Rule of twelfths

Approximation to a sine curve

Graph showing relationships between the rule of twelfths (coloured bars), a sine wave (dashed blue curve) and a clockface, if high tide occurs at 12:00

The rule of twelfths is an approximation to a sine curve. It can be used as a rule of thumb for estimating a changing quantity where both the quantity and the steps are easily divisible by 12. Typical uses are predicting the height of the tide or the change in day length over the seasons.

## The rule

[edit]

The rule states that over the first period the quantity increases by 1/12. Then in the second period by 2/12, in the third by 3/12, in the fourth by 3/12, fifth by 2/12 and at the end of the sixth period reaches its maximum with an increase of 1/12. The steps are 1:2:3:3:2:1 giving a total change of 12/12. Over the next six intervals the quantity reduces in a similar manner by 1, 2, 3, 3, 2, 1 twelfths.

Period | Rule or
actual
values  | Increment | Cumulative

|
|
|

Exact value | Decimal | Relative size | Exact value | Decimal | Relative size
1  | Rule | 1 / 12 | 0.08333  | 0.08333

| 1 / 12 | 0.08333  | 0.08333

Actual | (cos 0° - cos 30°) / 2 | 0.06699  | 0.06699

| (1 - cos 30°) / 2 | 0.06699  | 0.06699

2  | Rule | 2 / 12 | 0.16667  | 0.16667

| 3 / 12 | 0.25  | 0.25

Actual | (cos 30° - cos 60°) / 2 | 0.18301  | 0.18301

| (1 - cos 60°) / 2 | 0.25  | 0.25

3  | Rule | 3 / 12 | 0.25  | 0.25

| 6 / 12 | 0.5  | 0.5

Actual | (cos 60° - cos 90°) / 2 | 0.25  | 0.25

| (1 - cos 90°) / 2 | 0.5  | 0.5

4  | Rule | 3 / 12 | 0.25  | 0.25

| 9 / 12 | 0.75  | 0.75

Actual | (cos 90° - cos 120°) / 2 | 0.25  | 0.25

| (1 - cos 120°) / 2 | 0.75  | 0.75

5  | Rule | 2 / 12 | 0.16667  | 0.16667

| 11 / 12 | 0.91667  | 0.91667

Actual | (cos 120° - cos 150°) / 2 | 0.18301  | 0.18301

| (1 - cos 150°) / 2 | 0.93301  | 0.93301

6  | Rule | 1 / 12 | 0.08333  | 0.08333

| 12 / 12 | 1  | 1

Actual | (cos 150° - cos 180°) / 2 | 0.06699  | 0.06699

| (1 - cos 180°) / 2 | 1  | 1

## Applications

[edit]

A regular dodecahedron (grey) approximated with a lattice polygon using the rule of twelfths (red)

In many parts of the world the tides approximate to a semi-diurnal sine curve, that is there are two high- and two low- tides per day. As an estimate then each period equates to 1 hour, with the tide rising by 1, 2, 3, 3, 2, finally 1 twelfths of its total range in each hour, from low tide to high tide in about 6 hours, then the tide is decreasing by the same pattern in the next 6 hours, back to low tide. In places where there is only one high and one low water per day, the rule can be used by assuming the steps are 2 hours. If the tidal curve does not approximate to a sine wave then the rule cannot be used.[1][2] This is important when navigating a boat or a ship in shallow water, and when launching and retrieving boats on slipways on a tidal shore.[3]

The rule is also useful for estimating the monthly change in sunrise and sunset and thus day length.[4]

## Example calculations

[edit]

### Tides

[edit]

If a tide table gives the information that tomorrow's low water would be at noon and that the water level at this time would be two metres above chart datum, and that at the following high tide the water level would be 14 metres, then the height of water at 3:00 p.m. can be calculated as follows:

* The total increase in water level between low and high tide would be: 14 - 2 = 12 metres.
* In the first hour the water level would rise by 1 twelfth of the total (12 m) or: 1 m
* In the second hour the water level would rise by another 2 twelfths of the total (12 m) or: 2 m
* In the third hour the water level would rise by another 3 twelfths of the total (12 m) or: 3 m
* This gives the increase in the water level by 3:00 p.m. as 6 metres.

This represents only the increase - the total depth of the water (relative to chart datum) will include the 2 m depth at low tide: 6 m + 2 m = 8 metres.

