Rational Dimensia

R. Hanush Physics Phor Phun, Paso Robles, CA 93446∗ (Received) Rational Dimensia is a comprehensive examination of three questions: What is a dimension? Does direction necessarily imply dimension? Can all physical law be derived from geometric principles? A dimension is any physical quantity that can be measured. If space is considered as a single dimension with a plurality of directions, then the spatial dimension forms the second of three axes in a Dimensional Coordinate System (DCS); geometric units normalize unit vectors of time, space, and mass. In the DCS all orders n of any type of motion are subject to geometric analysis in the space- time plane. An nth-order distance formula can be derived from geometric and physical principles using only algebra, geometry, and trigonometry; the concept of the derivative is invoked but not relied upon. Special relativity, general relativity, and perihelion precession are shown to be geometric functions of velocity, acceleration, and jerk (first-, second-, and third-order Lorentz transformations) with v2 coefficients of n! = 1, 2, and 6, respectively. An nth-order Lorentz transformation is extrapolated. An exponential scaling of all DCS coordinate axes results in an ordered Periodic Table of Dimensions, a periodic table of elements for physics. Over 1600 measurement units are fitted to 72 elements; a complete catalog is available online at EPAPS. All physical law can be derived from geometric principles considering only a single direction in space, and direction is unique to the physical quantity of space, so direction does not imply dimension.

I. INTRODUCTION analysis can be applied to all. The power and efficiency of dimensional analysis make it an extremely valuable tool for the sciences.3–5 It is the proposition of this author What is a dimension? Does, as Edwin Abbott puts that all physical measurements can be expressed and an- forth in his 1884 novella Flatland, “direction imply alyzed in the terms of the Periodic Table of Dimensions. dimension”?1 Can all physical law be derived from ge- ometric principles as Minkowski predicts?2 To answer these questions one must first consider the myriad of myr- II. CONTRACTION OF THE SPACE METRIC iad of different measurements mankind has used through- out history. Which are unique dimensions and which are redundant? Not only are there scores of dimensia (n, pl Pluralitas non est ponenda sine necessitate: 1: the plural of sets of dimensions, i.e., money, monies): Plurality should not be posited without neces- SI, mks, cgs, etc., but within each system there are nu- sity. merous quantities that express the same dimensionality. For length one could measure a wavelength, or a path -Occam’s Razor length, or a radius, or a diameter—and in the SI system one could even measure an energy and choose to call it a temperature. It is this dimensia (n, pl 2: the minutiae A. Physical metrics of dimensions) that frustrates the researcher of geometric relations between the dimensions. It is the thesis of this Consider the Pythagorean distance formula article that these dimensia can be rationalized, and with 2 2 2 that rationalization, the underlying geometric relations x + y = d , (1) are revealed. illustrated in Fig. 1(a), which gives the magnitude of the By characterizing space as a single dimension with space metric d in two directions. Given three spatial a plurality of directions and building on its one-to-one axes, this Euclidean metric is expanded to express the equivalence with time in the space-time plane, a Dimen- magnitude sional Coordinate System (DCS) composed of the three physical dimensions mass m, space d, and time t can be x2 + y2 + z2 = d2 (2) constructed. Within that construction, kinematic and relativistic effects are examined and expanded beyond of a spatial three-vector. The spatial three-vector is current knowledge, based purely on geometric principles shown in comparison to the spatial two-vector in Fig. and the constancy of the speed of light. Validation of m 1(b). The scalar magnitude d2 is the starting point of as a third axis is provided in the form of an exponential most commonly cited discussions of relativity.2,6–11 The- scaling of the coordinate axes, resulting in an ordered Pe- oretically, the space metric can be expanded to include riodic Table of Dimensions. Within this table over 1600 any number of components, but practical geometric rep- units of measurement have been fit to 72 elements, and resentations of four or more mutually orthogonal compo- both arithmetic and geometric methods of dimensional nents seem to be nonexistent. 2

y in front c y in front c d d l l z igh z igh tlik tlik d2 d3 e spacelike d2 d3 e spacelike y y d y beforetimelike timelike after d y beforetimelike timelike after x t x t spacelike spacelike d2 ike d2 ike htl htl lig lig

(a) x (b) x behind (a) x (b) x behind

FIG. 1: The space metric d in (a) two and (b) three directions. FIG. 2: The timelike and spacelike hyperbolae of space-time separated by lightlike asymptotes.

In consideration of physical phenomena, the distance given by the space metric can be set equal to the distance Whether infinitesimal or not, when s2 is greater than light travels in one second, i.e., 3×108 meters. Algebraic zero (positive), the interval is called timelike and can be manipulation of the dimensional relation for light speed anywhere in the areas labeled timelike in Fig. 2; when s2 c = d/t gives is less than zero (negative), the interval is called space- like and can be anywhere in the areas labeled spacelike 2 2 2 d = c t (3) in Fig. 2.2,6,12,18,25–28 Physical interactions can occur in timelike intervals; only physical separation can occur over so that spacelike intervals. 2 2 2 2 x2 + y2 + z2 = c2t2, (4) Consideration of a spacelike interval c t − x − y − z2 = −s2 implies imaginary coordinates. Some have as presented by Einstein and others.8,9,12,13 Note that in suggested2,9,10,22 that this imaginary coordinate origi- Eq. (3), t2 could just as easily be the isolated variable, nates as a coefficient of either the term for space, i.e., leaving d2/c2 expressed in square seconds.14,15 (id)2 = i2(x2 + y2 + z2), or the term for time, i.e., Collecting both the three-component term for space (ict)2. Others, such as Misner, Thorne, and Wheeler and the term for time from Eq. (4) on either side of the point out, “This imaginary coordinate was invented to equal sign gives two equally valid expressions: make the geometry of space-time look formally as lit- tle different as possible from the geometry of Euclidean x2 + y2 + z2 − c2t2 = 0 (5a) space.”6,25 This is demonstrated by the similarity of the two general forms for a spatial two-vector in imaginary and space, (ct)2 + (id)2 = s2 and (ict)2 + (d)2 = s2, to the Pythagorean distance formula in Eq. (1). c2t2 − x2 − y2 − z2 = 0. (5b) But space-time is not imaginary; space-time is real and 2,10,13,16,19,26–30 These two forms impart antisymmetric metric signa- hyperbolic. Intervals are not given by the y in front c sum of squares, but by the difference of squares—the d tures of (+, +, +, −) and (+, −, −, −), respectively, to z l 2,6,9,10,12,13,16 squares of real numbers. The only imaginary term oc- igh the components. For the purposes of this d d tlik discussion, the (+, −, −, −) metric is employed unless curs in s, which is imaginary only when the difference of 2 3 e spacelike otherwise noted; both are prevalent throughout the real coordinates is negative, i.e., spacelike, resulting in 2,6,7,9–24 a 90-degree rotation of the hyperbolic axes. Figure 2 il- y y literature. d beforetimelike timelike after lustrates the hyperbolic asymptotes of time and space, x t When these expressions are equal to zero, as in Eqs. (5a) and (5b) and as shown by the line labeled c in Fig. and their relation to the asymptotic light speed axis. spacelike 6,12,25,26 d2 ike 2, the interval is called lightlike. Only phenom- When the difference of the squares is negative, the re- htl 2 lig ena propagating at the speed of light can take place over sult is −(s) , the root of which is is. This does not result lightlike intervals. Intervals other than those accessible from either the term for space or the term for time being (a) x (b) x behind only at light speed can be considered by letting the re- less than zero, but from the ratio of d/t referred to in Eq. sult of Eqs. (5a) or (5b) vary. Designating the interval (3) exceeding the value of light speed c. given by these expressions as s, and limiting the discus- Equation (6) is the rectangular form of the space- sion to infinitesimal local frames of reference, gives the time metric, which is equivalent to Eq. (5a) when the 2,6,7,9–14,16,19,20,22 scalar length of a worldline in the familiar form of the (+, +, +, −) metric is employed. Other space-time metric: forms of the space-time metric frequently found in the literature include the metric tensor gµν introduced by c2dt2 − dx2 − dy2 − dz2 = ds2. (6) Einstein in the general theory. This is commonly pre- 3 sented in matrix form, and z as equivalents, to the point of melding them into a single space-time continuum.8 Both Minkowski and Weyl  1 0 0 0  also contend that time is on equal footing with the three 2 0 −1 0 0 directions of space,2,11 which permits identical treatment ds = gµν =   , (7)  0 0 −1 0  of the four coordinates x, y, z, and t. And in Edwin Ab- 0 0 0 −1 bott’s Flatland, A. Square, the two-dimensional narrator of the story, remarks to Lord Sphere of three-space that or with the shorthand notation diag(+1, −1, −1, −1), “dimension implies direction and measurement.”1 Con- with all signs reversed in the case of the (−, +, +, +) sidering the context, this last statement begs the ques- 6,10,12,17,18,21–23 metric. (By convention time is assigned tion, exactly how many directions does dimension imply? to the zeroth of a 0-3 index here, rather than the fourth Two, as in Flatland? Three, as in Lord Sphere’s world? of 1-4.) In tensor notation the interval of the space-time Typically, relativistic systems6–9,12,13,18,25,33,36 consist metric is given by the relation of the three directions, x, y, and z. When three directions 2 µ ν are given, almost without exception, y and z are simply ds = gµν dx dx , (8) set to zero and all events are considered to occur along the x-axis. As can be seen from the Lorentz transformations where the superscripts µ and ν are indices of 0, 1, 2, 3 in three directions, rather than exponents, and Einstein’s summation con- vention is applied.6,7,9–12,20,21,31,32 vx 0 t − c2 Two significant polar forms of the space-time metric t = (11) q v2 also warrant mention. The first is the Schwarzschild met- 1 − c2 ric equation

