Units in Multiple Dimensions

Although units are most often used in a 1-dimensional way (for example using "feet" to measure a length), units can also be used in multiple dimensions to measure things such as an area (2 dimensions), or a volume (3 dimensions). AutoMathic supports specifying arbitrary units and measurements in arbitrary numbers of dimensions, but it has special support for several common 2 and 3-dimensional measurements.

Multi-Dimensional Units
Multi-Dimensional Unit Phrases
Multi-Dimensional Unit Conversion

Finding Unknowns

Multi-Dimensional Units

Since we perceive a universe with only three spatial dimensions (and one time dimension), in practice most multi-dimensional measurements are limited to 2 or 3 dimensions:

2-dimensional measurements are usually for spatial measurements of area.
3-dimensional measurements are usually for spatial measurements of volume.

Higher-dimensional measurements of time (e.g. "seconds squared" as in measurements of acceleration, or "seconds cubed" as in measurements of jerk) are less common... They almost always appear in the denominator for a different type of unit (such as length or mass) to measure "rates of change" over time. However, multi-dimensional time units are fully-supported generically as well as with some special vocabulary.

Higher-dimensional measurements of more exotic quantities (e.g. "amperes squared" as in measurements of capacitance) are even less common, but can still be expressed generically.

Just as with 1-dimensional units, AutoMathic requires that a few special conventions be followed for units in higher dimensions:

Multi-dimensional units are intrinsically compound units (i.e. multiple units combined).
To ensure that multi-dimensional units are interpreted correctly, it is best to enclose them in parentheses or square brackets, like most unit phrases!

2-dimensional units

Unit x Unit	A 2-dimensional unit (such as "square feet" for an area) is a 1-dimensional unit multiplied by a 1-dimensional unit:
e.g.	(feet * feet) [foot x foot] [feet by feet] [ft(ft)] [in x in] (second * second) [m/(sec * sec)]
Unit squared	A 2-dimensional unit can be a 1-dimensional unit followed by "squared" (or an abbreviation "sq", "sqr", "sqrd", or "sqd") when the unit of measurement is the same in both dimensions or directions:
e.g.	(feet squared) (inches sq) (yards sqr) (in sqrd) (s squared) [ft/(sec sq)]
Unit ^ 2	The special "^2" suffix is also a synonym for "squared". Note that spaces are optional around the caret symbol "^", but the caret symbol is NOT otherwise usable as a generic mathematical operator:
e.g.	[feet ^ 2] [feet^2] [in^2] (sec^2) [meters/(second^2)]
Unit2	Although this usage is somewhat questionable, appending "2" directly to a unit's name is a common shorthand for "squared". AutoMathic has special vocabulary words supporting that usage for several common macroscopic units (inch, foot, yard, mile, nautical mile, millimeter, centimeter, meter, kilometer, second) in all their singular, plural, and abbreviated forms:
e.g.	[inch2] [in2] [foot2] [feet2] [ft2] [miles2] [nmi2] [cm2] [kms2] [km's2] (sec2) [m/s2]
square Unit	A 2-dimensional unit can be the word "square" or "squared" (or their abbreviations "sq", "sqr", or "sqrd") followed by a small set of macroscopic length units, allowing prefixed forms of some common 2-dimensional units of area. Length units recognized with prefixes are: inches, feet, yards, miles, nautical miles, millimeters, centimeters, meters, and kilometers (in all their singular, plural, and abbreviated forms)
e.g.	`[square inch], [square foot], [square yard], [square mile], [square nautical mile][square inches], [square feet], [square yards], [square miles], [square nautical miles]` `[square millimeter], [square centimeter], [square meter], [square kilometer][square millimeters], [square centimeters], [square meters], [square kilometers]` `[sq in], [sq ft], [sq yd], [sq mi], [sq nmi]` `[sq ins], [sq ft], [sq yds], [sq mis], [sq nmis]` `[sq mm], [sq cm], [sq m], [sq km]` `[sq mms], [sq cms], [sq ms], [sq kms]`

