Geometrical Dimensional Analysis ABSTRACT 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. BACKGROUND 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. MAP THE PHYSICS 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. DIMENSIONS PLUS 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 visualizing...in 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. IN CONCLUSION 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. --------------------------------------------------------------------