2 0 x − vt 2 2 2 dr x = (12a) ds = c dt (1 − 2m/r) − q v2 (1 − 2m/r) 1 − c2 −r2(dθ2 + sin2 θ dφ2), (9) 0 where r, θ, and φ are spherical coordinates, and m = y = y (12b) GM/c2 where G is the gravitational constant and M is the total mass of the system; the condition where z0 = z, (12c) r = 2m, so that the term 1 − 2m/r goes to zero, is the Schwarzschild radius; an object of mass m smaller than considering y and z has absolutely no effect on the physics its Schwarzschild radius is a black hole.7,33–35 The second whatsoever.8–10,12,37 The limitation lies in the difficulty polar form that should be mentioned is of geometrically representing four directions, all at right angles to each other. Ohanian6 addresses this limitation ds2 = c2dt2 − dr2 − r2dφ2 − dz2, (10) by simply omitting the z-coordinate in constructing the light cone t2−x2−y2 = 0, while Muller38 finds it revealing where r, θ, and z are cylindrical coordinates of flat to assign y to the spin axis and proceed in t, x, and z. Minkowski space-time in string theory.33 For the system of one spatial direction the space axis is No matter the form, the scalar interval given by s2 is simply d, the composition of the spatial axes of a multi- the space-time metric. In a grammatical sense all these directional system as defined in Eqs. (1) and (2) and il- forms are synonymous; they are many ways of saying the lustrated in Fig. 1. Cook says it most succinctly when same thing. The s2 resulting from any one form of the he notes that it is the proportion of d`, a composition of space-time metric is the same s2 given by any other form the x, y, and z directions, to time that places the speed for the same interval, and when any transformation s0 of light at the invariant value c in all directions, and that conforms to the postulates of relativity is applied, gives the space-time metric in one spatial direction as s2 = (s0)2. The space-time metric expresses space-time ds2 = −c2dt2 + d`2, albeit with the (−, +, +, +) metric.7 increments equivalently with respect to any inertial ref- Some, including Minkowski, disregard y and z for the erence frame, a property called invariance. While direc- sake of simplicity.2,12,24,26,27 Others, including Pierseaux, tion can be determined from initial values, the space-time who quotes Poincar´eextensively on this subject, point to metric gives the scalar length of a worldline without re- the arbitrariness of the x-axis and use the convention of gard to direction, while the linear velocity vector, with assigning to space a single direction defined by the linear tangent d/t, always gives the direction. velocity vector.16,36 In the end it all boils down to one direction in space at any one instant in time. The addition of directions to the dimension of space B. Space-time equivalence conserves the dimensionality of each direction in the nonlinear proportion of the sum of squares. Since all In the special theory of relativity, Einstein sets space forms of the space-time metric are equivalent, consid- and time equal, t = x = y = z = 0, treating t, x, y, eration of the rectangular form in Eq. (6) is equivalent 4 to consideration of any of the other forms, including has been done that can identify an x direction indepen- those presented in Eqs. (7) through (10). Extrapolat- dently of the y or z directions. However, when involv- ing the pattern given by the rectangular forms for spa- ing kinematics, special relativity shows that there exists tial dimensions of one, two, and three directions gives a geometrically preferred direction along the axis of uni- d2 = x2, = x2 + y2, = x2 + y2 + z2, = x2 + y2 + z2 + ··· . form linear translation, also known as the velocity vector. The relationship of the terms for space and time are such Convention assigns this to the x-axis in the Lorentz trans- that the sum of the squares of the spatial directions, i.e., formations for space, Eqs. (12a) through (12c), but by Eq. (2), is proportionate to the square of the single term Lorentzian definition the dynamics of space and time are for time c2t2. If x, y, and z were each individually of the not affected along the y and z axes only because motion same physical nature as time, then the proper propor- does not occur along the y and z axes, i.e., y = 0, z = 0. tions would be ct : x, ct : y, and ct : z rather than ct : d. But if no particular direction can be identified as the x Geometrically, the composition of directions is Euclidean, direction, then applying the velocity vector specifically but the composition of space-time is hyperbolic. Numer- to the x-axis is completely arbitrary.16,36 ically, this inequivalence of direction to time varies with Relativistic contraction in space occurs only in the the number of directions considered and reaches the fol- direction of linear motion. Neither x, nor y, nor z lowing maxima when distances in each of the directions matters—only the direction of velocity v contracts. Then considered are set equal to each other: any space-time metric taken in the direction d, defined by the velocity vector, can be said to be in contracted ct : d(x) = 1 : 1 (13a) form. The contraction is also grammatical in nature in that the term for space is “shortened by omission” (Web- 1 ster’s: contraction) of extraneous directions. The space- ct : d(x, y) = 1 : √ = 1 : 0.707 (13b) time metric then becomes a simple two-vector in a space- 2 time plane that conforms to hyperbolic plane geometry as the magnitude 1 ct : d(x, y, z) = 1 : √ = 1 : 0.577 (13c) 2 2 2 2 3 s = c t − d (14) with a slope given by the velocity 1 ct : d(w, x, y, z) = 1 : √ = 1 : 0.5 ... (13d) 4 d v = . (15) t It is a fundamental proposition of this paper that space is a single dimension, equivalent to time, with three (more Pythagorean contraction of the space metric should not or less) directions. Corollary to that, direction is not be confused with a tensor contraction of the space-time equivalent to dimension, as demonstrated in the Lorentz metric as discussed by Jackson and Ohanian, in that transformations, Eqs. (11), and (12a) through (12c). Ge- the rank of the remaining space-time matrix is only re- ometrically, the space-time metric represents the length duced by one.6,12 This Pythagorean contraction derives of a worldline in a two-axis coordinate system composed its name purely from the physical phenomenon associ- of a single time axis and a single space axis. The space ated with it. The space metric d, on its own, is reduced axis is a Pythagorean composition of the three (more or in rank from a 32 matrix to a 30 matrix, and accordingly less) spatial directions x, y, and z, as in Eq. (2), aligned some directional information, i.e., x, y, z, but not v, is along the axis of linear motion. Only the axis paral- lost. However, no dimensional information is lost. The lel to linear translation is affected in the same manner full tensor contraction of the space-time metric from a as time. If space is considered as a single dimension of 42 matrix to a 40 matrix results in the scalar s2; d and t equal standing to time, with three (more or less) direc- have direction, s does not. tions, the hyperbolic identity of distance x2 − y2 = s2 holds true for a worldline in two-dimensional space. Brill 28 and Jacobson provide a very enlightening discussion C. Orientation of this x2 − y2 = s2 geometry that also includes a dif- ference of squares geometric proof in the manner of the When using the contracted space metric, i.e., Eq. (2), Pythagorean prooof illustrated as x2 + y2 = d2 in Fig. 1. to express the velocity vector in Eq. (15), some rational This is the geometry of relativity. tenets of geometric analysis require revisiting. First and The composition of spatial directions is independent of foremost, the slope of a line is its rise over run. If v is to any absolute direction in an inertial reference frame con- be the slope of a line as in Eq. (15), then the rise is d, sidered to be at rest. As Einstein notes in one popular and the run is t. The general geometric statement, with exposition on relativity, “the most careful observations the slope given as m in the xy-plane, is have never revealed such anisotropic properties in ter- restrial physical space, i.e., a physical nonequivalence of y 9,39 m = , (16) different directions.” In other words, no experiment x 5 or as expressed in slope-intercept form, where b is the D. The mass axis y-intercept: Contraction of the space metric reduces a four dimen- y = mx + b. (17) sional problem to one of two dimensions. As discussed in Sec. II A, practical geometric representations of four Substituting t for x, d for y, and v for m in Eq. (16) or more components orthogonal to each other seem to be gives Eq. (15), also equivalent to Eq. (17) when b is set to nonexistent. However, the representation of three orthog- zero. Another frequently invoked property of the velocity onal components as x, y, and z is very familiar. Contract- vector is that its angle θ from the t-axis is given by the 25,26,30,40 ing the space metric not only simplifies the space-time inverse tangent of the velocity, or conversely metric, but also results in a free axis within the confines d of familiar three-vector mathematics. It would be nice tan θ = , (18) if there were a third vector quantity to be represented t by that free z-axis. Preferably this quantity would be and dimensional so as to obey certain, yet to be determined, fundamental physical laws of dimensions also applicable y tan θ = . (19) to space and time. It should also be related to space-time x in a right-hand manner just as space is related to time. Given Eqs. (16), (18), and (19), it seems more rational Also, there must be some transformation, such as Eq. to propose that t goes to the x-axis and d goes to the (3), that relates units on this new axis to the geometric y-axis than the contrary. units on the t- and d-axes. Traditionally, for reasons unspecified, time has been The Planck units provide the transformation necessary assigned to the y-axis.2,6,12,15–17,19,25–30,41,42 Some excep- to express mass as a geometric quantity on the third axis. tions have been noted, but no rationale seems to be given The Planck units for time tP, space dP, and mass mP are 42–44 for these exceptions. However, in a chapter on kine- G matics in one dimension, Ohanian conducts an analysis t2 = ~ (20a) P c5 of worldlines that includes approximations of a curved worldline as the derivative v = dx/dt, which is a tan- gent line with slope (x − x )/(t − t ) = ∆x/∆t taken G 2 1 2 1 d2 = ~ (20b) between any two infinitesimally separated points on the P c3 worldline.45 In order to show the algebraic relation of the slope, the tangent, and the derivative to the space-time 2 ~c plane, a.k.a. the xy-plane, the t-axis is made the horizon- mP = , (20c) tal axis. This is the standard that should be practiced G with the same implications as maintaining the orienta- where ~ is Planck’s constant h divided by 2π, and G is tion of the y-axis to the x-axis in any Cartesian system. the gravitational constant. The relations between these Any system can consider itself to be at rest. Any sta- are tionary system, i.e., v = 0, will have a velocity vector mPG parallel to the resting system’s t-axis. The direction of ctP = dP = , (21) the t-axis defines the state of a system at rest. Since c2 any system may be considered at rest, every system has 2 2 2 so that dP/tP = c as given in Eq. (3), and the unit vec- a defined t-axis. When v = 0 the space-time metric in tors relating a three-axis system of physical dimensions Eq. (14) and the t-axis are parallel because only the t on the Planck scale are17 term contributes to the s term; every measurement takes some finite amount of time. Any system in uniform mo- tP = ctP (22a) tion, i.e., v > 0, will have a velocity vector at an angle to the t-axis of the resting system. The velocity vector in the resting system is parallel to the t-axis of the system dP = dP (22b) with velocity v; any system can consider itself at rest. The space metric of the system with velocity v, in its mPG proper orientation, is always taken to be in the direction mP = , (22c) c2 of motion; the t-axis of the (resting) system measuring v is by definition at an angle of tan−1v clockwise from which go to the axes traditionally labeled as x, y, and z, the direction of motion. The d-axis is always perpen- respectively. The Pythagorean magnitude d of the space dicular to the t-axis.18,21 Making d the functional axis, metric given by d2 = x2 + y2 + z2 is then analogous to secondary to time, in a right-handed system, facilitates the magnitude r of the Pythaogrean dimensional metric: the treatment of space-time in accordance with all the 2 accepted rules and methods of geometric analysis in any mG r2 = (ct)2 + (d2 + d2 + d2) + ; (23) xy-plane. x y z c2 6

