Geometrical Dimensional Analysis
Formulas of physics, represented as vectors, are structured into
vector spaces simultaneously recognizable as multi-dimensional use
of mathematical language and logic, concept maps, and dimensional
analysis maps.  With the values of fundamental physical constants
inserted as common logarithms, the results are multi-dimensional
logarithm tables, structured systems of natural units, and periodic
tables of physics definition and physical law which yield new
perspectives on the contention over terminology and unit systems
central to all of science.  The metaformula format not only offers
increased efficiency and density of information display that is the
calculating device, combination of these structures, as vector space
multiplication, amounts to mathematics of an order above and beyond
customary use, portraying whole new arrays of relationships derived
all at once.
To learn, and especially to teach knowledge without structure is a
mistake, for evolution has hardwired our brains to become familiar
with our place in a topology, concrete or abstract, like any fish's
spawning ground or any troop's social order.  Structure provides the
common ground for communication, it facilitates the extrapolation of
pattern, it lets the student see the road ahead to knowledge, and it
helps the taught to organize and remember what has been learned.
26, 106, 266, 275, 276, 277.
Students of history fill in a timeline, students of biology flesh
out the tree of life, those of chemistry internalize the periodic
table of the elements.  In physics, child of mathematics, wed to the
quantity calculus, that which structures and maps must be as logical
as these subjects themselves.
Happ, who sought such structure in 1954, produced trees, branchings
of line segments representing formulas of physics. 84  Seen as
correlating displacement to changing factor of unit and numerical
value, they are disarrayed, unorganized groupings of concepts as
nodes, linked by lines that represent the formulas, the statements
of mathematical relationship.  Though rudimentary, they do show that
statements with truth independent of position or orientation can be
combined into structures or patterns that project more truth.
In 1951, seeking structure for his purposes, Corrsin had produced a
dimension space for the terms and concepts of mechanics as exactly
the tool his title claims: "A Simple Geometric Interpretation of the
Buckingham Pi Theorem." 6, 125, 126, 127  Connect the dots, as it
were, and if the formula makes a closed path it is dimensionally
sound.  His dimension space is not only one of the mappings produced
by this author in searching for a method to organize the expressions
of physics, it is also Happ's disorganized groupings meshed, woven
into a three dimensional lattice of logic, a physics concept map as
also a formula map.  With each location seen as the endpoint of the
various paths to it and displaying the composing factors, it is a
factor map, and since a particular unit or group of units fits each
concept, the concept/formula/factor map is a units map as well.
As Happ and Corrsin each note, the mathematical change of value to
be associated with change of position in these depictions is
logarithmic, and that reminds of another, older approach to the
structuring of physics knowledge.  Archimedes spoke of expressing
numerical values as logarithms in a common base as a way to allow
multiplication through the addition of exponents, Napier, in 1614,
facilitated practical use of the principle with first publication of
a table of logarithms, and Briggs published his table of common logs
with the important 10^0 = 1 a short time after that. 1, 11, 19, 71
With Gunter's production of his logarithmic line of numbers, mapping
the linear structure of these collections on a scale to make their
use a mechanical affair through adjustment and movement of a pair of
calipers, Oughtred and his student Delamin were able to alter the
technique and produce the first modern slide rules, one linear and
one circular. 14, 15, 71, 259  The slide rule's convenience of two
scales, Gunter's overt mapping of a collection on a line, and the
need to display only the range of values between 10^0 and 10^1 that
was already used in existing log tables, these factors all combined
to provide human calculators with the handy, holstered weapon of
choice for centuries to come.
The word overt is used above because it is important to remember
that, while the actual number of values in a log table was more or
less limited, the patterns of change in the listing allowed for
extrapolation of values not actually seen.  In other words, even in
columns and rows, a log table itself can be seen as a map, Gunter's
logarithmic line of numbers is surely a map, and it might be argued
that three hundred years ago, without development of the slide rule,
the linear depictions would have been extended in length, combined
with additional number lines in independent directions, and then
recognized as formats similar to those being proposed here.
Over three hundred and fifty years of slide rule and log table use
has not only entrenched the form of numerical expression we now call
scientific notation, tailored to be translated into characteristic
and mantissa, it has fueled argument for expression of numerical
values as logs wherever possible.  Logarithms are more compact and
easier to remember, they emphasize the magnitude of a value by
placing it first, they simplify conversion to and comparison with
reciprocals, and they facilitate multiplication in a pursuit where
one might employ addition, but will always use multiplication (by a
unit of measure, even if dimensionless unity). 12, 17, 33, 34  As
electronic calculators have almost eliminated log table and slide
rule use, one might also say, now, that the multiple step, multiple
context scientific notation is outmoded, meant for use with obsolete
technology.  Ironically, while an electronic calculator can be had
for only a small fee, offering a multitude of mathematical functions
and able to display more values than a human could count, only one
or two at a time are visible.  Since any structures, any patterns to
be found in the value collections are seen only by the little robot,
some opportunity for the use of our intelligence has been lost.
