Declock

Introducing Declock, a timekeeping system that displays time in decimal days without the need for hours, minutes, or seconds.

0 Declock

My website provides many examples of the Quarto publishing and the Dec measurement systems in action. I leverage Quarto support for the Observable data analysis and visualization system to create animated and interactive graphics like the bar📊charts clocks🕓, solar☀️terminator map🗺, Earth🌍orbit diagram, and daylight area chart below.

Dec times are measured in fractional days, often using metric prefixes like deci, centi, or milli. The top, middle, and bottom bars indicate the decidays, millidays, and centimillidays, respectively, of the time since the start, , or until the end, , of the day in the Zone , the Dec time zone selected by the red⭕️circle on the map beneath the bars.

To rotate the globe🌐in the Earth🌏orbit diagram, drag the red circle horizontally↔︎️or slide the red🔴dot on the daylight area chart vertically↕. The red circle’s vertical position determines the yearly daylight pattern visualized by the area chart. Shift the vertical red—line on the area chart horizontally to move the globe along the ellipse of the Earth🌎orbit.

Bar chart clocks

Longitude latitude map

Daylight area chart

1 Annual day aggregate (ada)

The red line indicates a “day of year” (doy): , which identifies a day in a year like a Gregorian calendar month and “day of month” (dom). The red dot denotes a “time of day” (tod): , which specifies a point in a day like an “hour minute second” (hms) triplet. Together, a doy and tod can form an “annual day aggregate” (ada): + ÷ 1000 = .

\[ \begin{split} \text{doy} &= \lfloor\text{ada}\rfloor \\ &= \text{ada} - \text{tod} \end{split} \tag{1}\]

As their names suggest, doy and ada are measured in days. The measurement unit of a tod can be a day or a submultiple of a day with a metric prefix. By changing how a decimal tod is measured, we can shift its decimal separator or turn it into an integer. The tod along the y-axis of the area chart are integers because they have three digits and are measured in millidays.

2 Epochal day aggregate (eda)

To obtain a tod from an ada, we can keep the remainder after dividing by one to isolate the decimal part of the quotient: mod 1 = 0.. We can use this same approach to separate a tod from an “epochal day aggregate” (eda): mod 1 = 0.. The current eda tells us how many days have passed since the Dec epoch: Year 0 Day 0 Tod 0.

\[\begin{split} \text{tod} &= \text{ada} \bmod 1 \\ &= \text{eda} \bmod 1 \end{split} \tag{2}\]

UNIX time equation

Similarly, UNIX time tallies the seconds since the UNIX epoch, which is exactly 719468 days after the Dec epoch. To get the tod in Zone 0, the Dec time zone that is in between the two leftmost vertical lines on the map, we can divide UNIX time by the number of seconds in a day and then keep the remainder after dividing the resulting days by 1 as in Equation 3.

\[\text{tod} = \text{unix} \div 86400 \bmod 1 \tag{3}\]

Julian date equation

Julian dates track the days since the beginning of the Julian period and thus are akin to eda. We can produce a Zone 5 tod from a Julian date simply by keeping the remainder after dividing by one. If we want a Zone 0 tod instead, we should add 5 decidays to the Julian date, divide the sum by one, and then keep the remainder as in Equation 4 below.

\[\text{tod} = (\text{julian} + 0.5) \bmod 1 \tag{4}\]

3 Hour minute second (hms)

We can also obtain a Zone 0 tod from a Coordinated Universal Time (UTC) hms triplet by summing its components after converting them to fractional days, as shown in the equation below. The computer programming code in the tabset panel beneath the equation compares tod derived from UTC and UNIX time as Quarto was rendering this webpage.

\[\text{tod} = \dfrac{\text{hour}}{24} + \dfrac{\text{minute}}{1440} + \dfrac{\text{second}}{86400} \tag{5}\]

Julia

using Dates
hms = now(UTC)

2026-10-30T14:47:26.076

hour(hms) / 24 +
minute(hms) / 1440 +
second(hms) / 86400 +
millisecond(hms) / 864e5

0.6162740277777777

datetime2unix(hms) / 86400 % 1

0.616274027775944

Observable JavaScript

hms = new Date();
hms.getUTCHours() / 24 +
hms.getUTCMinutes() / 1440 +
hms.getUTCSeconds() / 86400 +
hms.getUTCMilliseconds() / 864e5;

hms.getTime() / 864e5 % 1;

Python

from datetime import datetime, timezone
hms = datetime.now(timezone.utc)
hms.hour / 24 + \
hms.minute / 1440 + \
hms.second / 86400 + \
hms.microsecond / 864e8

0.6162818051388889

hms.timestamp() / 86400 % 1

0.6162818051379872

R

hms <- as.POSIXlt(Sys.time(), tz = "UTC")
hms$hour / 24 +
hms$min / 1440 +
hms$sec / 86400

[1] 0.616282

(as.numeric(as.POSIXct(hms)) / 86400) %% 1

[1] 0.616282

Equations 6 to 10 below convert UNIX time or a Zone 0 tod into the three components of an hms triplet: the “hour of day” (hod), “minute of hour” (moh), and “second of minute” (som), using a “daily second aggregate” (dsa) and “hourly second aggregate” (hsa). While both count seconds, dsa start at midnight (Tod 0) and hsa begin at the top of the hour.

\[\begin{split} \text{dsa} & = \text{tod} \ast 86400 \\ & = \text{unix} \bmod 86400 \end{split} \tag{6}\] \[ \text{hsa} = \text{dsa} \bmod 3600 \tag{7}\] \[ \text{hod} = \lfloor \text{dsa} \div 3600 \rfloor \tag{8}\] \[ \text{moh} = \lfloor \text{hsa} \div 60 \rfloor \tag{9}\] \[ \text{som} = \lfloor \text{hsa} \bmod 60 \rfloor \tag{10}\]