whereas the Minkowskian dimensional metric is of the m form m m d s2 = t2 − d2 ± m2, (24) m d d with the ± nature of m to be left to future explorations. d All of the rules applying to three-directional geometric t t t t analysis, as well as the rules applying to dimensional analysis, are equally applicable in this three-dimensional system of coordinates. FIG. 3: The mutually perpendicular unit vectors of time t, space d, and mass m on their respective t-, d-, and m-axes.

III. THE DIMENSIONAL COORDINATE d v-1 c SYSTEM d v-1 c v The whole universe is seen to resolve itself v into similar world-lines, and I would fain an- ticipate myself by saying that in my opinion t physical laws might find their most perfect ex- t pression as reciprocal relations between these world-lines. FIG. 4: Light speed c, and the velocity v and inverse velocity v−1 vectors in the space-time plane. -H. Minkowski2 unit on the t-axis equals 1 second, a measurement of 1 A. The prime dimensions second has a geometric description 1 unit long and par- allel to the t-axis; a measurement of 3 × 108 meters has a geometric description 1 unit long and parallel to the Physical units can be assigned to the coordinate axes d-axis, perpendicular to the t-axis; and a measurement by the following relations, when the measurements of t, of 4 × 1035 kilograms has a geometric description 1 unit d, m, and the physical constants c and G are in SI units: long and parallel to the m-axis, perpendicular to both t = ct meters (25a) the t- and d-axes. Other magnitudes of mass, space, and time are described by scaling of the unit vectors.

q 2 2 2 d = dx + dy + dz meters (25b) B. First order motion

Gm Consideration of the special case of uniform linear mo- m = meters. (25c) tion requires examination of the first-order rate of change c2 of position, velocity v. The geometric description of ve- These relations express the scale for the physical inter- locity is a straight line of slope v with respect to the t-axis pretation of geometric analyses in the Dimensional Co- as shown in Fig. 4 and given by Eq. (15); v = c = 1 de- ordinate System (DCS). The quantities t, d, and m are scribes light speed. Qualitatively, with d as a function of unit vectors analogous to i, j, and k, respectively, in a v, Cartesian coordinate system. Using each of the funda- d(v) = vt. (26) mental dimensions as the unit dimension, three different m scale factors can be derived, as shown in Table I along Quantitatively, d as a function of constant velocity is also with the Planck scale. Each of these physical scale factors given by Eq. (26). m d is equally valid and will lead to the same conclusions, as Both the space-time metric and the Lorentz transfor- mations describe the length of the velocity vector with d will any proper interpolation between them. Therefore, all of the following sets of physical relations are referred respect to the resting system. This is the basis of the t t to as geometric units.17 If there is a need, the unit axis, invariance of s2 with respect to the Lorentz transforma- i.e., t = 1, d = 1, or m = 1, can be so designated. tions. From Eqs. (14) and (15), There is a unique geometric description in the Dimen- s2 sional Coordinate System for every measurable quantity. 2 2 c − v = 2 , (27) The graph of each unit quantity is a straight line, either t coincident with or parallel to its coordinate axis, with and therefore d v-1 each unit vector mutually perpendicular to the others as v2  s 2 c 1 − = . (28) shown in Fig. 3. Designating t as the unit axis, i.e., 1 c2 ct v t 7

