}}{146096} \rfloor}{365} \biggr \rfloor \tag{21}\]

\[\text{yoe} = \text{yoc} + \text{coe} \ast 400 \tag{22}\]

\[\text{pid} = \biggl \lfloor \text{doc} - \text{yoc} \ast 365 - \lfloor \frac{\text{yoc}}{4} \rfloor + \lfloor \frac{\text{yoc}}{100} \rfloor \biggr \rfloor \tag{23}\]

Dates generated by the Dec date equations are guaranteed to be in the standard yoe+pid format. Therefore, we can standardize Dec dates by using the Dec doe and date equations consecutively to perform a round-trip “date to doe to date” conversion and thus ensure that the yoe is an integer and the pid is a positive integer less than the dyl: 0 ≤ pid < dyl.

5 Day of week (dow)

Even though Decalendar functions best with x, Dec dates can display dow by splitting a pid into a “beginning of week” (bow) and the POSIX zero-based dow. To obtain the bow, we subtract the dow from the pid: bow = pid - dow. According to the current Zone 0 yoe+bow+dow date, , the most recent Dow 0 was on d and today is Dow .

\[\text{yoe} + \frac{\text{pid}}{\text{dyl}} = \text{yoe} + \frac{\text{bow + dow}}{\text{dyl}} \tag{24}\]

A Dec bow date can have a countup or a countdown bow. The countdown equivalent of is -. Like nid, countdown bow can be useful for calculations. We can add up to 52 weeks to any countdown bow without having to take into account the length of the year. The sum of 52 weeks and the last bow+dow of this year is 52 × 7 + + = + .

Based on the calculation above, the Dec bow date that is 52 weeks after -+ is ++. When we see the same dow in two dates, we know that difference between them is a multiple of 7. The bow can be used to refer to the current week in phrases like “the week of d” or “the week that begins with d” and thus can function like a woy.

6 Week of year (woy)

We can convert between the bow+dow in Dec bow dates and the 7×woy+dow-boydow in Dec woy dates. The boydow is the dow of the first day of the Dec year. To obtain the current woy, we sum the boydow with the current bow and then divide by 7: = ( + ) ÷ 7. Dec truncates woy dates so that they only display the yoe, woy, and dow: ++.

\[\text{yoe} + \frac{\text{bow + dow}}{\text{dyl}} = \text{yoe} + \frac{7\ast\text{woy + dow}-\text{boydow}}{\text{dyl}} \tag{25}\]

If dow were like doy and reset at the boy, the boydow would always be zero and conversion between woy and bow dates would be straightforward but the reality is that we need the boydow to convert a woy date to a bow date. Dec pads the left side of the woy and bow in dates with zeros so that we can distinguish between two-digit woy and three-digit bow.

To find the boydow, we first divide the bow by 7 and subtract the remainder from 7. Then, we divide the resulting difference by 7 and keep the remainder. We could also get the boydow without a bow by turning the yoe into a doe and then the doe into a dow. Dec woy dates obfuscate🫣doy much more than bow dates, but may be useful for week-based accounting🧾.

\[\text{boydow = (7} - \text{bow}\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod}7)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod}7 \tag{26}\]

7 Day of month (dom)

Dec dates can be expanded to display Dec month and POSIX dom numbers. The current Dec dom date is . Dec month numbers are the last doy of the previous month because POSIX dom are one-based. For zero-based dom, Dec represents each month with its first doy: . This way, Dec can support both zero- and one-based dom.

Dec dom dates replace the doy d from Dec dates with d-m+m. We evaluate the subtraction to get d-m, the Dec month number, but not the addition, so we can see m, the dom. If we combine the dom and dow patterns above, we can create hybrid dom+dow Dec dates: , where is d-m-w, the doy of the last Dow 0 before the beginning of the month.

We can obtain Dec month numbers using only a pair of hands🤲by counting index☝️and ring💍fingers as 30 days and other fingers as 31 days. For zero-based dom, we start counting from 0. For one-based dom, we start counting from -1, as shown in the image below. To spread 12 months across 10 fingers, the first and last fingers each represent 2 months.

Wikimedia

The finger🖐mnemonic described above is similar to the knuckle👊and musical keyboard🎹mnemonics. These mnemonics attempt to make sense of the irregular pattern of month lengths in the Gregorian calendar️. As opposed to months, we do not need mnemonics, tables, or mental calculations to use x, because all of the required information is plainly visible in the doy.