Julia

using Dates
dsa = datetime2unix(now(UTC)) / 86400 % 1 * 86400

53246.80799993221

hsa = dsa % 3600

2846.807999932207

map(x -> floor(Int, x), (dsa / 3600, hsa / 60, hsa % 60))

(14, 47, 26)

Observable JavaScript

dsa = Date.now() / 864e5 % 1 * 86400;
hsa = dsa % 3600;
[dsa / 3600, hsa / 60, hsa % 60].map(Math.floor)

Python

from datetime import datetime, timezone
dsa = datetime.now(timezone.utc).timestamp() / 86400 % 1 * 86400
hsa = dsa % 3600
tuple(map(int, [dsa // 3600, hsa // 60, hsa % 60 // 1]))

(14, 47, 26)

R

dsa <- (as.numeric(as.POSIXct(Sys.time())) / 86400) %% 1 * 86400
hsa <- dsa %% 3600
sapply(c(dsa %/% 3600, hsa %/% 60, hsa %% 60), as.integer)

[1] 14 47 26

4 Universal time offset (uto)

The Global Positioning System, BeiDou, and Galileo global navigation satellite systems along with most — if not all — programming languages do not account for leap seconds, which appears to be for the best given that leap seconds can cause problems and will be abolished no later than 2027+081 according to the International Bureau of Weights and Measures.

On or before the deadline to abolish leap seconds, the tolerance for disagreement between UTC and Universal Time (UT) will increase from 1 Dec second (s) to 3750 s. Since 1972+122, only 27 leap seconds have been inserted, which is 0.75% of 3750 s, and at this rate it will take over five millennia for the disparity between UT and UTC to reach 3750 s.

Instead of leap seconds, Dec matches UT using a “universal time offset” (uto). With the leap second insertion dates provided by the International Earth Rotation and Reference Systems Service, we can approximate the uto that yields UT when added to the Zone 0 tod on the Dec date chosen by the range🎚️inputs below: ÷ 8640 = ÷ 90000 = .

5 Second interval difference (sid)

The equation above divides leap seconds by the number of International System of Units (SI) seconds (sis) or Dec deciseconds (ds) in a deciday (dd): 8640 sis = 90000 ds = 1 dd. To avoid repeating decimals when turning sis into ds, we can put a Dec quadravigesimal (base24) digit in place of the ones digit and decimal part of the ds: = × 10 + ÷ 2.4.

Known as a “second interval difference” (sid), the base24 (b24) character at the end of is a letter of the Greek alphabet. To display sis in ds, we can concatenate the integer part of the s we obtain from Equation 11 with the b24 representation of the sid we get from Equation 12. The table beneath these equations lists base10 (b10) and b24 equivalents.

\[ \text{s} = \dfrac{\text{ds}}{10} = \dfrac{\text{sis}}{0.96} \tag{11}\] \[ \begin{split} \text{sid} &= \text{sis} \bmod 24\\ &= (\text{s} - \text{sis}) \ast 24 \bmod 24 \end{split} \tag{12}\]

Base24

b10	b24	b10	b24	b10	b24	b10	b24
0	α	6	η	12	ν	18	τ
1	β	7	θ	13	ξ	19	υ
2	γ	8	ι	14	ο	20	φ
3	δ	9	κ	15	π	21	χ
4	ε	10	λ	16	ρ	22	ψ
5	ζ	11	μ	17	σ	23	ω

b10	b24	b10	b24
0	α	12	ν
1	β	13	ξ
2	γ	14	ο
3	δ	15	π
4	ε	16	ρ
5	ζ	17	σ
6	η	18	τ
7	θ	19	υ
8	ι	20	φ
9	κ	21	χ
10	λ	22	ψ
11	μ	23	ω

b10	b24
0	α
1	β
2	γ
3	δ
4	ε
5	ζ
6	η
7	θ
8	ι
9	κ
10	λ
11	μ
12	ν
13	ξ
14	ο
15	π
16	ρ
17	σ
18	τ
19	υ
20	φ
21	χ
22	ψ
23	ω

Table 1

Base9

Whereas b24 is like glue that allows us to combine s and sid, Dec nonary (base9) enables us to append s to a tod. The table below shows the astronomical symbols used by Dec for base9 (b9). When we replace the “decimilliday” (dmd) digit in a b10 tod with a b9 digit, the b9 digit counts tens of s instead of dmd and the next digit tracks s instead of “centimillidays” (cmd).

b10	b9	b10	b9	b10	b9
0	☉	3	♁	6	♄
1	☿	4	♂	7	♅
2	♀	5	♃	8	♆

b10	b9
0	☉
1	☿
2	♀
3	♁
4	♂
5	♃
6	♄
7	♅
8	♆

Table 2

A tod with four b10 digits, one b9, and one b24 digit can match the time represented by any hms triplet, because the product of the bases of its digits is evenly divisible by the number of sis in a day: 10000 × 9 × 24 ÷ 86400 = 216000 ÷ 86400 = 25. For hod, one b24 digit is sufficient. The hms triplet selected by the range inputs below corresponds to Hod and Tod .

In Table 3 below, the quotidie (q) column shows how many multiples of the unit in each row can fit into one day and the md column uses endings called Dec terminators to transform the fraction of a day that each unit represents from a repeating decimal into a terminating decimal. The b9 digit in the terminators in Table 3 represents 6 decaseconds (Ds): 6 ÷ 9 = 0.♄.

Time units	q	md
hour	24	41.♄
minute	1440	0.♄η
sis	86400	0.0☿β
s	90000	0.0☿

Table 3

6 Coordinated Universal Time (UTC)

Rather than use a terminator as in Table 3, Dec expresses one hour in dd as a single b24 digit: 0.41♄ = β. If we add β dd to the Zone 0 tod, we get the tod associated with the UTC+1 offset: 0. + β ÷ 10 = . Currently, the Zone 0 Dec time is −0 and the UTC+1 Dec time is −β. Dec times consist of a tod, a minus sign (−), and an offset.