TABLE I: Geometric units of the Dimensional Coordinate System (DCS). Units time (t) space (d) mass (m) Planck seconds (s) 1 3.3 × 10−9 2.5 × 10−36 5.4 × 10−44 meters (m) 3.0 × 108 1 7.4 × 10−28 1.6 × 10−35 kilograms (kg) 4.0 × 1035 1.3 × 1027 1 2.2 × 10−8 Planck−1 1.9 × 1043 6.2 × 1034 4.6 × 107 1

When compared to the γ-factor from the Lorentz trans- (1 ,1 ) c formations (1 ,1 ) c d BB d BB A v-1 r A v-1 B v2 B γ = 1 − , (29) c2 -1 -1 (tA , γ ) (tA , γ ) (0,1) it is seen that γ = s/ct, so that γ is dimensionless and (0,1) normalized to the unit time. Remember that in Eq. (3) v = t the choice to express t in light-seconds as a function of ct v = t tB B B tB B B -1 instead of d in seconds as a function of d/c was arbitrary. -1 (γ ,dA ) The velocity vector is a timelike interval; v = c is light- (γ ,dA ) like. As discussed in Sec. II C, the velocity vector in the (γ-1 ,0) resting system is the t-axis of the system with velocity v. (γ-1 ,0) θ −1 θ t The geometric description of inverse velocity v is a t (0,0) (1,0) A −1 (0,0) (1,0) A tA γ (γ,0) straight line with slope v with respect to the t-axis, as tA γ (γ,0) shown in Fig. 4 and given by the relation v−1 = t/d, the transpose of t and d in Eq. (15). Qualitatively, with d as a function of v−1, FIG. 5: The resting coordinate system of A, and the Lorentz transformed coordinate system of B with velocity vB. The −1 t oblique triangle with vertices (0,0), (γ,0), and (γ , dA) has d v−1
= . (30) v−1 one side formed by the B-unit interval tB and a second by B’s contracted measurement of the A-unit interval tAγ. The −1 Quantitatively, d as a function of constant inverse veloc- rhombi (0,0), (1,0), (1,1), (0,1) and (0,0), (γ , dA), (1B, 1B), −1 ity is also given by Eq. (30). (tA, γ ) always enclose the same area. The inverse velocity vector has the same length, given

by the space-time metric or the Lorentz transformations, −1 as the velocity vector. The inverse velocity vector of the intersecting B’s t- and d-axes at intervals of γ A-units resting system is a spacelike interval that forms the d- define B’s unit intervals and form its coordinate grid. To verify this relation, the linear unit interval of tB axis of the system with velocity v. Figure 5 shows the −1 resting coordinate system of A with the velocity and in- from (0,0) to (γ ,dA) along B’s t-axis is determined from verse velocity vectors of a moving system B forming the the relationship t- and d-axes, respectively, from which the contracted tA tB = , (31) coordinate grid of B, moving with velocity vB, is con- γ cos θ structed. This is the geometric description of the Lorentz transformations.9,37,46 where tA is the resting unit interval from (0,0) to (0,1) To construct A’s resting coordinate grid, lines parallel and θ is the angle of the velocity vector from Eq. (18); to the d-axis intersect the t-axis at unit intervals, and tB is congruent to B’s d-unit interval dB. With B’s unit lines parallel to the t-axis intersect the d-axis at unit in- intervals known, application of the law of sines to the −1 tervals. The resting system A will measure a contraction oblique triangle (0,0), (γ,0), (γ ,dA) in Fig. 5 shows (1BB ,1 ) c in the moving system B of magnitude γ from Eq. (29). that dA -1 vB For A to measure what B calls a unit interval, the mov- sin( π − 2θ) −1 t γ = t 2 , (32) ing unit must be γ long in resting units to result in A B sin( π + θ) a measurement of exactly one unit. The moving unit in 2 (t , γ -1 ) A B is defined by A as longer than its own resting unit, where (π/2 − 2θ) and (π/2 + θ) are the angles opposite (0,1) −1 so that after a contraction of magnitude γ is accounted the sides (0,0) (γ,0) and (0,0) (γ ,dA), i.e., tAγ and tB, −1 −1 for, they are equal. To construct B’s coordinate grid, the respectively; the triangles (0,0), (γ ,0), (γ ,dA) and t v = t −1 −1 B B B axes for B’s coordinate system are coincident to A’s ve- (γ,0), (γ ,0) (γ ,dA) are similar triangles. In words, (γ -1 ,d ) A locity and inverse velocity vectors for B’s motion (vB and tAγ, from B’s resting perspective, is B’s properly con- −1 2 2 1/2 vB are B’s t- and d-axes, respectively). Parallel gridlines tracted unit interval of (1 − v /c ) moving A-units. θ (γ-1 ,0) (0,0) (1,0) tA tA γ (γ,0) 8

tiderivative of both sides of Eq. (26) with respect to time n = 2 n = 2 and normalizing to the unit time vector. This results in the general form taught in first term physics, d n = 1 d n = 1 n = 3 n = 3 1 d(a) = at2. (35) 2 n = 4 n = 4 Change in acceleration is jerk j.48–51 The geometric description of jerk is a line of third-order curvature with n = 5 n = 5 respect to the t-axis, as shown by n = 3 in Fig. 6 and n = 6 n = 6 expressed by the relation t t d v = = j. (36) FIG. 6: Worldlines of first- through sixth-order curvature. t3 t2 Qualitatively, with d as a function of j, Since space and time have been defined as orthogonal 3 quantities, it is worth noting that B’s perpendicularity is d(j) ∝ jt . (37) not Euclidean from A’s perspective. It is determined by the constancy of the speed of light; the angle (π/4 − θ), Quantitatively, d as a function of constant jerk, when between c and t, is always equal to the angle between c starting from the origin with v = 0, is found by taking the and d. This looks much the same way a circle viewed antiderivative of both sides of Eq. (35) and normalizing 48 sideways turns into an ellipse - consider B’s system to to the unit time vector, resulting in the general form be at Euclidean right angles but viewed from an oblique 1 angle. If the moving coordinate system is truly perpen- d(j) = jt3. (38) dicular and the unit distances are correct, then the area 6 −1 −1 AB of B’s rhombus (0,0), (γ ,dA), (1B,1B), (tA,γ ) Change in jerk is snap ψ, change in snap is crackle χ, in Fig. 5 and the unit area AA of A’s rhombus (0,0), and change in crackle is pop φ.50 The geometric descrip- (1,0), (1,1), (0,1) in the same figure should both be 1 tions of snap, crackle, and pop are lines of fourth-, fifth-, square unit—AB is invariant with respect to AA. The and sixth-order curvature, respectively, as shown in Fig. length of each side of B’s rhombus is equal to tB from 6 and expressed by the relations Eq. (31). The acute angle between B’s t- and d-axes is 2(π/4 − θ) = (π/2 − 2θ). Then the area AB of B’s d v 2 = = ψ, (39) rhombus is AB = tB sin(π/2 − 2θ) = 1. The area AB is t4 t3 Poincar´e’s area invariant as discussed by Pierseaux; Mer- min reduces all of space-time to these rhombi of constant 16,27,47 d v area. Due to the relation of tB to γ as a function = = χ, (40) of v from Eq. (31) and thus to θ from Eq. (18), this is t5 t4 true for all values of θ derived from velocities less than c; the proof is left to the reader. d v = = φ. (41) t6 t5 n = 2 C. Greater orders of motion Qualitatively, with d as a function of snap, crackle, and d n = 1 pop, n = 3 Consideration of the general case of any motion re- quires the examination of second and greater orders of d(ψ) ∝ ψt4, (42) the rate of change of position. Change in velocity is ac- n = 4 celeration a. The geometric description of acceleration is a line of second-order curvature with respect to the d(χ) ∝ χt5, (43) n = 5 t-axis, as shown by n = 2 in Fig. 6 and expressed by the n = 6 relation45 d(φ) ∝ φt6. (44) t d v = = a. (33) t2 t Quantitatively, in general form, d as a function of con- Qualitatively, d may be expressed as a function of a: stant snap, crackle, or pop is given by the normalized antiderivatives, from the origin with v = 0, of Eqs. (38), d(a) ∝ at2. (34) (45), and (46),