The calculation can be simplified by adding twelfths together and reducing the fraction beforehand:

Rise of tide in three hours  = ( 1 12 + 2 12 + 3 12 ) × 12 m = ( 6 12 ) × 12 m = ( 1 2 ) × 12 m = 6 m {\displaystyle =\left({1 \over 12}+{2 \over 12}+{3 \over 12}\right)\times 12\ \mathrm {m} =\left({6 \over 12}\right)\times 12\ \mathrm {m} =\left({1 \over 2}\right)\times 12\ \mathrm {m} =6\ \mathrm {m} }

### Daylength

[edit]

If midwinter sunrise and set are at 09:00 and 15:00, and midsummer at 03:00 and 21:00, the daylight duration will shift by 0:30, 1:00, 1:30, 1:30, 1:00 and 0:30 over the six months from one solstice to the other. Likewise the day length changes by 0:30, 1:00, 1:30, 1:30, 1:00 and 0:30 each month. More equatorial latitudes change by less, but still in the same proportions; more polar by more.

## Caveats

[edit]

The rule is a rough approximation only and should be applied with great caution when used for navigational purposes. Officially produced tide tables should be used in preference whenever possible.

The rule assumes that all tides behave in a regular manner, this is not true of some geographical locations, such as Poole Harbour[5] or the Solent[6] where there are "double" high waters or Weymouth Bay[5] where there is a double low water.

The rule assumes that the period between high and low tides is six hours but this is an underestimate and can vary anyway.

## Refinement

[edit]

The rule relies on the approximation of tan 60° or √3 (~1.732) with 5/3 (~1.667) yielding 3.77% error. The next best rational approximation, 7/4 (1.75) yields 1.04% error. The steps are 1:3:4:4:3:1 giving a total change of 16/16:[7]

Period | Rule or
actual
values  | Increment | Cumulative

|
|
|

Exact value | Decimal | Relative size | Exact value | Decimal | Relative size
1  | Rule | 1 / 16 | 0.0625  | 0.0625

| 1 / 16 | 0.0625  | 0.0625

Actual | (cos 0° - cos 30°) / 2 | 0.06699  | 0.06699

| (1 - cos 30°) / 2 | 0.06699  | 0.06699

2  | Rule | 3 / 16 | 0.1875  | 0.1875

| 4 / 16 | 0.25  | 0.25

Actual | (cos 30° - cos 60°) / 2 | 0.18301  | 0.18301

| (1 - cos 60°) / 2 | 0.25  | 0.25

3  | Rule | 4 / 16 | 0.25  | 0.25

| 8 / 16 | 0.5  | 0.5

Actual | (cos 60° - cos 90°) / 2 | 0.25  | 0.25

| (1 - cos 90°) / 2 | 0.5  | 0.5

4  | Rule | 4 / 16 | 0.25  | 0.25

| 12 / 16 | 0.75  | 0.75

Actual | (cos 90° - cos 120°) / 2 | 0.25  | 0.25

| (1 - cos 120°) / 2 | 0.75  | 0.75

5  | Rule | 3 / 16 | 0.1875  | 0.1875

| 15 / 16 | 0.9375  | 0.9375

Actual | (cos 120° - cos 150°) / 2 | 0.18301  | 0.18301

| (1 - cos 150°) / 2 | 0.93301  | 0.93301

6  | Rule | 1 / 16 | 0.0625  | 0.0625

| 16 / 16 | 1  | 1

Actual | (cos 150° - cos 180°) / 2 | 0.06699  | 0.06699

| (1 - cos 180°) / 2 | 1  | 1

The following best approximations are 19/11 (0.276% error) with steps 3:8:11:11:8:3, and 26/15 (0.074% error) with steps 4:11:15:15:11:4.[7]

## References

[edit]

1. ^ "Rule of Twelfths for quick tidal estimates". DIY Wood Boat. Retrieved 19 December 2017.
2. ^ Getchell, David R. (1994). The Outboard Boater's Handbook: Advanced Seamanship and Practical Skills. International Marine. p. 195. ISBN 978-0-07-023053-8.
3. ^ Sweet, Robert J. (16 September 2004). The Weekend Navigator: Simple Boat Navigation with GPS and Electronics. McGraw Hill Professional. p. 162. ISBN 978-0-07-143035-7.
4. ^ McAdam, Marcus. "The Rule of Twelfths". Mc2Photography.com. Retrieved 2021-03-11. "The same Rule of Twelfths can be applied to the duration of the days."
5. ^ a b Heritage, Trevor. "Poole Harbour and its tides" (PDF). Shrimperowners. Retrieved 19 December 2017.
6. ^ Ridge, M J, FRICS MCIT. "English Channel double tides". Bristol Nomads windsurfing club. Archived from the original on 22 August 2009. Retrieved 19 December 2017.`{{cite web}}`: CS1 maint: multiple names: authors list (link)
7. ^ a b Bizony, Marcus Bizony (1 June 2014). "Finding rational approximations to square roots". Learning and Teaching Mathematics. 2014 (16). Retrieved 25 March 2024.