There are 12 UTC offsets that become a terminating decimals after being converted from hours to days and therefore could be displayed in either b10 or b24. Although it has only ten time zones, Dec can express any UTC offset using one or two non-negative b24 digits. There are 27 UTC offsets that require only one b24 digit and 11 that need two b24 digits.

Out of the 38 UTC offsets, 14 are negative (<UTC) and 24 are non-negative (≥UTC). The UTC time zone with the most negative offset is completely uninhabited. The bar chart below visualizes the population of each UTC time zone based on 2020 Socioeconomic Data and Applications Center data. The vast majority of all people live in UTC time zones with ≥UTC offsets.

Negative UTC offsets only exist in the Americas and islands in the Atlantic and Pacific Oceans. Therefore, the bar chart above is essentially comparing the populations of the Americas and the rest of the world. According to 2021 United Nations Department of Economic and Social Affairs data, roughly 1 billion out of a total of almost 8 billion people live in the Americas.

Dec forbids the association of dates with negative offsets. Therefore, Dec will add ten decidays to any negative offset and add one day to the date associated with that offset. Negative offset handling has no effect on the tod or the conversion of an hms triplet to a tod and is typically only necessary after a “year month day” (ymd) triplet is converted into a Dec date.

If we do know whether Day 364 belongs to a common or leap year, adding one day to it yields either Day 0 or Day 365 and this uncertainty is expressed as a multiplication sign over the last digit of 365̽. The sign hints that 365̽ is short for 365×leap, where 365̽ is either 365 if leap is 1 or 0 if leap is 0. We can find leap by passing any year (y) to Equation 13 below.

\[ \text{leap}=\begin{cases} 1&{\begin{aligned} &\text{ if } (\text{y} + 1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} \ \ \ \ 4=0\\ &\href{https://en.wikipedia.org/wiki/Logical_conjunction}{\land}(\text{y} + 1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 100\neq0\\ &\href{https://en.wikipedia.org/wiki/Logical_disjunction}{\lor}(\text{y} + 1)\href{https://en.wikipedia.org/wiki/Modulo#:~:text=returns%20the%20remainder%20or%20signed%20remainder%20of%20a%20division}{\bmod} 400=0\end{aligned}}\\\\ 0&{\text{ otherwise}}\end{cases} \tag{13}\]

7 Day of week (dow)

Adding one day to the doy has no effect on the tod, but it increments the dom and “day of week” (dow). Dec meets the expectations of Gregorian calendar users who live in UTC time zones with negative offsets by essentially deriving the previous dom or dow from the current doy if a location in the Americas is provided without an offset or with a negative offset.

8 Time zone offset (tzo)

In Dec, “time zone offsets” (tzo) are derived solely from longitude. Dec measures longitude in wěi (w), arcbeats (a), or submultiples thereof. If we travel East or West until we reach our starting point, we will cover 1 w or 100000 (10⁵) a. A tzo is essentially a deciwěi (dw) longitude that became a single-digit b10 integer via truncation or flooring: ⌊ ÷ 100⌋ = .

The Equator is approximately (~) 1 taur (c) or ~10⁵ kilozem (kz) long. Near the Equator, a longitude that has a precision of 0.1 microwěi (µw) or 10 milliarcbeats (ma) is accurate to within ~0.1 microtaur (µc), ~10 “zone equatorial meters” (zem), or 4 meters, which is about the width of a U-shaped living room layout or the length of a typical subcompact car.

For precise geopositioning, we may want to express geographic coordinates in ma, but we are unlikely to need s or cmd when displaying the solar time or estimates of what the tod will be when the Sun rises, reaches its highest point of the day, or sets. By default, Dec uses md precision to display a solar time and the “solar time offset” (sto) used to produce it.

9 Solar time offset (sto)

In Dec, any offset can be called a sto or a uto, because there will always be a longitude at which the sum of the given offset and the Zone 0 tod will yield solar time or UT. In contrast, Dec has only ten tzo, one for each of its ten time zones. Dec translates UTC to Zone 0 and UTC−12 or UTC+12 to Zone 5, but does not categorize any other UTC offset as a tzo.

Dec permits any non-negative offset that is less than one day and can be written with a finite number of b9, b10, and b24 digits. When a tod or an offset does not have a decimal separator, its measurement unit is a submultiple of a day equal to 10^-til days, where til is its “time integer length”, its level of precision, or its digit count including any leading zeros.

Stockholm, the capital city of Sweden🇸🇪, is located at ~1 dw, almost exactly on the border between Zones 0 and 1. A longitude of 1 dw corresponds to a sto of 05. The til of 05 is 2 and therefore its measurement unit is the centiday (cd), which is 10^-2 days. We can use a multiplicative conversion factor of 10^-2 to convert cd to d: 5 × 10^-2 = 0.05.

Outside of Dec, longitudes are often provided in Greenwich degrees (g°). Equation 14 converts g° to w and Equation 15 does the opposite. To obtain a sto, we can pass w to Equation 16 or g° to Equation 17 below. As shown in Equation 18 below, we can sum an sto with the Zone 0 tod to get a yearly average of solar time called “mean solar time” (mst).

\[\text{w} = (\text{g}^\circ + 378) \div 360 \bmod 1 \tag{14}\]

\[\text{g}^\circ = (\text{w} + 0.45) \bmod 1 \times 360 - 180 \tag{15}\]

\[\text{sto} = (0.95 + \text{w}) \bmod 1 \tag{16}\]

\[\text{sto} = (\text{g}^\circ + 360) \div 360 \bmod 1 \tag{17}\]

\[\text{mst} = (\text{tod} + \text{sto}) \bmod 1 \tag{18}\]

10 Equation of time (eot)

If we want to tailor our solar time calculations to a specific ada instead of an average for the entire year, we can calculate “apparent solar time” (ast) by summing mst with the “equation of time” (eot), as shown in Equation 19. Obtaining the eot requires a “time of year” (toy), which is an ada that was converted from days to years as displayed in Equation 20.

