returns the possible combinations of the list of physical quantities pqi that are dimensionless.


returns the possible combinations of the list of physical quantities pqi that match the dimensions of physical quantity dim.

Details and Options

  • Physical quantities can be valid QuantityVariable objects, "PhysicalQuantity" entities or physical quantity strings.
  • dim can be a QuantityVariable object. It can also be a combination of QuantityVariable objects or their derivatives.
  • Solutions are determined by the physical quantity components in unit dimensions purely mathematically and have no guarantee of physical significance.
  • Physical dimensions include: "AmountUnit", "AngleUnit", "ElectricCurrentUnit", "InformationUnit", "LengthUnit", "LuminousIntensityUnit", "MassUnit", "MoneyUnit", "SolidAngleUnit", "TemperatureDifferenceUnit", "TemperatureUnit", and "TimeUnit".
  • Dimensionless physical quantities will not be used in the solution.
  • The following options can be given:
  • GeneratedParametersChow to name parameters that are generated
    IncludeQuantities{}additional quantities to include
  • GeneratedParameters takes the option None, which returns a list of parameter-free solutions.
  • IncludeQuantities allows quantity values and constants to be included in the combinations.
  • The setting "PhysicalConstants" for IncludeQuantities includes the quantities Quantity["BoltzmannConstant"], Quantity["ElectricConstant"], Quantity["GravitationalConstant"], Quantity["MagneticConstant"], Quantity["PlanckConstant"], and Quantity["SpeedOfLight"].


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Basic Examples  (1)

Determine the combination of physical quantities that are dimensionally equivalent to energy:

Find all combinations of physical quantities that result in a dimensionless expression:

Discover if a dimensionless expression is possible with a set of physical quantities:

Scope  (3)

Use any combination of QuantityVariable objects or physical quantity strings:

The target physical dimensions can be specified as a combination of physical quantities:

Derivative objects may also be included in the expression:

"PhysicalQuantity" entities, including in QuantityVariable expressions, can also be used:

Options  (5)

GeneratedParameters  (3)

Use a different symbol for parameters:

By default, a generic solution is returned:

Use GeneratedParameters->None to get specific solutions:

GeneratedParameters->None works with IncludeQuantities to allow mixtures of QuantityVariable and Quantity objects:

IncludeQuantities  (2)

Include additional constants and Quantity objects in the result:

Use the setting "PhysicalConstants" to include a standard set of physical constants:

Applications  (4)

Find the missing physical constants in the formula E^2 - p^2 == m^2:

Solve for the value of the constants:

Insert the correct exponents:

Eliminate unnecessary constants:

Find the dimensions of the constant needed to balance Kleiber's law :

Solve for the value of the mass exponent:

Estimate the power of a bomb blast by using only these physical quantities:

Construct a dimensionless combination:

Given the values of the parameters at a given time, estimate the energy of an explosion:

Determine possible dimensionless price impact functions depending on stock price, size and cost of bets, trading volumes and the volatility of the stock:

Find the general dimensionless combination:

Determine specific instances:

Properties & Relations  (1)

Formulas for dimensionless constants can be constructed from physical quantities:

Possible Issues  (5)

Only valid physical quantities can be used:

Dimensionless quantities will be omitted from the result:

Only valid constants will be used:

Angular units and physical quantities are not treated as dimensionless:

While returned combinations are dimensionless, they do not necessarily have a magnitude of one:

Interactive Examples  (1)

Examine all possible dimensionless combinations for a set of physical quantities and constants:

Neat Examples  (2)

Explore the possible dimensionless combinations of electromagnetic physical quantities:

Derive the factor for the fine structure constant from physical quantities:

Wolfram Research (2014), DimensionalCombinations, Wolfram Language function, https://reference.wolfram.com/language/ref/DimensionalCombinations.html (updated 2018).


Wolfram Research (2014), DimensionalCombinations, Wolfram Language function, https://reference.wolfram.com/language/ref/DimensionalCombinations.html (updated 2018).


Wolfram Language. 2014. "DimensionalCombinations." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2018. https://reference.wolfram.com/language/ref/DimensionalCombinations.html.


Wolfram Language. (2014). DimensionalCombinations. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DimensionalCombinations.html


@misc{reference.wolfram_2022_dimensionalcombinations, author="Wolfram Research", title="{DimensionalCombinations}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/DimensionalCombinations.html}", note=[Accessed: 14-August-2022 ]}


@online{reference.wolfram_2022_dimensionalcombinations, organization={Wolfram Research}, title={DimensionalCombinations}, year={2018}, url={https://reference.wolfram.com/language/ref/DimensionalCombinations.html}, note=[Accessed: 14-August-2022 ]}