8 Month of year (moy)

To convert doy to or from POSIX month and dom numbers, we can use the month of year (moy) equations from the civil_from_days and days_from_civil algorithms. Unlike POSIX month numbers, moy are zero-based and start from Moy 0 instead of Moy 10. As shown in the first moy equation below, we can obtain a moy from a doy or a POSIX month number.

\[\text{moy} = (5 \ast \text{doy} + 2) \div 153 = \begin{cases}\text{month}-3&{\text{if }\text{month}\gt 2;}\\\text{month}+9&{\text{otherwise.}}\end{cases} \tag{27}\]

\[\text{month} = \begin{cases}\text{moy}+3&{\text{if }\text{moy}\lt 10;}\\\text{moy}-9&{\text{otherwise.}}\end{cases} \tag{28}\]

\[\text{dom} = \text{doy} - (153 \ast \text{moy} + 2) \div 5 + 1 \tag{29}\]

\[\text{doy} = (153 \ast \text{moy}+ 2) \div 5 + \text{dom} - 1 \tag{30}\]

A moy and its equivalent POSIX month number differ by 9 in Moy 10 or 11 and -3 in any other moy because the Dec epoch, 0000+000, is 2 months later than the Gregorian calendar epoch: -0001+306. To convert years, we add 1 to the yoe and subtract 1 from the Gregorian calendar️ common era year (cey) if the doy is greater than 305, the moy is greater than 9, or the month is less than 3:

\[\text{yoe} = \begin{cases}\text{cey}-1&{\text{if }\text{doy}\gt 305\href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}\text{moy}\gt 9\href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}\text{month}\lt 3;}\\\text{cey}&{\text{otherwise.}}\end{cases} \tag{31}\]

\[\text{cey} = \begin{cases}\text{yoe}+1&{\text{if }\text{doy}\gt 305\href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}\text{moy}\gt 9\href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}\text{month}\lt 3;}\\\text{yoe}&{\text{otherwise.}}\end{cases} \tag{32}\]

Summary

This article describes Dec and how it can interoperate with the Gregorian calendar by minuend expanding Dec dates into dom and dow dates or dividend expanding dow dates into woy dates. On its own, Dec minuend expands dates into span equivalents, but has no use for dividend expansion, because doy display x and dox without any expansion.

Diagram 8 below visualizes the conversion of the UNIX epoch Dec date 1969+306 into a Gregorian calendar️year, month, dom and dow. To obtain its associated dow, we first need to convert a Dec date into a doe. Outside of its interoperability with the Gregorian calendar️, Dec converts dates into doe to find the number of days in between two dates.

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '20px'}}}%%
flowchart LR
   L~~~Y
   L-->Y
   L-->D
   L-->E
   R-->Y
   M---R[  ]:::empty
   D---A
   D~~~B[  ]:::empty
   A-->G
   G---R
   Y~~~L[  ]:::empty
   Y---A[  ]:::empty
   Y~~~A
   B-->M
   B~~~N
   B-->N
   B-->D
   B~~~D
   N~~~B
   E-->W
   subgraph decdate[Decalendar]
   Y[yoe<br>1969]
   D[doy<br>306]
   E[doe<br>719468]
   end
   subgraph gregdate[Gregorian calendar]
   W[dow<br>4]
   G[year<br>1970]
   M[month<br>1]
   N[dom<br>1]
   end
   classDef empty width:0px;
   click E "#doe"
   click W "#dow"
   click Y "#yoe"
   click D "#doy"
   click G "#eya"
   click M "#moy"
   click N "#dom"

Diagram 8

For simplicity, the flowchart above does not show conversion byproducts, such as the coe, yoc, and doc generated during Dec doe to date conversion or the moy that we need in order to split a doy into a month and a dom or combine a month and a dom into a doy. The arrows in the flowchart represent equations adapted from chrono-Compatible Low-Level Date Algorithms.

Instead of converting Dec dates, we can expand them to view the information we need as part of the math-inspired notation of Dec. At its heart❤️, Dec is a simple system that uses only years and days for time measurement. Thanks to Dec expansion, Dec can work with other units. In this way, Dec expansion bridges the gap between the Dec and Gregorian calendars.