\[\text{ast} = (0.95 + \text{tod} + \text{w} + \text{eot}) \bmod 1 \tag{19}\]

\[\text{toy} = \text{ada} \div \text{syl} \tag{20}\]

\[\text{syl} = \text{leap} + 365 \tag{21}\]

To obtain a toy, we divide an ada by a “solar year length” (syl), which is the number of days in the year. Dec passes the toy directly from Equation 20 to the eot function in Equation 22 without first multiplying it by 2𝜋 or 𝜏 to convert it from turns (t) to radians (rad) because the sin and cos trigonometric functions used by Dec are designed for t instead of rad.

\[\begin{split} \text{eot} & = \beta_0 \\ & + \beta_1 \ast \text{cos(toy)} \\ & + \beta_2 \ast \text{sin(toy)} \\ & + \beta_3 \ast \text{cos(2} \ast \text{toy)} \\ & + \beta_4 \ast \text{sin(2} \ast \text{toy)} \end{split} \tag{22}\]

By comparing the sincost and sincos functions, the code below demonstrates that using t rather than rad can yield more accurate results. Whereas sincost expects to receive t and is always correct, sincos assumes that it will get rad and generates the slightly incorrect E notation values below because it cannot produce zero unless its input is zero.

Julia

sincost(x) = sincospi(2 * x)

sincost (generic function with 1 method)

sincost(0.25)

(1.0, 0.0)

sincost(0.5)

(0.0, -1.0)

sincost(0.75)

(-1.0, 0.0)

sincost(1)

(0.0, 1.0)

sincos(0.25 * 2 * pi)

(1.0, 6.123233995736766e-17)

sincos(0.5 * 2 * pi)

(1.2246467991473532e-16, -1.0)

sincos(0.75 * 2 * pi)

(-1.0, -1.8369701987210297e-16)

sincos(1 * 2 * pi)

(-2.4492935982947064e-16, 1.0)

Observable JavaScript

sincospi = require(
  'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-sincospi@umd/browser.js'
  )
function sincost(x) {
  return sincospi(2 * x);
}
function sincos(x) {
  return [Math.sin(x), Math.cos(x)]
}
sincost(0.25)

sincost(0.5)

sincost(0.75)

sincost(1)

sincos(0.25 * 2 * Math.PI)

sincos(0.5 * 2 * Math.PI)

sincos(0.75 * 2 * Math.PI)

sincos(1 * 2 * Math.PI)

Python

from math import cos, sin, pi, cospi, sinpi

def sincost(x):
    x *= 2
    return sinpi(x), cospi(x)

def sincos(x):
    return sin(x), cos(x)

sincost(0.25)

(1.0, 0.0)

sincost(0.5)

(0.0, -1.0)

sincost(0.75)

(-1.0, 0.0)

sincost(1)

(0.0, 1.0)

sincos(0.25 * 2 * pi)

(1.0, 6.123233995736766e-17)

sincos(0.5 * 2 * pi)

(1.2246467991473532e-16, -1.0)

sincos(0.75 * 2 * pi)

(-1.0, -1.8369701987210297e-16)

sincos(1 * 2 * pi)

(-2.4492935982947064e-16, 1.0)

R

sincost <- function(x) {
  x <- 2 * x
  c(sinpi(x), cospi(x))
}

sincos <- function(x) {
  c(sin(x), cos(x))
}

sincost(0.25)

[1] 1 0

sincost(0.5)

[1]  0 -1

sincost(0.75)

[1] -1  0

sincost(1)

[1] 0 1

sincos(0.25 * 2 * pi)

[1] 1.000000e+00 6.123234e-17

sincos(0.5 * 2 * pi)

[1]  1.224647e-16 -1.000000e+00

sincos(0.75 * 2 * pi)

[1] -1.00000e+00 -1.83697e-16

sincos(1 * 2 * pi)

[1] -2.449294e-16  1.000000e+00

The values below are the coefficients of a harmonic model, adapted from the National Oceanic and Atmospheric Administration (NOAA) General Solar Position Calculations. Before fitting the model to NOAA yearly solar data, Dec divides eot values by 1.44 to convert minutes to md, converts ymd triplet dates and hms triplet times into ada, and sorts by ada.

{0: 0.0011114386869002235,
 1: -4.155124400404918,
 2: -2.9710873225303756,
 3: -4.635570414590801,
 4: 5.116176940364264}

Source: Bonus examples

In the line📈chart above, the eot values range from -9.8 to 11.4 md and the range of doy, -365 to 365, represents a common year followed by a leap year. The absolute value of the eot reaches about one centiday (cd) on Days 245 and 349, but passes through zero on Days 45, 103, 184, and 299. Therefore, mst and ast are equivalent four times per year.

Apart from transforming an mst into an ast, we can also use eot values to more accurately estimate the tod of solar noon, sunrise, and sunset. Equation 23 below creates a solar noon tod by keeping only the decimal part of the result of subtracting a w longitude and an eot value that is measured in days from the sum of 9.55 and a tzo that is measured in days.

\[\text{solarnoon} = \left(9.55 + \text{tzo} - \text{w} - \text{eot}\right) \bmod 1 \tag{23}\]

11 Cambridge and Cambridge

To compare the solar noon tod in two cities, we can plug the longitude of each city into Equation 23 above. If all of the variables other than longitude in Equation 23 are set to zero, the resulting solar noon tod is almost five decidays for Cambridge, England in the United Kingdom and nearly seven decidays for Cambridge, Massachusetts in the United States.

\[4.99635 = (9.55 - 0.050365) \bmod 1 \ast 10 \tag{24}\]

\[6.97517 = (9.55 - 0.852484) \bmod 1 \ast 10 \tag{25}\]

The two homonymous cities are about two dw apart and therefore will always differ by around two decidays in solar time, regardless of what time zone we use as our frame of reference. England is in Zone 0 and Massachusetts is in Zone 8. From the perspective of Zone 8, solar noon is roughly Tod 5 in Cambridge, MA and about Tod 3 in Cambridge, UK.

