CITE THIS NOTEBOOK: Analyzing the Planck length and other natural units with Linear Algebra by Peter Burbery. Wolfram Community DEC 24 2022.
I deployed a function to the Wolfram Cloud named AugmentedMatrixFromCoefficientMatrixAndColumnVector. I am waiting for approval for it to be included in the Function Repository in the meantime.
In[]:=
AugmentedMatrixFromCoefficientMatrixAndColumnVector//ClearAll​​AugmentedMatrixFromCoefficientMatrixAndColumnVector[coefficientMatrix_,columnVector_]:=MapThread[Append,{coefficientMatrix,columnVector}]/;ArrayDepth[columnVector]==1​​AugmentedMatrixFromCoefficientMatrixAndColumnVector[coefficientMatrix_,columnVector_]:=MapThread[Append,{coefficientMatrix,Flatten[columnVector]}]/;ArrayDepth[columnVector]==2

Neat Examples

The dimensions of the Planck natural unit system are time (T), length (L), mass (M), and temperature (Θ). The Planck system uses the Planck constant h or the reduced Planck constant ℏ (read “h bar”, keyboard alias hb) which is h/2π, the speed of light c, the Boltzmann constant k, and the gravitational constant G. Each one has certain dimensions:
In[]:=
ResourceFunction["CanonicalDimensionalProduct"][
h
]
Out[]=
-1
T
2
L
1
M
In[]:=
ResourceFunction["CanonicalDimensionalProduct"][
ℏ
]
Out[]=
-1
T
2
L
1
M
In[]:=
ResourceFunction["CanonicalDimensionalProduct"][
c
]
Out[]=
-1
T
1
L
In[]:=
ResourceFunction["CanonicalDimensionalProduct"][
k
]
Out[]=
-2
T
2
L
1
M
-1
Θ
In[]:=
ResourceFunction["CanonicalDimensionalProduct"][
G
]
Out[]=
-2
T
3
L
-1
M
Make a coefficient matrix with rows for time, length, mass and temperature and columns for h, c, k, and G or ℏ, c, k, and G and row reduce the augmented matrix with the unknown with the dimensions
1
L
:
In[]:=
AugmentedMatrixFromCoefficientMatrixAndColumnVector
-1
-1
-2
0
2
1
2
3
1
0
1
-1
0
0
-1
0
,
0
1
0
0
//MatrixForm
Out[]//MatrixForm=
-1
-1
-2
0
0
2
1
2
3
1
1
0
1
-1
0
0
0
-1
0
0
In[]:=
RowReduceAugmentedMatrixFromCoefficientMatrixAndColumnVector
-1
-1
-2
-2
2
1
2
3
1
0
1
-1
0
0
-1
0
,
0
1
0
0
//MatrixForm
Out[]//MatrixForm=
1
0
0
0
1
2
0
1
0
0
-
3
2
0
0
1
0
0
0
0
0
1
1
2
The leftmost column represents the powers to raise the constants to:
In[]:=
MapThreadPower,
ℏ
,
c
,
k
,
G
,Map[Last]RowReduce
Augmented matrix

Out[]=

1
ℏ
,
1
/
3/2
c
,1,
1
G

Multiply the constants:
In[]:=
Times@@MapThreadPower,
ℏ
,
c
,
k
,
G
,Map[Last]RowReduce
Augmented matrix

Out[]=
1
G
ℏ
/
3/2
c
Verify this has dimensions of length:
In[]:=
ResourceFunction["CanonicalDimensionalProduct"]Times@@MapThreadPower,
ℏ
,
c
,
k
,
G
,Map[Last]RowReduce
Augmented matrix

Out[]=
1
L
Convert the length of the Airbus A321neo in the Wolfram Cloud to the Planck length:
In[]:=
EntityRegister[ResourceData["https://www.wolframcloud.com/obj/burbery1/DeployedResources/Data/Airbus-Aircraft-Data/"]]
Out[]=
{Airbus}
In[]:=
EntityList["Airbus"]
Out[]=

A321neo
,
A320neo
,
A319neo
,
A321ceo
,
A320ceo
,
A319ceo
,
A220-100
,
A220-300
,
A330-800
,
A330-900
,
A330-200
,
A330-300
,
A350-900
,
A350-1000
,
A321P2F
,
A330-200F
,
A330P2F
,
BelugaST
,
BelugaXL

In[]:=
A321neo
AIRBUS
["overall-length"]
Out[]=
44.51
m
In[]:=
UnitConvert
A321neo
AIRBUS
["overall-length"],Times@@MapThreadPower,
ℏ
,
c
,
k
,
G
,Map[Last]RowReduce
Augmented matrix

Out[]=
2.7539×
36
10
G
ℏ
/
3/2
c
This is basically the definition of the Planck length. We can verify this by looking at the equivalent forms of the Planck length:
In[]:=
MapAtAssociation,ReapMemberQSowActivate
Planck length
PHYSICAL CONSTANT
["EquivalentForms"],"forms of the Planck length",SowTimes@@MapThreadPower,
ℏ
,
c
,
k
,
G
,Map[Last]RowReduce
Augmented matrix
,"the form of the Planck length that I derived",_,Rule,2
Out[]=
True,forms of the Planck length
1
ℏ/(
m
P
c
)
,
1
2π
h
/(
m
P
c
)
,
1
G
ℏ
/
3/2
c
,
1
2π
G
h
/
3/2
c
,
1
t
P
c
,the form of the Planck length that I derived
1
G
ℏ
/
3/2
c