Next

After reading this article, you should be able to understand the examples in my filter, include, and script articles, but you may want to start with my Quarto article. To see the full extent of the benefits that Dec provides, I recommend that you continue through the Dec section of my site to the [time](/dec/time, [snap](/dec/snap, and span🌈articles. Dec has a lot more to offer than just dates!

%%{init: {'theme': 'default', 'themeVariables': { 'fontSize': '32px'}}}%%
flowchart LR
   A[Dec]-->B[date]-->C[time]-->D[snap]-->E[span]
   Z[  ]:::empty~~~F[Quarto]-->G[filter]-->H[include]
   classDef empty width:0px;
   click A "/dec"
   click B "/dec/date"
   click C "/dec/time"
   click D "/dec/snap"
   click E "/dec/span"
   click F "/quarto"
   click G "/quarto/filter"
   click H "/quarto/include"
Diagram 9

In addition my Dec and Quarto articles, many other articles on my site discuss Dec. Notably, my Jupyter article compares the code underlying Dec in several programming languages, my Reveal article features a presentation on Dec time measurement, and my Observable article describes how I demonstrate Dec in action with interactive and animated visualizations.

Thank you for your interest in Dec. You will find citation information for this article below. Please note that the original source of the algorithms underlying the conversion of Dec year+day dates and doe is Hinnant, Howard. 2021+184. “chrono-Compatible Low-Level Date Algorithms.” . https://howardhinnant.github.io/date_algorithms.html.

Cite

Of the bibliography file formats supported by Quarto, I recommend yaml. The yaml bibliography file shown below contains bibliographic records (metadata) about the article you are currently reading and the article entitled chrono-Compatible Low-Level Date Algorithms in which Howard Hinnant (2021+185) describes the algorithms underlying Dec dates.

ref.yml
references:
- id: hinnant2021date
  author:
    - family: Hinnant
      given: Howard
  title: [<code>chrono</code>]{.nocase}-Compatible Low-Level Date Algorithms
  url: https://howardhinnant.github.io/date_algorithms
  issued:
    literal: 2021+185
- id: laptev2026date
  author:
    - family: Laptev
      given: Martin
  title: Decalendar
  url: https://maptv.github.io/dec/date
  issued:
    literal: 2026+097

Quarto configuration files, such as _quarto.yml and _metadata.yml, are written in yaml. Quarto input files, including Quarto markdown, Jupyter notebook, markdown, and specially formatted script files, can start with a yaml header. Therefore, we could put the metadata above directly into a Quarto configuration or input file rather than into a bibliography file.

As an alternative to yaml, I suggest the BibTeX format. The BibTeX bibliography file below can be used by Quarto equally as well as the yaml bibliography file above. Regardless of the bibliography file format we choose, Quarto configuration and input files require that we store the path to our bibliography file, or our list of bibliography file paths, in yaml format.

ref.bib
@misc{hinnant2021date,
  author = "Howard Hinnant",
  title = "\texttt{chrono}-Compatible Low-Level Date Algorithms",
  url = "https://howardhinnant.github.io/date_algorithms",
  year = 2021+185
}
@misc{laptev2026date,
  author = "Martin Laptev",
  title = "Decalendar",
  url = "https://maptv.github.io/dec/date",
  year = 2026+097
}

In addition to storing metadata in a bibliography file, we can keep instructions regarding how to style citations and references in a Citation Style Language (csl) file. If we do not provide a csl file, Quarto will follow the Chicago Manual of Style when processing parenthetical citations: (Hinnant 2021+185), narrative citations: (2021+185), and references:

Hinnant, Howard. 2021+185. chrono-Compatible Low-Level Date Algorithms. https://howardhinnant.github.io/date_algorithms.html.

When provided with nature.csl, american-medical-association.csl, or a similar csl file, Quarto will produce superscript numeric citations, which look just like Quarto footnotes: 1. Unlike Quarto citations, Quarto footnotes do not require any additional files or configuration. A Quarto output file can have both a Footnotes and References section.