Quantitatively, from the origin with v = 0, d as a func- 1 d(ψ) = ψt4, (45) tion of constant acceleration is found by taking the an- 24 9

The concept of average velocity has no physical meaning d a = 1 d d d a = 1 d d at any single instant in time, and the concept of position as a function of instantaneous velocity has no physical 0.5 vavg a 0.5 vavg a meaning for orders of n greater than one. v v The significance of the average and instantaneous ve- 0.5 1.0 a -( a) a -( a) (- a) -(- a) 0.5 1.0 a -( a) a -( a) (- a) -(- a) locities of nth-order kinematic functions is in their pro- t t t vθ t t t vθ (a)(b) (c) (d) (a)(b) (c) (d) portion to each other. Equation (48) shows that a factor of 1/n! appears in the expression for d(n), while differ- FIG. 7: (a) A worldline of acceleration with its tangent in- entiating Eq. (50) the prescribed number of times results stantaneous velocity vector v, and its secant average velocity in the cancellation of that factor, so that the proportion vector vavg. (b) A worldline with a period of acceleration a of vavg to v is 1 : n!; v is n! times larger than vavg. This followed by an equal period of deceleration −(a). (c) A world- is demonstrated graphically in Fig. 7(a) by the relative line with the four permutations of linear acceleration a, −(a), slopes of vavg and v. This proportion has a profound (−a), and −(−a), and a worldline of centripetal acceleration effect on the Lorentz transformations, where the method (a sine curve) offset slightly above it. (d) Circular motion of measuring the velocity plays a crucial role for orders with its v and a vectors. θ of n greater than one. Decreasing velocity is deceleration −(a). Deceleration 1 is a permutation of acceleration during which the slope d(χ) = χt5, (46) 120 of the worldline is positive, but the change in slope is negative. The geometric description of deceleration is an 1 inverted curve of second-order with respect to the t-axis, d(φ) = φt6, (47) 720 as shown in Fig. 7(b) and given by45 respectively. From these it is evident that the general, d v quantitative statement for d as a function of all orders n − 2 = − = −(a). (51) of any motion is given by the formula t t Qualitatively, d as a function of deceleration is given by 1 d n d(n) = n t . (48) n! t d[−(a)] ∝ −(a)t2. (52) Only for worldlines of zero and first orders, the geomet- ric descriptions of distance and velocity respectively, are Quantitatively, d as a function of constant deceleration the average velocity vavg and the instantaneous velocity is given by v equal; v = v = 0 for the zeroth-order, and v = v avg avg 1 for the first.45 For worldlines of second and greater orders d[−(a)] = [−(a)]t2. (53) 2 vavg and v no longer share the same geometric descrip- tion. Figure 7(a) shows the relation of vavg and v to the There are two descriptions for motion of this nature. second-order curve for acceleration. The description of A description of d in absolute terms is consistent with vavg from t = 0 to any time t is given by everything discussed to this point. With the addition d(n) of the operator ±, relative motion in exact opposition vavg = , (49) to d can be uniquely described. Continuously repeat- t ing motions of these types, described either absolutely while v is given by the slope of the tangent line to the or relatively, encompass the class of physical phenomena curve of acceleration. Thus, the instantaneous velocity known as harmonic motion. The absolute description is at any time t is given by the derivative of position as a representative of phenomena such as wave propagation. function of acceleration. The instantaneous acceleration The relative description is representative of phenomena is given by the derivative of position as a function of such as springs, orbits, and, by extension, pendula. jerk, and therefore the instantaneous velocity is given by d a = 1 d d Deceleration −(a) is not negative acceleration (−a); the second derivative of position as a function of jerk. negative deceleration −(−a) is not necessarily accelera- 0.5 vavg a For orders up to and including the sixth, instantaneous tion (a). In the strictest sense, deceleration is only that v jerk is given by the derivative of position as a function component of acceleration during which there is a reduc- 0.5 1.0 a -( a) a -( a) (- a) -(- a) of snap, instantaneous snap is given by the derivative t t t vθ tion in velocity in the direction of d. During negative (a)(b) (c) (d) of position as a function of crackle, and instantaneous acceleration there is an increase in velocity, and during crackle is given by the derivative of position as a function negative deceleration there is a decrease in velocity, but of pop. Likewise, v is given by the third, fourth, and both are in the negative direction of d. During negative fifth derivatives of position as a function of snap, crackle, acceleration, both the slope and the change in slope are and pop respectively. Then v is given by the normalized negative. During negative deceleration, the slope is neg- derivative ative while the change in slope is positive. In all cases, dn−1 the term acceleration refers to a worldline of second-order v = tn−2 d(n). (50) dtn−1 with the slope and the change in slope of like sign. The 10 term deceleration refers only to second-order worldlines is obtained. Like all good constant velocities, this one too with slope and change in slope of opposite sign. Upon is subject to special relativistic effects.52 Comparison of substituting the term for negative acceleration into Eq. the two perspectives vθ and v = 0, the resting system, (53), one can see that deceleration −(a) and negative ac- reveals a difference in the measurements of ω. The sys- celeration (−a) are described by the same curve. This tem with vθ measures ωθ due to Lorentzian contraction is also true of acceleration (a) and negative deceleration in the proportion −(−a). By alternating Eqs. (35) and (53), a close ap- r v2 proximation of the sine curve, ω = ω 1 − θ , (58) θ c2  π  d = sin t − , (54) 2 while the system with v = 0 measures ω. Circular motion requires a centripetal acceleration a with the appropriate scaling and offsets, is constructed. with a linear velocity component vθ as shown in Fig. The sine curve in Eq. (54) describes the worldline of 7(d). The geometric description of centripetal accelera- an object in circular motion under centripetal accelera- tion is the same line of second-order curvature with re- tion a, from a plane-edge view, when d is parallel to the spect to the t-axis as shown in Fig. 7(a). A constant cen- plane of the circular motion. The circular velocity re- tripetal acceleration can be described by the sine curve mains constant, but the measured linear velocity along Eq. (56), symmetric about a line parallel to the t-axis the plane of the motion varies as the direction of the as shown in Fig. 7(c). Qualitatively, linear acceleration velocity varies with respect to d. Circular motion on- and centripetal acceleration are synonymous dimensions; edge describes a sine curve. Linear motion describes the both are given by Eq. (34). As described in Einstein’s second-order curves given in Eqs. (35) and (53). The principle of equivalence, centripetal acceleration is in- two worldlines shown in Fig. 7(c) are not the same circle distinguishable from any other form of acceleration.52,53 from two different perspectives. These are the worldlines Quantitatively, the centripetal acceleration is related to of two different physical phenomena: linear acceleration the linear velocity component in square proportion: and centripetal acceleration, each with a = 1 c/s. 2 Consideration of the case of circular motion requires vθ = ar. (59) the examination of angular velocity. If an event can be 2 described as repeating in angular units θ, then it may be Substituting Eq. (59) for v in Eq. (29) gives described as having an angular velocity ω. If an event r ar can be described as repeating in temporal units, then it γθ = 1 − . (60) c2 may be described as having a frequency f. The geomet- ric description of continuous angular velocity is the sine The principle of equivalence is geometrically demon- curve shown parallel to the t-axis of the reference system strated by the correspondence of the curves for the dif- in Fig. 7(c). The sine curve has amplitude 2r where r ferent kinds of acceleration. Centripetal acceleration is is the radius of curvature of the path of travel. Quali- a composition of linear accelerations from varying direc- tatively, angular velocity and frequency are synonymous tions. Once corrected for, a worldline of the same accel- dimensions; both are given by eration from a single direction is geometrically indistin- guishable. In practice this means that gravity is just as θ f ∝ ω = . (55) good a centripetal force as any other. Then the gravita- t tional acceleration given by Quantitatively, d as a function of constant angular veloc- GM a = (61) ity is given by r2 d(ω) = r − r cos(ωt), (56) can be substituted into Eq. (60), to arrive at an expres- sion and ω and f are related in the proportion r : 2πr; angular r GM velocity is measured in radial units along the circumfer- γ = 1 − (62) ence per unit time, and frequency can be measured as g c2r circumferential units per unit time. Wavelength λ is a for a special relativistic effect in proportion to the grav- closely related measurement. A wavelength times a fre- itational field. However, as Einstein found, and others quency gives a velocity, λf = v. Wavelength is a spatial have discussed, this is only half of the gravitational effect unit in 1 : 1 proportion to the temporal unit f −1, sym- measured; there is a missing factor of two.10,20,34,35,53–57 metric about the velocity c. By taking the proportion of the circumferential path length `, given by ` = 2πr, to the period T , given by D. Factors of n!: a geometric description T = 2π/ω, a constant velocity vθ, where ` For Einstein the development of the tensor field of v = = ωr (57) θ T general relativity revealed this factor of two, and the 11