Made all the more impressive, then, the Gunter Table submitted with
this material is a compilation of quantities and numerical values
expressed as common logarithms and arranged in increasing absolute
value order.  With a sequencing germane to the subject and a reduced
context mode of expression that is also designed to require minimal
space, the list makes it possible, even desirable, for every product
of every quantity multiplication ever performed in any investigation
to be compared to and included in an ever growing list of values: a
dictionary for the vocabulary of nature.  All that is required next
is a method for discrimination within the unit of measure context of
quantities, and, without loss of the advantageous sequencing, that
can be supplied by expansion into more dimensions.
Stipulate, for instance, three different logarithmic number scales,
and that change of position along each is to correlate to change of
numerical factor and unit factor.  Further, stipulate that they all
have their origin located at the same place and are oriented in
three orthogonal directions.  Finally, adopt the conventions that:
(1) displacement along the up and down axis correlates to increasing
and decreasing factor of length, (2) displacement along the left and
right axis corresponds to increasing factors of time and frequency,
respectively, and (3) displacement along the axis in and out of the
plane of the page corresponds to increasing and decreasing factor of
mass.  Then, treating the displacements as vectors (gectors for use
here), gector addition produces the logarithmic numerical value and
unit value for each location in the quantity space as the sum of the
component gectors parallel to the reference axes.  The result should
be seen as a three dimensional log table and quantity table which is
identical to the dimension space discussed above. Further, if the
quantities chosen to insert as base units are from a natural system,
the concept, formula, factor, and units map becomes recognizable as
as a multidimensional structured natural unit system projecting laws
of nature as mathematical relationships.
Beginning with Stoney's as the first, in 1881, a number of those
natural systems of units have been proposed, where the dimensional
analysis of experimentally derived constants of nature supplied the
base units. 33, 51, 63, 76, 137, 138, 148, 272  Using experimental
values of his day, Stoney inferred a mass, length, and time set of
base units that can be compared to today's Planck mass, [PM], Planck
length, [PL], and Planck time, [PT].  A later natural system found
in literature and discussion is called the atomic system of units.
It is based on mass of the electron as atomic unit of mass, [Me],
charge of the electron as atomic unit of charge, [Q0], radius of the
Bohr atom as atomic unit of length, [a0], and ([2Pi][f0]), product
of two Pi and the frequency in that idealized orbit as atomic unit
of frequency. 23, 75, 76, 120
Current values of both these unit systems are included in online and
journal publications of the fundamental constants by the National
Institute of Standards and Technology, with emphasis on the fact
that they are non-SI units.  It is not only true that the Bohr model
of the hydrogen atom has been relegated to simply a starting point
for more modern description, the atomic units of mechanics and the
atomic units of electromagnetism had their beginnings before the
adoption and popular acceptance of the International System of Units
(or SI), and these units, as shown, do not mesh into a coherent set.
23, 27, 77, 154  However, if charge is replaced by the product of
charge and the square root of the electrical interaction constant,
{[Q0]([Ke]^[1/2])}, in each formula presented by NIST to define an
atomic unit of electromagnetism, then the elements of the system can
again all be viewed in a single context. 223, 261, 262, 263
To depict the four MKSA core concepts of the SI in three dimensional
gector spaces and present the density of information and proposed
structure in common two dimensional media, some of this material to
date has employed the debatable convention of treating permittivity,
[k], as dimensionless and associated with no amount of displacement.
27, 32, 36, 41, 52, 56, 79, 124, 136, 188, 237, 246, 256, 265  As
for a helicopter pilot flying into Manhattan with only a street map,
the contention has been that even a map of insufficient dimension is
better than none at all, especially if the GDA technique is seen as
an alternative evolution of log tables and multidimensional use of
mathematical language.  With the four dimensional mappings included
in this writing, it is hoped that the advantages of the various
formats will be even easier to recognize.
Perhaps more controversial than insufficiency of bases in mapping
will be use of what will be called the ad hoc symbol system and set
of unit labels, consisting entirely of alphanumeric characters.