3-dimensional units

Unit x Unit x Unit	A 3-dimensional unit (such as "cubic feet" for a volume) can be a 1-dimensional unit multiplied by a 1-dimensional unit multiplied by a 1-dimensional unit:
e.g.	(foot * foot * foot) [feet x feet x feet] [foot by foot by foot] [ft(ft)ft] [in(in)in]
Unit cubed	A 3-dimensional unit can be a 1-dimensional unit followed by "cubed" (or the abbreviation "cbd") when the unit of measurement is the same in all three dimensions or directions:
e.g.	(feet cubed) (feet cbd) (in cubed)
Unit ^ 3	The special "^3" suffix is also a synonym for "cubed". Note that spaces are optional around the caret symbol "^", but the caret symbol is NOT otherwise usable as a generic mathematical operator:
e.g.	[feet ^ 3] [feet^3] [in^3] (sec^3) [meters/(second^3)]
Unit3	Although this usage is somewhat questionable, appending "3" directly to a unit's name is a common shorthand for "cubed". AutoMathic has special vocabulary words supporting that usage for several common macroscopic units (inch, foot, yard, mile, nautical mile, millimeter, centimeter, meter, kilometer, second) in all their singular, plural, and abbreviated forms:
e.g.	[inch3] [in3] [foot3] [feet3] [ft3] [inches3] [nmi3] [cm3] [kms3] [km's3] (sec3) [m/s3]
cubic Unit	A 3-dimensional unit can be the word "cubic" or "cubed" (or their abbreviations "cu", or "cbd") followed by a small set of macroscopic length units, allowing prefixed forms of some common 3-dimensional units of volume. Length units recognized with prefixes are: inches, feet, yards, miles, nautical miles, millimeters, centimeters, meters, and kilometers (in all their singular, plural, and abbreviated forms)
e.g.	`[cubic inch], [cubic foot], [cubic yard], [cubic mile], [cubic nautical mile][cubic inches], [cubic feet], [cubic yards], [cubic miles], [cubic nautical miles]` `[cubic millimeter], [cubic centimeter], [cubic meter], [cubic kilometer][cubic millimeters], [cubic centimeters], [cubic meters], [cubic kilometers]` `[cu in], [cu ft], [cu yd], [cu mi], [cu nmi]` `[cu ins], [cu ft], [cu yds], [cu mis], [cu nmis]` `[cu mm], [cu cm], [cu m], [cu km]` `[cu mms], [cu cms], [cu ms], [cu kms]`

Multi-Dimensional Unit Phrases

Multi-dimensional unit phrases combine units with properties or scalars into a cohesive whole that defines a measurement. The measurement can be of the whole space, or of the separate measurements of the space's boundaries:

Counting units of area or volume	A multi-dimensional measurement can be expressed as a measurement of an entire area or volume:
e.g.	1,200[ft sq] 40 x 30[ft sq] 5x7[in^2] 350[in3] Area measured in ft sq is 1,200. Area is 1,200[ft2]. 350 is the volume in terms of in^3. Volume is 350[in^3].
Multiplying independent measurements	A multi-dimensional measurement can be expressed as independent measurements multiplied by each other. The measurements for the various dimensions do NOT have to use the same units:
e.g.	40[ft] by 30[ft] 5[in] x 7[in] 4[in]x6[in]x8[in] 1[m] x 2[in] x 25[ft] Pi x 1.44[ft^2] x 5.6[ft] is the volume Area is 2[m] x 1.2[ft]

Multi-Dimensional Unit Conversion

Since different units can be mixed freely in unit phrases, multi-dimensional unit conversions can be performed, and in all the same ways that 1-dimensional unit conversions can be performed.

Statements / Questions

One way to do unit conversions is to have a statement defining a measurement, while asking for the measurement in different units:

The area is 5x7[in^2]. What's the area in [cm^2]?
What's the area in [cm^2] when the area is 5x7[in^2]?
If the area is 5x7[in^2], what's the area in [cm^2]?

Direct Questions

A more direct way of doing conversions is to eliminate the statement, asking for the conversion directly:

What's 5x7[in^2] measured in [cm^2]?
What's 5x7[in^2] in terms of [cm^2]?
5x7[in^2] in [cm^2]

Conversion Statements

Another way to do unit conversions is by using a "conversion statement". A "conversion statement" is based on (or inspired by) the verb "convert":

Convert 8[ft] x 3[ft] x 7[ft] into [meters cubed].
8[ft] x 3[ft] x 7[ft] to [meters cubed].

Convert 168 from [feet cubed] to [meters cubed].
168 from [ft3] to [m3].

168 converted from [ft^3] to unit [m^3].
168 from [ft^3] to [m^3].

168[ft3] converted to [m3].
168[ft3] to [m3].

How many [m3] does 168[ft3] convert to?

Finding Unknowns

Portions of a multi-dimensional measurement can be unknown, and therefore solved for:

3,200[ft^2] is 40[ft] x how many unit yards?

How many [ft sq] is 2x6[m^2]?
4[in] x 6[in] x 8[in] is what percentage of 1[meter3]?
How many times 2[ft cubed] is 2[in] x 1[m] x 25[ft]?

Note that although AutoMathic can accept arbitrary multi-dimensional unit specifications, numeric solutions are only guaranteed for purely 1-dimensional problems since solutions for some multi-dimensional problems could involve taking square roots, cube roots, etc.! Despite this limitation, AutoMathic is still able to solve most common problems involving multi-dimensional units by using indirect-solution strategies that try to avoid situations that might lead to solving equations for non-linear variables.