\[2.99635 = (9.55 + 0.8 - 0.050365) \bmod 1 \ast 10 \tag{26}\]

\[4.97517 = (9.55 + 0.8 - 0.852484) \bmod 1 \ast 10 \tag{27}\]

12 Light arc whole (law)

The path in the sky that the Sun appears to follow from a sunrise to a sunset is a “light arc whole” (law). Solar noon, the midpoint of a law, is halfway between sunrise and sunset. A law can be evenly split into two “light arc halves” (lah). Whereas tod are time points, law and lah are durations. The difference between a solar noon tod and a lah is a sunrise tod.

\[\text{law} = \text{lah} \ast 2 \tag{28}\]

\[\begin{split} \text{sunrise} & = \text{solarmidnight} + \text{nah} \\ & = \text{solarnoon} - \text{lah} \\ & = \text{sunset} - \text{law} \end{split} \tag{29}\]

\[\begin{split} \text{sunset} & = \text{solarmidnight} + \text{nah} + \text{law} \\ & = \text{sunrise} + \text{law} \\ & = \text{solarnoon} + \text{lah} \\ \end{split} \tag{30}\]

Night arc whole (naw)

The Sun continues its path after it disappears below the horizon, moving from sunset to sunrise along a “night arc whole” (naw). Like conjugate angles, a law and a naw form a full circle that represents one day. Likewise, a lah and a “night arc half” (nah) are like two supplementary angles that form a semicircle from solar noon to the naw midpoint: solar midnight.

\[\text{naw} = \text{nah} \ast 2 \tag{31}\]

\[\begin{split} \text{day} & = \text{law} + \text{naw} \\ & = (\text{lah} + \text{nah}) \ast 2 \end{split} \tag{32}\]

\[\begin{split} \text{solarmidnight} &= \text{sunrise} - \text{nah} \\ &= \text{solarnoon} - \text{lah} - \text{nah} \\ &= \text{sunset} - \text{law} - \text{nah} \end{split} \tag{33}\]

The range input below controls the yellow law and the blue naw in the diagram beneath it. At the Equator, the sun always rises about a quarter turn from North and sets around three quarter turns from North, resulting in a law of roughly half a day: 75% − 25% = 50%. If the law is zero, a polar night occurs. A day without a naw is called a polar day.

The numbers in the diagram above are “solar azimuth angle” (saa) milliturns (mt). At solar noon, the saa is 0 mt in the Southern Hemisphere and 500 mt in the Northern Hemisphere. The radio📻input above controls whether the midpoint of the law is 0 or 500 mt. To measure the saa during the daytime, point a compass🧭at the spot on the horizon below the Sun.

During an equinox at the Equator, the Sun goes from East to West directly overhead, the saa flips like a switch from 0.25 t to 0.75 t at solar noon, the “solar elevation angle” (sea) peaks at 0.25 t, and ast is 0.5 − sin(saa) × (0.25 − sea). To obtain the sea, we can use the height (h) of a vertical rod on a level surface and the length (L) of its shadow: sea = atan2(h, L).

In addition to determining the sea, the atan2 function can also be used to calculate ast, as shown in Equation 34, where j is a latitude measured in turns called jīng (j). In Dec, inverse trigonometric functions like atan2 yield t as opposed to rad. At the Equator (j = 0), the latitude is zero and Equation 34 simplifies to ast = 0.5 + atan2(-sin(saa)cos(sea), sin(sea)).

\[ \text{ast} = 0.5 + \text{atan2}\!\!\left( \begin{aligned} & \text{-sin(saa)cos(sea),} \\ & \text{sin(sea)cos(j) } - \\ & \text{cos(saa)cos(sea)sin(j)} \end{aligned} \right) \tag{34}\]

\[ \text{ast} = \begin{cases} \text{saa} & \text{j} = 0.25 \\ \text{atan2(-sin(saa)cos(sea), sin(sea))} & \text{j} = 0 \\ (1.5 - \text{saa}) \bmod 1 & \text{j} = -0.25 \end{cases} \tag{35}\]

At the North Pole (j = 0.25), the latitude is 0.25 j and Equation 34 simplifies to ast = saa. At the South Pole (j = -0.25), the latitude is -0.25 j and Equation 34 simplifies to ast = 0.5 − saa before solar noon or ast = 1.5 − saa after solar noon. In between the Equator and the South or North Pole (-0.25 < j < 0 ∨ 0 < j < 0.25), we should use Equation 19, 34, or 36.

\[ \text{ast} = 0.5 + \text{arccos}\!\!\left(\dfrac{\text{sin(sea)} - \text{sin(j)sin(sda)}}{\text{cos(j)cos(sda)}}\right) \ast \begin{cases} \text{-}1 & 0 \le \text{saa} < 0.5 \\ 1 & 0.5 \le \text{saa} < 1 \end{cases} \tag{36}\]

Unlike Equation 34, Equation 36 requires a “solar declination angle” (sda). Whereas latitude is the angle from the Equator in the geographic coordinate system, declination is the angle from the celestial equator in the equatorial coordinate system. For simplicity, we can use Equation 22 for both the eot and sda. The line plot below shows sda over the course of a year.

In the line plot above, zero crossings are equinoxes and stationary points are solstices. From the northward to the southward equinox, the sda is positive and every latitude in the Northern Hemisphere has a law that is longer than the law at the Equator, whereas every latitude in the Southern Hemisphere has a shorter one. This pattern is reversed when the sda is negative.

The chart below splits the Socioeconomic Data and Applications Center population data into two bars representing the Northern and Southern Hemispheres and then further divides each bar by UTC offset. Nearly 7 billion out of a total of almost 8 billion people live in the Northern Hemisphere and thus experience the longest law of the year at the northern solstice.