These have come into being not only because they are a better fit to
the density of information display in graphics and value tables,
their sometimes cumbersome and space consuming use in text is offset
by the fact that they allow the public at large, who do make it all
possible, to take part in the discussion without the multiple fonts,
italics, subscripts, superscripts, foreign language characters and
graphics which are demanded by authorities and publishers of print.
Furthermore, the English language alphanumeric set can be exported
to calculation and optical character recognition software, and both
in and out of the internet's worldwide forum it makes the subject
more easily searchable for word, symbol, formula, or numerical value
while remaining backward compatible even to the typewriter.
With these declarations in mind, consider the example of the gector
space in GJN020A.  Planck mass, Planck length and Planck frequency
[Pf], serve as unit gectors to structure quantities that would be of
interest where two Planck masses are separated by a distance equal
to Planck's length in an idealized planetary model similar to the
Bohr atom.  Placed at their respective locations, then, are Planck
force, Planck energy, Planck momentum, the speed of light [c], the
rationalized Planck constant [hB], and the constant of gravitation
[G]. 167, 240  In contrast, GJN016A is both different and the same.
While GJN020A is numerical, GJN016A is analytical, displaying with
appropriate labels instead of numerical values.  While both figures
deal with the same concepts in the same system of units, they are of
distinctly different shape.  GJN020A is a more visible portrayal of
the conceptual statements:
[G] = ([PM]^[-1])([PL]^3)([Pf]^2) ,
[hB] = ([PM]^1)([PL]^2)([Pf]^1) , and
[c] = ([PM]^0)([PL]^1)([Pf]^1) ,
and the nature of the statements:
[PM] = ([G]^[-1/2])([hB]^[1/2])([c]^[1/2]) ,
[PL] = ([G]^[1/2])([hB]^[1/2])([c]^[-3/2]) , and
[PT] = ([G]^[1/2])([hB]^[1/2])([c]^[-5/2])
is more evident in GJN016A, though all these declarations are true.
In GJN017A, and in GJN018A, where the unit gectors are mass of the
electron [Me], the atomic unit of length [a0], and ([2Pi][f0]), the
atomic unit of frequency, various atomic units of mechanics fall in
place, this time with the analytical and numerical diagrams of the
same shape, to enable consideration of the Bohr atom environment.
One aspect of the GDA technique that is noteworthy is visible in
each numerical view above as the plane of unity which is shown to
bisect the diagram.  Since a displacement and its associated change
of factor in a uniform space is the sum of components both parallel
to and proportional to the unit gectors, any occurrence of a given
numerical value will be found in such a plane.  Moreover, all the
planes of value in a uniform space will have the same slope.  This
further example of pattern allows one to search for those locations
where significant numerical values approach or coincide with some
specific plot point.
Another advantage in use of three dimensions to portray the MKSA
system, keeping permittivity dimensionless, lies in how readily one
can show that gector spaces may be displayed in alternative shapes,
surviving vector space transformation.  Aside from the two views of
the Planck unit system already depicted, consider a charge, length,
time space in GJN014A, a current, length, time space in GJN015A, and
the mass, length, charge and mass, charge, time spaces of GJN015R.
Since the mathematical relationships survive such transformation and
all views are equivalent as far as accuracy and coherence, choice is
a matter of convenience and pertinence, which, when combined with
history, impelled the overall conceptual representation in GJN03DA.
The most impressive results, however, will still be seen with the
expansion into four dimensions.
In the atomic units maps, the value of (hB), the rationalized Planck
constant, plots out at ([M]^1)([L]^2)([f]^1).  From there, one unit
to the right and one unit up, as multiplication by the atomic unit
of velocity, the location for ([hB][v0]) is accepted as equal to
{([Q0]^2)[Ke]}, product of the square of the fundamental charge and
the electrical interaction constant. 23, 65  Halfway back to the
origin, as the square root of ([hB][v0]), [Q0]([Ke]^[1/2]) will be
found to be in context instead of charge alone, as explained above,
and makes clear the motive for choosing ([Ke]^[1/2]) as that unit
gector representing the fourth dimension.  Symbolized by a wedge, as
in GJN164A, each unit gector into the fourth dimension displaces out
from the 2D depiction plane as a factor to be divided out, or it
displaces in from the depiction plane as a factor to be multiplied
in: divide out and multiply in.  With reference to the numbered
locations found in the alternative atomic units views of GJN163A
and GJN165A, the validation table exhibits both verification for the
approach and the rationale for the more general maps in GJN166A,
GJN167X, and GJN175X.