the small triangle. The side opposite the angle θ of the −1 d v-1 d v-1 smaller triangle is γ − γ, so that vavg of the smaller 2v-1 c 2v-1 c triangle is given by the ratio of the side opposite to the side adjacent: 2θ 2θ γ−1 − γ θ θ v = . (63) v v avg vγ−1 vγ-1 vγ-1 θ θ θ θ This is the vavg that, if doubled, will give the correct 2 γ 1γ-1 t γ 1γ-1 t description for v. For legibility c = 1 is omitted in the following, and v2 is considered dimensionless due to the implied division by c2. FIG. 8: The similar triangles generated by the Lorentz trans- formation, where v is the average velocity. The length of the Substituting 2v = 2vavg = v for vavg, the composition base of the large triangle is γ−1, and the side opposite the of v as a function of γ proceeds as follows: angle θ is vγ−1. The sides opposite and adjacent the corre- sponding angle in the small triangle are γ−1 − γ and vγ−1, 2v2 1 − 1 + 2v2 2v = = respectively. The hypotenuse of the small triangle is parallel v v −1 to the inverse velocity vector, and 2v is the inverse instan- 1 − (1 − 2v2) 1 1 − 2v2 taneous velocity vector. = = − v v v 1 √ √ 2 2 1 − 2v = 1−2v − √ 1 √ 1 Schwarzschild metric depends on it. Schiff concludes that v 2 v 2 1−2v √ 1−2v this factor of two arises from the sum of two separate √ 1 2 2 − 1 − 2v terms, one for space and one for time. Strandberg de- = 1−2v ; (64) v √ 1 rives it in a vector field by considering both tangential 1−2v2 and radial vectors. However, in a dimensional field with only one direction, such extravagance is unnecessary. In- compare this to Eq. (63). From the final forms of Eqs. stead, simple geometry is used to describe the same phe- (62) and (64) it can be seen that γ as a function of a nomenon revealed to Einstein in the tensor field. The from Eq. (61), and properly corrected for v = 2v with c2 physical reality of this factor of two has been quantita- reinserted, is actually given by tively confirmed to a 99% precision by the redshift ex- r periments of Pound and Rebka (1959) and Pound and GM γ = 1 − 2 . (65) Snider (1965) in a 22.6 meter tower. The body of phys- a c2r ical evidence in support of the exact magnitude of these 54,58 general relativistic effects is impressive and growing. The effects of Eq. (65) apply to space-time in any field The definition of velocity is the distance traveled per of acceleration. An object static under a gravitational time traveled. In measuring the distance traveled over acceleration is affected equally to an object static with re- any period of acceleration, time must pass. The distance spect to a linear acceleration, i.e., in a rocket ship. When between two spatial points separated in time constitutes, motion within the field of acceleration occurs, third-order by definition, the measurement of an average velocity motion called jerk occurs. Perihelion precession is a phys- vavg; by definition the instantaneous velocity is instanta- ical manifestation of jerk. neous. However, as discussed in Sec. III C, the instanta- Einstein derived the magnitude of planetary perihelion d v-1 neous velocity v is the velocity the relativist must con- -1 precession ∆φ as 2v c tend with. Since v is the correct measure to use, but vavg is the measure represented by the worldline, a modified 24π3r2 rad 2θ ∆φ = a = 5.018 × 10−7 (66) Lorentz transformation must be derived to determine the T 2c2(1 − e2) orbit θ contribution of the acceleration. Geometrically, this con- v −1 struction must result in perpendicular axes of v and v . when values for Mercury are used.10,52,55 This is often -1 vγ If the slope of v is twice that of vavg as discussed in Sec. presented in terms of the product GM as θ θ III C, then the slope of its perpendicular must be half γ γ-1 t −1 1 that of vavg. 6πGM ∆φ = (67) The key is to express the Lorentz transformation in 2 2 c ra(1 − e ) terms of v. This is accomplished by taking advantage of the similar triangles discussed in Sec. III B and shown by substituting Kepler’s expression for the square of the in Fig. 8. The length of the base of the large triangle is period, γ−1. This is the side adjacent to the angle θ and coinci- dent with the t-axis. The side opposite the same angle is 2 3 4π ra −1 T 2 = , (68) given by vavgt, where t = γ , and also forms the base of GM 12 into Eq. (66). In his discussion on the perihelion preces- contributes to precession due to constant velocity in a sion, Strandberg gives circular orbit. The second factor, (1 − e2), contributes to precession due to the eccentricity of the orbit. These 2π 6πGM ∆φ = − 2π ≈ (69) two motions produce jerk—jerk due to the change in di- q 2 GM c r rection of the acceleration as a circular orbit varies in 1 − 6 2 c r position, and jerk due to a change in the magnitude of as an approximation [although there does seem to have the acceleration as an elliptical orbit varies in its radius been a negative sign in the exponent omitted from r. Schot gives the magnitude of the jerk generated by a 48,49 Strandberg’s Eq. (50)]. This is appropriate for a circular central force proportional to distance as 2 orbit of e = 0, as the factor (1 − e ) scales the result to j = −ω2v. (77) the semi-latus rectum r` as a function of eccentricity. To show the relationship of Einstein’s approximation To establish the relationship of the jerk to the perihelion to the Lorentz contraction, beginning with Eq. (67) in precession, the discriminating term, i.e., Eq. (75), in the radians per orbit, the number of radians given is the dif- central expression of Eq. (69) must be expressed in terms ference of the number of radians in a Euclidean orbit of Eq. (77) to arrive at the same precession predicted by from the number of radians in a relativistic orbit. The the general theory. number of radians in a Euclidean orbit is 2π. The num- Expressing ω in terms similar to those of the discrim- ber of radians in a relativistic orbit is greater than 2π. inant, From Eq. (67), multiplying by 1/2π orbits per radian, 2 GM ω = 3 , (78) 6πGM rad 1 orbits GM orbits ra 2 × = 3 2 , (70) c r` orbit 2π rad c r` orbit and taking note of the fact that ω2 must be equal to 2 vorb for the product in Eq. (77) to be equivalent to Eq. where r` is given by r` = ra(1 − e ), leaves the orbital (76), it is found that Eq. (77) is in excess by a factor of excess. This is the portion of the relativistic orbit in 2 excess of the Euclidean orbit. The expression for the vorb/ra. Furthermore, vorb/2πra = T , so in terms of the complete relativistic orbit is simply one Euclidean orbit first-order Lorentz transformation, plus the excess, or 2 jraT v = 2 (79) ∆φ GM 2π(1 − e ) 1 + = 1 + 3 2 . (71) 2π c r` for the third-order Lorentz transformation. Substituting this term into the center expression of Eq. (69) and using Keeping in mind that, prior to the advent of computers, values for the planet Mercury gives approximation methods such as log tables, trig tables, and slide rules were common, Einstein’s application of 2π −7 rad q − 2π = 5.018 × 10 (80) two popular approximations, jraT orbit 1 − 6 c2 2π(1−e2) 1 1 + x ≈ , (72) as the correct value for the perihelion precession from Eq. 1 − x (66), in terms of the jerk. In the final form, the contri- and butions of the circular component (2π) and the eccentric component (1−e2) are clearly visible in the determinant. 1 √ 1 − x ≈ 1 − x, (73) The geometric components (2π) and (1 − e2) tie di- 2 rectly to the spatial component ra, which in turn is the can be extracted.10,20,52,57 The result is that related measure of d in the Lorentz transformation. The period T can also be generalized as a measure of t. Ex- GM 1 1 + 3 ≈ . (74) trapolating from the first- through third-order Lorentz 2 q c r` GM transformations, a general form of the nth-order Lorentz 1 − 6 2 c r` transformation is proposed. For the first through third orders v, a, and j Two factors contributing to precession can be isolated r in the discriminating term of the Lorentz factor on the v2 right-hand side of Eq. (74), γ = 1 − (81) v c2 GM GM 6 = 6 . (75) r c2r c2r (1 − e2) a · d ` a γ = 1 − 2 (82) a c2 The first factor, r GM j · d · t = v2 , (76) orb γj = 1 − 6 2 (83) ra c 13 and a zeroth-order d transformation (the null transfor- the case of true uniform circular motion. Quantitatively, mation) with its four permutations of positive and negative, ac- r celeration and deceleration, Eq. (35) is all that is needed 0 to express spin. In actual practice, spin is commonly ex- γd = 1 − (84) c2 pressed as an integer proportion, from the set of rational is relatively safe to propose. Keeping in mind that the numbers Q, of Planck’s physical constant ~. determinant must be dimensionless, the general form of The dimensions of ~ are those of angular momentum the nth-order Lorentz transformation then appears to be L. These dimensions include factors of mass and of spe- cific angular momentum, the inverse of acceleration. The s v