* v
* t
* e

Physical oceanography

Waves|

* Airy wave theory
* Ballantine scale
* Benjamin–Feir instability
* Boussinesq approximation
* Breaking wave
* Clapotis
* Cnoidal wave
* Cross sea
* Dispersion
* Edge wave
* Equatorial waves
* Fetch
* Gravity wave
* Green's law
* Infragravity wave
* Internal wave
* Iribarren number
* Kelvin wave
* Kinematic wave
* Longshore drift
* Luke's variational principle
* Mild-slope equation
* Radiation stress
* Rogue wave
* Rossby wave
* Rossby-gravity waves
* Sea state
* Seiche
* Significant wave height
* Soliton
* Stokes boundary layer
* Stokes drift
* Stokes wave
* Swell
* Trochoidal wave
* Tsunami
* megatsunami
* Undertow
* Ursell number
* Wave action
* Wave base
* Wave height
* Wave nonlinearity
* Wave power
* Wave radar
* Wave setup
* Wave shoaling
* Wave turbulence
* Wave–current interaction
* Waves and shallow water
* one-dimensional Saint-Venant equations
* shallow water equations
* Wind setup
* Wind wave
* model

Circulation|

* Atmospheric circulation
* Baroclinity
* Boundary current
* Coriolis force
* Coriolis–Stokes force
* Craik–Leibovich vortex force
* Downwelling
* Eddy
* Ekman layer
* Ekman spiral
* Ekman transport
* El Niño–Southern Oscillation
* General circulation model
* Geochemical Ocean Sections Study
* Geostrophic current
* Global Ocean Data Analysis Project
* Gulf Stream
* Halothermal circulation
* Humboldt Current
* Hydrothermal circulation
* Langmuir circulation
* Longshore drift
* Loop Current
* Modular Ocean Model
* Ocean current
* Ocean dynamics
* Ocean dynamical thermostat
* Ocean gyre
* Overflow
* Princeton ocean model
* Rip current
* Subsurface currents
* Sverdrup balance
* Thermohaline circulation
* shutdown
* Upwelling
* Wind generated current
* Whirlpool
* World Ocean Circulation Experiment

Tides|

* Amphidromic point
* Earth tide
* Head of tide
* Internal tide
* Lunitidal interval
* Perigean spring tide
* Rip tide
* Rule of twelfths
* Slack tide
* Tidal bore
* Tidal force
* Tidal power
* Tidal race
* Tidal range
* Tidal resonance
* Tide gauge
* Tideline
* Theory of tides

Landforms|

* Abyssal fan
* Abyssal plain
* Atoll
* Bathymetric chart
* Coastal geography
* Cold seep
* Continental margin
* Continental rise
* Continental shelf
* Contourite
* Guyot
* Hydrography
* Knoll
* Oceanic basin
* Oceanic plateau
* Oceanic trench
* Passive margin
* Seabed
* Seamount
* Submarine canyon
* Submarine volcano

Plate
tectonics|

* Convergent boundary
* Divergent boundary
* Fracture zone
* Hydrothermal vent
* Marine geology
* Mid-ocean ridge
* Mohorovičić discontinuity
* Vine–Matthews–Morley hypothesis
* Oceanic crust
* Outer trench swell
* Ridge push
* Seafloor spreading
* Slab pull
* Slab suction
* Slab window
* Subduction
* Transform fault
* Volcanic arc

Ocean zones|

* Benthic
* Deep ocean water
* Deep sea
* Littoral
* Mesopelagic
* Oceanic
* Pelagic
* Photic
* Surf
* Swash

Sea level|

* Deep-ocean Assessment and Reporting of Tsunamis
* Future sea level
* Global Sea Level Observing System
* North West Shelf Operational Oceanographic System
* Sea-level curve
* Sea level rise
* Sea level drop
* World Geodetic System

Acoustics|

* Deep scattering layer
* Hydroacoustics
* Ocean acoustic tomography
* Sofar bomb
* SOFAR channel
* Underwater acoustics

Satellites|

* Jason-1
* Jason-2 (Ocean Surface Topography Mission)
* Jason-3

Related|

* Acidification
* Argo
* Benthic lander
* Color of water
* DSV Alvin
* Marginal sea
* Marine energy
* Marine pollution
* Mooring
* National Oceanographic Data Center
* Ocean
* Explorations
* Observations
* Reanalysis
* Ocean surface topography
* Ocean temperature
* Ocean thermal energy conversion
* Oceanography
* Outline of oceanography
* Pelagic sediment
* Sea surface microlayer
* Sea surface temperature
* Seawater
* Science On a Sphere
* Stratification
* Thermocline
* Underwater glider
* Water column
* World Ocean Atlas

* Oceans portal
* Category
* Commons

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template