Seeing these samples verify themselves as the visual embodiment of
quantity calculus, forced into sight by the mapping conventions,
consider again the ongoing publication of the fundamental constants.
Their values are decided by a process of successive approximations
called the method of least squares, and, in the words of the NIST
authors themselves, "...may be regarded as conventional values or
best estimates, depending on one's point of view." 23b, 118, 121
In evolution for two hundred years, the method is the clear cut
choice of present day authorities, but this has not always been the
case.  In 1930, and for years afterward, there was an alternative
under discussion that was quite literally a new perspective.  It was
an electron charge, electron mass, Planck constant space that came
to be called the Birge Eye View. 72, 149, 152, 224
As an aid for evaluation of approximations, the three dimensional
Birge-Bond diagram was soon found to be only slightly more desirable
than a reduced map, and a majority of portrayal devolved to the two
dimensional isometric consistency chart, but discussion of both was
present among metrologists and experts for many years to come.  One
concise opinion is Dumond's: "The author believes that a method for which to the greatest extent possible all reliable
original data are separately visible is... much to be preferred to
any method in which original data are concealed behind averages or
least-squares solutions.  No blind mathematical process of averaging
should in his opinion precede an opportunity for the exercise of
intelligent judgement." 152  With the passage of time, especially
as constants proliferated in number, interest in the diagrams waned.
Now, in this writer's opinion, there are good reasons to renew that
interest.  More devices for the labors of calculation, more progress
in theoretical development, and more visualization technologies make
it possible to produce diagrams in greater numbers and of greater
sophistication.  Further than that, it is now possible to see every
depiction discussed so far as only a snapshot taken during the use
of software written for dynamic investigation.  With the ability to
see the effects of incremental change in input values while they are
under way, with the ability to watch the reshaping or movement of
visible patterns, with the potential offered by the total immersion
illusion of virtual reality, it should now be possible to take Lord
Napier's "...numbers speaking for themselves..." to limits never
before dreamt of.  In light of these options, what reason can there
be to ignore such testimony, such second opinion, to give preference
and precedence only to values which must be called best estimates?
Compelling evidence for a point of view asking that question can be
seen through the use of comparison in the multiplicity of views, as
in the GDA technique entitled metaformula ratios. If a gector space
is seen as a metaformula, as a lattice of logic and mathematical
statements, then a metaformula ratio is the mapping produced if one
map is seen as numerator and another as denominator, and point by
point, part by part, and entirety by entirety they are combined.
Consider that in speaking of a one one hundredth scale map there are
three mappings involved, three topologies.  There is the topology of
the real terrain, there is the topology that is the representative
scale diagram, and then there is the mapping of the mathematical
relationships between points of real terrain and of the diagram.
This third map never needs depiction, for it is a trivial process to
show that at each point, in any part, and overall, it will be 1/100,
that is what is meant by one one hundredth scale, and it was created
that way.  With use of mappings like those under discussion here, a
multitude of combinations are possible that are not trivial, but are
seen as new gector spaces, as vector space multiplication, as new
multidimensional arrays of derivation.
As one such metaformula ratio, a projection of dimensionless numbers
because it involves two spaces with the same configuration, GJN023A
is the overlay of GJN020A on GJN018A from discussion above.  Here, a
Planck unit numerator to an atomic unit denominator at each point in
the space not only projects the line of intersection of the separate
planes of unity, where
([PM]^x)([PL]^y)([Pf]^z) = ([Me]^x)([a0]^y){([2Pi][f0])^z} = 1 ,
it creates, all at one time, a whole new lattice of logic, a whole
new collection of patterns made visible for inference, entire arrays
of formulas derived by inspection alone.
Referring to both GJN023A and its analytical counterpart, GJN024A,
it is a given that the origin must correlate to the value 10^0 = 1
in all three spaces.  With the plot point for action in each root
space equal to [hB], the overlay ratio at that point must also be
10^0 = 1.  Since the composite mapping is a vector space and EVERY
displacement of one unit to the right, one unit in, and two units up
must equate to multiplication by one, a glance can confirm that:
[10^0]/[10^0] = [hB]/[hB] ; [Pf]/([2Pi][f0]) = [PE]/[He] ,
[c]/[v0] = [G]([PM]^2)/{([Q0]^2)[Ke]} , etc.
Since, at velocity, [c]/[v0] equals 1/[aa], the inverse of the fine
structure constant, and this is one unit down and one unit out from
[hB]/[hB], every such displacement must be 1/[aa]:
(1/[aa]) x [hB]/[hB] = [c]/[v0] ,
(1/[aa]) x [G]([PM]^2)/{([Q0]^2)[Ke]} = [c^2]/([v0]^2) ,
(1/[aa]) x [PL]/[a0] = ([PM]/[Me])^[-1] , etc.