n−2 physical direction of L is perpendicular to both the t- and n−1 · d · t γ = 1 − n! t . (85) d-axes of the worldline of any point on or in any object n c2 with spin. This perpendicularity does not necessarily ex- ist for an object free in space. The spin axis of a free E. The origin and the axes object can have a nonperpendicular orientation to its ve- locity axis, and therefore to its t-axis. This quantitatively distinguishes a free object from free space-time, and one If the worldlines of all points on or in an object have free object from another, as applied by Pauli in his exclu- the same angular velocity about an axis within that same sion principle. Thus the L-axis of a free object is parallel object, the object is said to have spin. Angular velocity, (or antiparallel) to its m-axis. frequency, and spin are synonymous dimensions. They The geometric description of a free object in the DCS is are all related to a centripetal acceleration, which in turn that of the origin of a unique dimensional triad (t, d, m). is synonymous with linear acceleration. The linear veloc- Qualitatively, a free object has the dimensionless measure ity component vθ of any specific point on or in a spinning of number N. Quantitatively, N is real and can therefore object varies with its distance from the axis of spin, but be counted, is nonzero and positive, and is given by the the frequency, angular velocity, and spin are constant re- closed set of all natural numbers N. Large numbers of gardless of position. N may be expressed in proportion to a standard quan- Einstein proposed that theoretical consideration of lin- tity n. The SI quantity of n is the mole 6.0 × 1023 N. ear acceleration gives information about gravitational 53 Furthermore, when quantities with similar dimensional acceleration. This proposition is known as the principle qualities are expressed in proportion to each other, the of equivalence. Because the angular velocity, frequency, ratio is dimensionless. Qualitatively, n is any dimen- and spin are related to a centripetal acceleration, which sionless quantity. Some synonymous dimensions are the in turn is synonymous with linear acceleration, the prin- radian and the gross. Quantitatively, n is given by the ciple of equivalence extends to all of these cases. There set of all real numbers (which has as a subset). appears to be no reason to limit this extension to the di- R N The dimensions of the DCS axes need not be limited mension of acceleration. One linear velocity, irrespective to the triad (t, d, m). Any triad consisting of axes related of magnitude, is just as good as any other; any one force by the same proportions as ct = d, and dc2/G, such as is just as geometrically predictable as the rest. Therefore, (v, v2,F ) or (a, d2/t3, md/t3), will have the same geomet- a more universal statement is proposed as a fundamental ric relations as a triad of the prime dimensions. property of the Dimensional Coordinate System: theo- The scaling of DCS axes is not limited to linear in- retical consideration of the general dimensional qualities crementation: axes of second and greater order scaling of any physical process gives information about any syn- create curled up dimensional spaces; allowing the orien- onymous dimensions. tation of coordinate axes to vary from the perpendicular Linear and gravitational acceleration are synonymous creates Gaussian curved spaces. As discussed in Sec. II D, dimensions, and all worldlines for acceleration are similar all methods applicable to the geometric space (x, y, z) are functions of second order in the space-time plane. Angu- equally applicable to the dimensional space (t, d, m). One lar velocity and frequency are synonymous dimensions, such interesting scale is the exponentially scaled triad and all worldlines for angular velocity and frequency are (et, ed, em) where e is the base of the natural logarithm. similar functions. Any two units that express synony- The Taylor series expansion of the form mous dimensions also express similar functions, i.e., years ∞ and days, meters and yards, joules and kelvins. Geo- X xn ex = (86) metrically, this means that all second-order worldlines n! in the space-time plane are equivalent to acceleration, n=0 and therefore the same results are predicted for linear or gives the discrete components gravitational acceleration. ∞ The geometric description of spin S is that of angular X xn x2 x3 x4 = 1 + x + + + + ··· (87) velocity ω, the sine curve depicted in Fig. 7(c) and given n! 2! 3! 4! in Eq. (56). Qualitatively then, d as a function of spin, in n=0 its most basic form, is given by the expression for accel- for each respective DCS axis. The first term 1 is not eration in Eq. (34). The physical quantity spin describes only x0, but also, both qualitatively and quantitatively, 14 the origin N in the DCS as the point of intersection t0 = d0 = m0 = 1 = N. The factor 1/n! should be familiar TABLE II: The System of Prime Dimensions (SP). from the discussion in Sec. III C and its application to SP dimension SI unit the Lorentz transformation in Sec. III D. The remaining number (n) mole (mol) n factor x is the subject of the following section. polarity (±) n/a time (t) second (s) IV. THE PERIODIC TABLE OF DIMENSIONS space (d) meter (m) mass (m) kilogram (kg) The idea is that, armed with the fundamen- tal and derived dimensions, every numeri- cally expressible quantity in the universe can be formulated. Carried further, this means that every possible relation, encompassing ev- the units; in an infinitely-dimensional system everything ery facet of our knowledge about the physical is fundamental and therefore relates no common traits world can be—at least in theory—uniquely de- among the components. He concludes by recommend- fined. ing the SI system with its seven fundamental dimensions: length, time, mass, electric current, amount of substance, -T. Szirtes4 temperature, and luminous intensity. However, there are ambiguous and redundant physical qualities expressed by some fundamental SI units. Amount of substance n is di- A. The elements mensionless, and temperature can be derived as energy with the dimensions md2t−2. Electric current and lu- Consider a system composed of three exponentially minous intensity can also be resolved in terms of t±n, scaled, mutually perpendicular axes t, d, and m, with d±n, and m±n, so that the entire SI system may be re- N N N unit increments of t , d , and m , respectively. Within solved into a system of three primary dimensions t, d, and 1 this small domain the dimensional terms for time t , m, from which all other dimensions m±nd±nt±n may be 1 1 space d , and mass m are easily identified. Other terms derived.54 such as area A = d2 and volume V = d3 also appear as derived dimensions on these simple xN number lines. A system of prime dimensions, derived from SI units, Applying the ± operator to the exponent allows the establishes the fundamental relationships from which all representation of further components, such as frequency other physical qualities can be derived. The System of f = t−1, as a reflection across the origin. Prime Dimensions (SP) shown in Table II consists of the Now consider a coordinate system constructed of all primary natural qualities described by the DCS axes: points (t±n, d±n, m±n) when n is allowed to range freely mass, space, and time. To quantitatively evaluate the over all real values for each axis. Combination of the results of any geometric analysis, two more physical char- terms t±n, d±n, and m±n allows the representation of acteristics are needed: one to express magnitude, and one every possible derived combination of mass, space, and to express polarity. These are the scalar n and the arith- time as a product m±nd±nt±n (the order of the variables metic operator ±. Neither n nor ± are dimensions. They t, d, and m is reversed when expressing a product to help are included strictly to facilitate quantitative analysis. avoid confusion).5 One version of Occam’s razor wisely advises that “entities should not be multiplied unneces- Every quality of nature can be derived as products of sarily.” In The Harmony of the World I XX, Kepler ad- these prime dimensions. If every derived combination vises that “No operation of addition or subtraction gives of the three prime dimensions represented by the DCS rise to diversity, but all are equally related to their pair axes is considered, the number of possible derivations is of ‘terms’ or ‘elements.’”59 Algebraically, what Kepler is infinite. However, there are practical limits to the num- saying is that unlike terms cannot be added—they can ber of measurements necessary to conveniently describe only be multiplied, and multiplication derives diversity. every aspect of nature. Alternatively, Occam is saying that multiplication should be used sparingly, so it is also prudent to set some upper After considering the number of dimensions, Szirtes limit on n to keep it real. After all, physical quantities turns to the question of standard magnitudes. Since all are measurements of real events. quantities go to n, SP is not a system of units. It is a sys- Szirtes and Jackson both examine the question of how tem of dimensional relations with all quantity removed, many dimensions a dimensional system should have.4,60 and thus any system of units may be applied; SI is the Szirtes demonstrates the possibility, and impracticality, present standard. Both SI and SP are coherent systems of of both mono- and omni-dimensional systems. In a dimensions, in that the derived dimensions are expressed one-dimensional system everything is derived and there as combinations of the fundamental dimensions without is therefore no resolution, i.e., differentiation, between conversion factors other than 1.4,61 15