Glossary

  • a: arcbeat, a hundred thousandth of a circle, 0.0036 degrees, 0.216 arcminutes, 12.96 arcseconds
  • b: beat, centimilliday, a hundred thousandth of an day, 864 milliseconds
    • mb: millibeat, centimicroday, a thousandth of a beat, a hundred millionth of a day, 864 microseconds
  • bpc: a musical or heart beat per centiday, a tenth of a beat per milliday, 0.0694 beats per minute, 100 beats per day
  • bpm: a musical or heart beat per milliday, ten beats per centiday, 0.694 beats per minute, 1000 beats per day
  • bmi: body mass index, kilograins of body mass divided by height in zem squared (kg/z²)
  • c: taur, 𝜏r, 100000 kilozem, 40000 kilometers, nearly the circumference of the Earth, roughly the product of 𝜏 and the radius of the Earth, approximately the dividend of the surface area and the diameter of the Earth
    • mc: millitaur, m𝜏r, a thousandth of a taur, 100 kilozem, 40 kilometers
    • nc: nanotaur, n𝜏r, a thousandth of a taur, 100 millizem, 1 decizem, 4 centimeters
    • nc³: cubic nanotaur, n𝜏r³, 1 cubic decizem
  • d: day, a tenth of a decaday, a seventh of week, a fifth of a pentaday, 10 decidays, 24 hours, 100 centidays, 1000 millidays, 1440 minutes, 86400 seconds, 100000 beats, the inverse of a quotidie
    • dox: day of xún
    • dop: day of pentaday
    • dom: day of month
    • dow: day of week
    • doy: day of year, xún * 10 + dox
    • dd: deciday, a tenth of a day, 2.4 hours, 144 minutes
    • cd: centiday, a hundredth of a day, 0.24 hours, 14.4 minutes
    • md: milliday, a thousandth of a day, 1.44 minutes
    • cmd: centimilliday, a hundred thousandth of a day, 1 beat, 864 milliseconds
    • µd: microday, a millionth of a day, 86.4 milliseconds
    • nd: nanoday, a billionth of a day, 86.4 microseconds
  • °: degree, 1/360 turns, 180/𝜋 or 360/𝜏 radians
    • c°: compass degree
    • h°: hue degree
  • e: egg, 1000 grains, 2 ounces, 64 grams
  • ℓ: ell, cubit, 10/9 zem
  • f: foot, 0.75 zem, 75 millimeter
  • g: drop (gutta in Latin) or grain (granum in Latin), 64 microliters or 64 milligrams
    • kg: kilograin or kilodrop, 64 grams or 64 milliliters
    • Mg: megagrain or megadrop, 64 kilograms or 64 liters
  • h: a Dec season, represented by h, because Dec seasons, except for Season 3, are 1 hectoday, 10 decadays, or one hundred days long
  • hex: hexadecimal, base 16
  • hsl: hue saturation lightness
  • hsv: hue saturation value
  • i: inch, a sixteenth of a zem, 25 millimeter
  • k: keg, cubic zem, 64 liters, 1000 wine glasses, a million drops, half a barrel
  • kmph: kilometers per hour, thousands of meters per hour, 1 kmph = 0.6 mv
  • L: liter, 15625 drops, a cubic decimeter
    • mL: milliliter, a cubic centimeter, a thousandth of a liter, 15.625 drops
    • µL: microliter, a cubic millimeter, a millionth of a liter, 0.015625 drops
  • m: meridian, a full circle around the Earth moving North or South; used in the abbreviations a.m. (antemeridian) and p.m. (postmeridian); the letter “m” in meridian can be vertically flipped to get the letter “w” in wěi
    • dm: decimeridian, a tenth of a meridian
    • mm: millimeridian, a thousandth of a meridian
  • m²: square meter, 6.25 square zem
    • cm²: square centimeter, 6.25 square centizem
    • dm²: square decimeter, 6.25 square decizem
    • km²: square kilometer, 6.25 square kilozem
    • cm³: cubic centimeter, 1 millilter, a thousandth of a liter, 15.625 drops
  • p: pentaday, a group of five days, half a decaday
  • n: note, a specific frequency within an octave
  • o: octave, a two fold change in frequency
    • do: decioctave, a tenth of a two fold change in frequency
  • þ: perbeat, the inverse of a beat, 1/beat, once per beat, every beat, 100000 q; symbolized by thorn (þ), which looks like a combination of the letters “p” and “b”; not to be confused with a picobeat (pb)