The top range input below picks the doy that we will convert into a toy and then pass to our sda model. The two range inputs beneath the doy input choose the geographic coordinates needed to find the sunrise and sunset tod. While the sunrise and sunset tod rely on the doy and both geographic coordinates, the law depends on the doy and latitude but not on longitude. The default latitude, 118 mj, is that of Cambridge, Massachusetts. At the Equator, the law is about 0.505 d all year long, so the doy input changes the law only away from the Equator: − = .

The equation above can be summarized as sunset − sunrise = law or minuend − subtrahend = difference. If we replace the minuend (sunset tod) with the current tod and the subtrahend (sunrise tod) with the sunset tod, the difference will no longer be the law but rather the time until sunset if it is negative or since sunset if it is positive: = .

In general, the subtrahend is the time of an event such as a rocket🚀launch and the difference is the negative remaining time if the minuend is less than the subtrahend or the positive elapsed time if the minuend is greater than the subtrahend. When the minuend is the current time, the difference acts as a “T minus (T−) countdown” or “T plus (T+) countup”.

\[\begin{cases}\text{difference}<0&{\text{if } \text{minuend} < \text{subtrahend};}\\\text{difference}=0&{\text{if minuend} = \text{subtrahend};}\\\text{difference}>0&{\text{if minuend}>\text{subtrahend.}}\end{cases} \tag{37}\]

Dec uses the term “minuend expansion” to describe the replacement of a minuend with a subtrahend and a difference. We can replace the current tod with the sum of either zero and a “positive interval tod” (pit), which is the time since the start of the current day (the previous midnight), or 1 day (d) and a “negative interval tod” (nit), which is the negative of the time until the start of the next day (the next midnight), as shown in Equation 38 below. At Tod 7, the tod and pit are both 0.7 d and the nit is -0.3 d: 0.7 = 0 + 0.7 = 1 + (-0.3).

Likewise, a “positive interval ada” (pia) is the time since the start of the current year and a “negative interval ada” (nia) is the time until the start of the next year. A pia equals the current ada, and a nia is the pia minus the syl, just as a “negative integer doy” (nid) is a “positive integer doy” (pid) minus the syl. Equation 39 applies minuend expansion to the ada.

\[\text{tod} = 0 + \text{pit} = 1 + \text{nit} \tag{38}\]

\[\text{ada} = 0 + \text{pia} = \text{syl} + \text{nia} \tag{39}\]

Dec is not the first system to use decimal time. During the French Revolution, decimal time was mandatory for French public records for 197 days (d), from 1794+205 to 1795+037. On 1998+247, the Swatch corporation announced Swatch Internet Time, which divides the day into 1000 “.beats”, only has one time zone, and is displayed by Dec as tod−β.

Declock measures time in fractional days. A single decimal tod replaces the three parts of an hms triplet and can be shown at any precision by changing its metric prefix. With one b9 digit and one b24 digit after four b10 digits, a tod can match any hms triplet exactly, and a uto keeps Dec in step with ut without leap seconds. Dec derives its ten tzo from longitude and never uses negative offsets, but it lets anyone write any sto after a tod without forcing them to change it. With the eot and sda, we can turn a Zone 0 tod into solar time and estimate the tod of sunrise, solar noon, and sunset. Finally, minuend expansion turns any tod into a countdown or countup relative to an event, such as midnight, sunset, or the start of the year.

Next

The next article in the Dec section of my site shows how we can combine Dec dates and times into Dec snaps🫰, which are analogous to the combined date and time representations in the ISO 8601 international standard for dates and times. The final article in the Dec section demonstrates how Dec dates, times, and snaps🫰can be paired up to express time intervals called Dec spans🌈.

Please spread the good word about Dec using the citation information at the bottom of this article. You may also want to cite the Observable notebooks that I adapted into the clock🕓, bar📊chart, map🗺️, and daylight☀️plot visualizations in this article or the 2014 blog post which proposed a system of 20 decimal time zones, each 5 centidays wide, based on the Greenwich Meridian:

• Pearson, Tom. 2013+124. “Simple D3 clock.” +. https://observablehq.com/@d3/simple-clock.
• Heyen, Frank. 2021+246. “BarChart Clock.” +. https://observablehq.com/@fheyen/barchart-clock.
• Johnson, Ian. 2021+090. “Draggable World Map Coordinates Input.” +. https://observablehq.com/@enjalot/draggable-world-map-coordinates-input.
• Bridges, Dan. 2021+311. “Visualizing Seasonal Daylight.” +. https://observablehq.com/@dbridges/visualizing-seasonal-daylight.
• Clements, John. 2014+091, “Decimal Time Zones.” +. https://www.brinckerhoff.org/blog/2014/05/31/decimal-time-zones.

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: laptev2026time
  author:
    - family: Laptev
      given: Martin
  title: Declock
  url: https://maptv.github.io/dec/time
  issued:
    literal: 2026+242

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{laptev2026time,
  author = "Martin Laptev",
  title = "Declock",
  url = "https://maptv.github.io/dec/time",
  year = 2026+242
}