In displacing from [hB]/[hB] to [c^2]/([v0]^2), the value changes
from 1 to 1/([aa]^2), therefore each movement of one unit out and
one unit to the right must correspond to 1/([aa]^2):
{1/([aa]^2)} x [hB]/[hB] = [c^2]/([v0]^2) ,
{1/([aa]^2)} x [PM]/[Me] = [Pf]/([2Pi][f0]) , etc.
By having erected the formulas into the structure that is the gector
space, that is a vector space, deriving one formula illuminates a
single array of derivations, and in this way the entire space is a
multitude of relationships derived all at one time.
It is also true that, through the overlay mapping, any entity that
might be of particular interest can now be seen in the context of
all the entities in all three spaces. In GJN020A, [G] is already the
endpoint of all the possible pathways from all the possible starting
points in that gector space, and can be expressed in terms of any of
those quantities if one draws the inference there:
[G] = ([PL]^3)([Pf]^2)/[PM] = [hB][c]/([PM]^2) = [c^2][PL])/[PM]  ,
or makes some substitutions to expand the scope:
  = ([z0]/[d])([PL]^2)[Pf]/[PM] = [PL]/([PM][d][k])  ,
or in the Bohr atom maps of GJN018A and GJN165A:
[G] = {([Q0]^2)[Ke]([Fg]/[Fe])}/([Me][Mp])  ,
  = ([Gye]^2)[Ke]([Fg]/[Fe])([Me]/[Mp]){([Bm]^2)/([ue]^2)} ,
  = [B0][Fg]{([2Pi][f0])^2}([a0]^[9/2])/{[Fe][Mp]([Ke][Me])^[1/2]} ,
  = {[a0]([v0]^2)/[Mp]}([Fg]/[Fe]) ,
  = 4[Ke]{([Bm]/[hB])^2}([Fg]/[Fe])([Me]/[Mp]) ,
  = [Ke]{([2af]/[B0])^2}([Fg]/[Fe])([Me]/[Mp]) ,
  = [Ke]{([Eg0]^2)/([B0])^4}([Fg]/[Fe])([Me]/[Mp]) , etc.
Now, through the metaformula ratio of GJN023A, it is linked to all
those quantities, all those locations in all three gector spaces,
and thus to any substitutions that can be made at any of them:
[G] = ([Q0]^2)[Ke]/{([PM]^2)[aa]} = [Fe]([a0]^2)/{([PM]^2)[aa]} ,
  = [Me][a0][aa]/{([PM]^2)[d][k]} ,
  = [Me]([a0]^3){{[2Pi][f0])^2}/{([PM]^2)[aa]} ,
  = [Ke]([i0]^2)/{([PM]^2)([2af]^2)[aa]} ,
  = [Ke][Me]{([Eg0]^2)/([B0])^4}([2af])/{[PM][Pf]([aa]^3)} ,
  = ([w0]^2)([v0]^2)/{([PM]^2)([Pi]^2)[Ke][aa]} , etc.
Once again pattern recognition and extrapolation enhance the use of
mathematical logic, but, if one can speak of a thread of thought,
then here is fabric of even more elaborate weave.
It is not possible to imagine how the world would be without maps.
No trans-oceanic exploration, no military action, no weekend trip to
see grandma is free of one, directly or indirectly, sooner or later.
As mentioned, other sciences have a central, stay on the road, "turn
ye back lest ye fall off ye edge", "beyond here be dragons" diagram.
Yet here is this physics, this foundation science with the system of
units of measurement at its core, base and cornerstone for all the
other sciences, with no single, common ground for communication and
reference overview, no map.  There are many more graphics than those
already cited, some with aspects of GDA. 23d, 48, 111, 114, 267, 278
There are more alternatives to the matrices and set theory common in
discussions of dimensional analysis. 53, 64, 247, 279, 281  There
are diagrams specific to the controversial subject of defining units
and systems. 22, 93, 94, 96  There are unit systems tables and lists
too numerable to cite.  This report is made in the belief that the
GDA techniques have advantages in all these areas, that they have a
mathematical character as old as western science but as modern as
virtual reality, and that they combine into a truly elegant set of
physics tools.  If nothing else, the author hopes to have shown how,
in some cases, the greatest impediment to understanding can be the
choice of or use of language, the format of expression, or in how
the investigators view the question.