    • Tþ: teraperbeat, 1012 perbeat, the inverse of a picobeat, 1/picobeat, once per picobeat, every picobeat
  • q: quotidie, the inverse of a day, a hundred thousandth of a perbeat; the letter “q” in quotidie can be flipped vertically to produce the letter “d” in day
  • r: compass rose, a full circle along the horizon, 360 compass degress
    • mr: compass millirose, a thousandth of a circle along the horizon, .36 compass degress
  • rad: radian, \(1\over\tau\) turns, \(360\over\tau\) degrees, \(1\over 2\pi\) turns, \(180\over\pi\) degrees
  • rgb: red green blue
  • s: second, 1/90 millidays, 0.9 beats, 1 Dec second = 0.96 SI seconds
  • SI: International System of Units
  • sol: speed of light, 647.55170928 kiloomegars, 299792458 meters per second
  • sos: speed of sound, 735.048 milliomegars, 340.3 meters per second
  • 𝜏: 2𝜋 or approximately 6.2831853
  • Tenet: ten equal temperament
    • Xet: Tenet
    • 12et: twelve equal temperance
  • tod: time of day
  • t: turn, 360 degrees, 𝜏 or 2𝜋 radians
    • ct: centiturn, a hundredth of a turn, 3.6 degrees, 𝜏/100 or 𝜋/50 radians
    • dt: deciturn, a tenth of a turn, 36 degrees, 𝜏/10 or 𝜋/5 radians
    • mt: milliturn, a thousandth of a turn, .36 degrees, 𝜏/1000 or 𝜋/500 r \(\pi\over 500\) radians
  • tzo: time zone offset
  • u: ounce (uncia in Latin), 500 grains, 32 grams, 500 drops, 32 milliliters
  • utc: Coordinated Universal Time
  • US: United States
  • v: omegar, ωr, 1041.6 miles per hour, 1.6 megameters per hour, 0.4629 kilometers per second, roughly 1.36 times the speed of sound
    • kv: kiloomegar, kωr, 1.6 gigameters per hour, 0.4629 megameters per second, approximately 0.1544% of the speed of light
    • mv: milliomegar, mωr, 1.0416 miles per hour, 1.6 kilometers per hour, 0.4629 meters per second, approximately 0.136% the speed of sound
  • w: wěi (纬), parallel, a measure of longitude; can be thought of as a measure of the width of a meridian on Earth; the letter “w” in wěi can be vertically flipped to get the letter “m” in meridian
    • dw: deciwěi, a tenth of a wěi (纬), a tenth of a parallel
    • mw: milliwěi, a thousanth of a wěi (纬), a thousanth of a parallel
  • x: xún (旬), decaday, a group of ten days, 2 pentadays, represented by x like the Roman numeral X
  • y: year
    • my: milliyear, a thousandth of a year
    • yoe: year of era, integer years since the Dec epoch
  • z: zem, zone equatorial meter, 4 decimeters, 16 inches
    • kz²: square kilozem, a million square zem, megahexamilliare, Mx, hexakilare, 16 hectares, 1600 ares, 40 acres, 0.16 square kilometers, 0.0625 square miles
    • kz: kilozem, 1000 zem, 400 meters, a quarter mile
    • z²: square zem, hexamilliare, 16 square decimeters, 1.7 square feet, 256 square inches
    • Dz²: square decazem, 1 hexadeciare, 16 square meters, 19.75 square yards, 100 square zem
    • z³: cubic zem, 1 keg, 64 liters, 1000 wine glasses, a million drops, half a barrel
    • dz³: cubic decizem, 1000 drops, 64 milliliters, 2 ounces, 1 wine glass
    • cz³: cubic centizem, 1 drop, 64 microliters
    • dz: decizem, a tenth of a zem, 4 centimeters
    • cz: centizem, a hundredth of a zem, 4 millimeters
    • mz: millizem, a thousandth of a zem, 0.4 millimeters
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References

Hinnant, Howard. 2021+185. chrono-Compatible Low-Level Date Algorithms. https://howardhinnant.github.io/date_algorithms.
Jukic, A.M., D.D. Baird, C.R. Weinberg, D.R. McConnaughey, and A.J. Wilcox. 2013+215. “Length of Human Pregnancy and Contributors to Its Natural Variation.” Human Reproduction 28 (10): 2848–55. https://doi.org/10.1093/humrep/det297.

Footnotes

  1. Hinnant, Howard. 2021+185. chrono-Compatible Low-Level Date Algorithms. https://howardhinnant.github.io/date_algorithms.html.↩︎

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