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
  • ma: milliarcbeat, a thousandth of an arcbeat, a hundred millionth of a circle, 12.96 milliarcseconds
• b: beat, centimilliday, a hundred thousandth of a 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, 0.069♂ beats per minute, 100 beats per day
• bpm: a musical or heart beat per milliday, ten beats per centiday, 0.69♂ beats per minute, 1000 beats per day
• BMI: body mass index, kilograins of body mass divided by height in zem squared (kg/z²)
• b9: base9, nonary, 9 digits written as the astronomical symbols of the Sun and the eight planets, ☉ ☿ ♀ ♁ ♂ ♃ ♄ ♅ ♆; the first digit after a tod with Dec seconds appended counts tens of Dec seconds
• b10: base10, decimal, the digits 0 to 9
• b24: base24, 24 digits written as the lowercase Greek letters α to ω; counts sid after a number of Dec seconds or hours after a UTC offset
• 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
  • gc: gigataur, a billion taur
  • kc: kilotaur, a thousand taur, 10⁸ kilozem
  • mc: millitaur, m𝜏r, a thousandth of a taur, 100 kilozem, 40 kilometers
  • µc: microtaur, µ𝜏r, a millionth of a taur, 100 zem, 40 meters
  • nc: nanotaur, n𝜏r, a billionth 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 a 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
  • dom_♀: Decyther day of month, 0 to 28 in Venusian short months or 0 to 29 in Venusian long months
  • dow: day of week
  • doy: day of year, xún * 10 + dox
  • pid: positive integer day of year, days elapsed since the beginning of the year, 0 to 364 or 365
  • nid: negative integer day of year, pid − syl, -365 or -366 to -1, the negative of the days left in the year
  • mid: mixed integer day of year, an nid rewritten with negative leading digits (marked by a vinculum) and a positive last digit or two
  • doe: day of era, days since the Dec epoch
  • eda: epochal day aggregate, doe + tod, days since the Dec epoch including the time of day
  • edd: eda difference difference, an eda minus a difference, used to recover the year and ada of a past event
  • ada: annual day aggregate, doy + tod
  • pia: positive interval ada, the time since the start of the current year, equal to the ada
  • nia: negative interval ada, pia − syl, the negative of the time until the start of the next year
  • doc: day of cycle, days since the start of the current 400-year Gregorian cycle, 0 to 146096
  • doc_♀: Decyther day of cycle, days since the start of the current 467-day Venusian cycle, doe mod 467, 0 to 466
  • doh: day of hectoday, doy mod 100, the percent of a hectoday that has elapsed
  • dob: day of bimester, 0 to 58 in common bimesters or 0 to 59 in leap bimesters
  • dob_♀: Decyther day of bimester, 0 to 57 in antes and the post of Qoc 2 or 0 to 58 in the other posts
  • dot: day of tricennium, days since the start of the current 30-year lunar cycle, 0 to 10630
  • doq: day of quadrimester, days since the beginning of a Decyther quadrimester, 0 to 115 or 116
  • boq: beginning of quadrimester, the pid of Doq 0, pid − doq
  • qoc: quadrimester of cycle, zero-based index of a Decyther quadrimester, 0 to 3
  • bob: beginning of bimester, the pid of Dob 0, pid − dob
  • bom: beginning of month, the pid of the day before the first day of the month, pid − dom
  • moc: month of cycle, zero-based index of a Venusian month, 0 to 15
  • bow: beginning of week, the pid of Dow 0, pid − dow
  • boy: beginning of year, midnight at the start of Day 0
  • boe: beginning of era, midnight at the start of Day 0 of Year 0
  • boc: beginning of cycle, the start of Doc_♀ 0 and of the first ante in a Decyther cycle
  • eoy: end of year, midnight at the end of Day 364 or 365
  • JDN: Julian day number, days since the start of the Julian period; increments at noon UTC
  • lid: lunar intercalation difference, a lunar nid that only resets after lunar leap years, -1063 to -1
  • ved: Venusian extracalation difference, days until the end of Doc_♀ 349, the last day of the one-day-shorter Qoc 2, Bimester 5, and Moc 11, (doc_♀ + 117) mod 467 − 467, -467 to -1
  • mud: misaligned unit difference, pid − dom − dow
  • 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
  • dmd: decimilliday, a ten thousandth of a day, 10 beats, 8640 milliseconds
  • 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
• eot: equation of time, ast − mst as a function of toy, about -9.8 to 11.4 millidays
• ℓ: ell, cubit, 10/9 zem
• f: foot, 0.75 zem, 75 millimeters
• 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, 1 hectoday, 10 decadays, or one hundred days
  • pih: positive integer hectoday, ⌊pid ÷ 100⌋, 0 to 3
  • nih: negative integer hectoday, ⌊nid ÷ 100⌋, -4 to -1
• hex: hexadecimal, base 16
• HSL: hue saturation lightness
• HSV: hue saturation value
• i: inch, a sixteenth of a zem, 25 millimeters
• ISO: International Organization for Standardization, the body behind ISO 8601 dates such as 1970-01-01 (ISO month date) and 1970-W01-4 (ISO week date)
• j: jīng (经), a full circle around the Earth moving North or South through both Poles, a measure of latitude; short for 经圈 (jīngquān), “meridian circle,” to emphasize a full circle rather than a meridian, which can be a half circle; an uppercase “J” without a serif is a vertical line drawn from top to bottom, like a jīng running from the North Pole to the South Pole, with a curve at the bottom where the jīng turns to come back up the other side of the Earth; although not the same as the jīng in Běijīng (北京) and Nánjīng (南京), jīng (经) is related to North (Běi) and South (Nán); on a loom, 经 is the warp, the threads that run lengthwise
  • dj: decijīng, a tenth of a jīng (经)
  • mj: millijīng, a thousandth of a jīng (经)
• k: keg, cubic zem, 64 liters, 1000 wine glasses, a million drops, half a barrel
• km: kilometer, 1000 meters, 2500 zem, 2.5 kilozem
• 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
• 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 milliliter, a thousandth of a liter, 15.625 drops
• p: pentaday, a group of five days, half a decaday
  • pox: pentaday of xún, 0 for Dox 0 to 4 and 1 for Dox 5 to 9
  • pob: pentaday of bimester, ⌊dob ÷ 5⌋, 0 to 11
  • pir: pentaday interquintile range, Dop 1 to 3, the three days between two lim
  • lim: liminal interconnecting margin, the pair of days between two pir, Dop 4 and the following Dop 0
• n: note, a specific frequency within an octave
• NOAA: National Oceanic and Atmospheric Administration, source of the General Solar Position Calculations document
• 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 rotation along the horizon, 360 compass degrees
  • saa: solar azimuth angle, the direction of the Sun along the horizon measured clockwise from North
  • mr: compass millirose, a thousandth of a rotation along the horizon, .36 compass degrees
• rad: radian, 1/𝜏 turns, 360/𝜏 degrees, 1/𝜋 turns, 180/𝜋 degrees
• RGB: red green blue
• s: second, 1/90 millidays, 10/9 beats, 1 Dec second = 0.96 SI seconds
  • ds: decisecond, a tenth of a Dec second, 1/900 millidays, 1/9 beats, 96 milliseconds
  • sid: second interval difference, 1 SI second − 1 Dec second, 1/24 Dec second, 1/25 SI second, 40 milliseconds
  • sis: SI second, 25/24 Dec seconds, 1/86.4 millidays, 1/86400 days
• soi: spin orbit interval, a lunar day, 1.03505 days
  • soy: soi of year
  • sob: soi of bimester
• SI: International System of Units
• sol: speed of light, 647.55170928 kiloomegar, 299792458 meters per second
• sos: speed of sound, 735.048 milliomegar, 340.3 meters per second
• sda: solar declination angle, the latitude at which the Sun is directly overhead
• sea: solar elevation angle, the angle of the Sun above the horizon, negative when the Sun is below the horizon
• 𝜏: 2𝜋 or approximately 6.2831853
  • 𝜋: pi, 𝜏/2 or approximately 3.1415927
  • ϡ: sampi, the Greek numeral for 900, 𝜏/900 or approximately 0.0069813 radians, 0.4 degrees
• Tenet: ten equal temperament
  • Xet: Tenet
  • 12et: twelve equal temperament
• tod: time of day
  • pit: positive interval tod, the time since the start of the current day (previous midnight), equal to the tod
  • nit: negative interval tod, pit − 1, the negative of the time until the start of the next day (next midnight)
  • hms: hour minute second, a tod written as an hod, moh, and som triplet
  • dsa: daily second aggregate, seconds since midnight, tod × 86400
  • hsa: hourly second aggregate, seconds since the top of the hour, dsa mod 3600
  • hod: hour of day, ⌊dsa ÷ 3600⌋, 0 to 23
  • moh: minute of hour, ⌊hsa ÷ 60⌋, 0 to 59
  • som: second of minute, ⌊hsa mod 60⌋, 0 to 59
  • mst: mean solar time, solar time based only on longitude and the Zone 0 tod
  • ast: apparent solar time, mst + eot(toy), solar time based on the actual position of the Sun
  • law: light arc whole, the time from sunrise to sunset, 1 − naw, sunset − sunrise
  • lah: light arc half, law/2, the time from sunrise to solar noon or from solar noon to sunset
  • naw: night arc whole, the time from sunset to sunrise, 1 − law
  • nah: night arc half, naw/2, the time from sunset to solar midnight or from solar midnight to sunrise
• 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 radians
• tzo: time zone offset
• u: ounce (uncia in Latin), 500 grains, 32 grams, 500 drops, 32 milliliters
• uto: universal time offset, the offset in decidays that turns a Zone 0 tod into Universal Time; a tzo is an integer uto
  • sto: solar time offset, any offset added to a Zone 0 tod, written after the tod as tod-offset; represents a single longitude, whereas a tzo is one of the 10 single-digit offsets of the Dec time zones
• UTC: Coordinated Universal Time
• US: United States
• UT: Universal Time, a time standard based on the rotation of the Earth
• v: omegar, ωr, 1041.♄ miles per hour, 1.♄ megameters per hour, 0.4♃♄ kilometers per second, roughly 1.36 times the speed of sound
  • kv: kiloomegar, kωr, 1.♄ gigameters per hour, 0.4♃♄ megameters per second, approximately 0.1544% of the speed of light
  • mv: milliomegar, mωr, 1.041♄ miles per hour, 1.♄ kilometers per hour, 0.4♃♄ meters per second, approximately 0.136% of the speed of sound
• w: wěi (纬), parallel, a full circle around the Earth moving East or West at the same latitude, a measure of longitude; short for 纬圈 (wěiquān), “latitude circle”; can be thought of as a measure of the width of a jīng on Earth; “w” is written from left to right, like a wěi running from West to East across a map, starts the word “width,” as in the width of a map, and is one of the widest letters in proportional fonts; on a loom, 纬 is the weft, the threads that run crosswise
  • dw: deciwěi, a tenth of a wěi (纬), a tenth of a parallel
  • mw: milliwěi, a thousandth of a wěi (纬), a thousandth of a parallel
  • µw: microwěi, a millionth of a wěi (纬), a millionth of a parallel
• x: xún (旬), decaday, a group of ten days, 2 pentadays, represented by x like the Roman numeral X
  • pix: positive integer xún, ⌊pid ÷ 10⌋, 0 to 36
  • nix: negative integer xún, ⌊nid ÷ 10⌋, -37 to -1
  • mix: mixed integer xún, the first two digits of an nih mid
  • xir: xún interdecile range, Dox 1 to 8, the days between Dox 0 and 9
• y: year
  • my: milliyear, a thousandth of a year
  • yoe: year of era, integer years since the Dec epoch
  • eya: epochal year aggregate, yoe + toy, years since the Dec epoch including the fraction of the current year
  • toy: time of year, the fraction of the year that has elapsed, eya mod 1, ada ÷ syl
  • ymd: year month day, a Gregorian calendar date triplet
  • syl: solar year length, 365 or 366 days, pid − nid
  • lyl: lunar year length, 354 or 355 days
  • cey: common era year, an ISO 8601 (Gregorian calendar) year number
  • coe: cycle of era, 400-year cycles since the Dec epoch
  • yoc: year of cycle, years since the start of the current 400-year cycle, 0 to 399
  • yot: year of tricennium, lunar years since the start of the current 30-year lunar cycle, 0 to 29
  • moy: month of year, zero-based, Moy 0 is March
  • woy: week of year, (boydow + bow) ÷ 7, 0 to 53
  • wom: week of month
• 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.♅ square feet, 256 square inches
  • cz²: square centizem, a ten-thousandth of a square zem, 16 square millimeters
  • 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

References

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

Footnotes

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