B. The table In an empirical attempt to qualify the claim that every numerically expressible quantity in the universe The construction of a three-axis periodic table, based can be formulated in the periodic table, 1,727 unique on the prime dimensions of the DCS axes, and illustrated physical measurements have been identified at the time in Fig. 9, provides a visual medium for an investigation of this writing in a limited search including standard 4,63–65 of derived dimensions. Each axis in this Periodic Table references. Of the unique physical measurements of Dimensions has both positive (+) and negative (−) identified, 1,626 (94.2%) are defined by sources in terms coordinates, 180 degrees in opposition from the origin; of fundamental units of time, space, mass, and electric each coordinate represents an exponent n. Positive co- charge. Only those measurements with clear definitions ordinates express positive exponents in the numerator, by outside sources in terms of the prime dimensions or while negative coordinates express positive exponents in electric charge and those used in the context of this pa- the denominator, according to convention. Expressing per were considered. Once resolved to prime dimen- a derived dimension as a product of factors with either sions, each measurement is cataloged with its synony- positive or negative exponents is equally valid. Unity, mous dimensions. This catalog is also available online 62 represented by the unit scalar N, is placed at the origin. at EPAPS. A representative sample demonstrating the Each step away from the origin along any axis represents statistical extremes and the diversity of the measure- an exponentially incremented, or decremented, factor of ments, units, and sources is presented in Table III. In the dimension assigned to that axis. Linear increments of general, web resources were avoided. Russ Rowlett’s Dic- 65 the exponent of each prime dimension proceed along each tionary of Units of Measurement is a notable exception. axis. Just as in any other three-axis coordinate system, Certainly, the fact that “dozen” is defined in only one every coordinate triplet (t, d, m) represents a unique po- source simply reflects the lack of cookbooks in the survey. sition; in the Periodic Table of Dimensions each unique A definite bias towards physics has been exercised, but position represents a unique physical dimension. the citation of the dimensions of thermal conductivity As a table in three directions, the Periodic Table of Di- 30 times seems to be statistically significant. This may mensions is best viewed in 3-D rather than on a 2-D page be due in part to a consensus on nomenclature; not one or computer screen. An interactive html table is available other dimension synonymous with thermal conductivity online at EPAPS, but a 3-D construction makes an excel- was identified. In all, the number of times a measure- lent classroom tool for presentations.62 Abbreviated ver- ment is defined according to the standards of this survey sions are frequently sufficient. For example, a discussion has little bearing on the results beyond the lack of a sup- on Newtonian mechanics requires no more than the lower porting reference in 19 cases. left-hand quadrant of the table presented here, while the There are 72 unique resolutions cataloged by SP di- ideal gas law requires approximately one-third of the ta- mensions at the time of this writing. All are within the ble to include both pressure and volume. Construction limits of t−6 to t3, d−5 to d4, and m−2 to m1. The 66 drawings are also available at EPAPS. physical quantities within the limits of t−4 to t2, d±4, and m±1 fit inside the 7×9×3 rectangular volume of 189 pos- sible triplet combinations presented in Fig. 9. Many of C. Dimensional resolutions the measurements are dimensional inversions of other de- rived dimensions. A simple example is frequency, whose The universal principle of equivalence provides the the- inverse is time. Elimination of these redundancies would oretical basis to resolve every numerically expressible reduce the number of unique dimensions, but would pro- quantity in the universe to the prime dimensions of time, mote ambiguity. Furthermore, no clear axial symmetry space, and mass. The universal principle of equivalence is of inversion is apparent within the scope of identified di- demonstrated beyond general relativity by the superpo- mensions. There is some symmetry about the axis in sition of forces and the superposition of energies. Math- the space-time plane containing velocity dt−1 and spe- ematically, superposition is demonstrated by addition of cific energy d2t−2, and even less about an approximate like terms. Addition and subtraction do not alter the axis through all three planes including fluidity m−1d1t1 dimensions of any physical equation, but multiplication and dynamic viscosity md−1t−1, but there is no one-to- and division provide dimensionally diverse results. This one correspondence. Many of the possible derivations general principle describes the limited diversity neces- have no associated measurements yet, but just as with sary to describe any derived dimension. Rather than re- the Periodic Table of Elements, applications of the yet- quiring a separate description for accelerations caused to-be-filled elements of the Periodic Table of Dimensions by nonuniform linear translation, uniform rotation, and may just be waiting to be discovered. gravitation, it is only necessary to describe the general One such application is that of the physical constant. case of acceleration, to which a scalar is applied for the When the Planck units are applied to the elements of description of the specific case; an expression of polarity this table, the magnitudes of the derived dimensions are can be used to further differentiate, i.e., deceleration vs. physical constants. By definition, t, d, and m in Planck acceleration. This is true of every dimensional quality of units are the Planck time tP, the Planck length dP, and nature, including forces and energies. the Planck mass mP. From these the derived magnitude 16 The Periodic Table of Dimensions t m

d N

first hyper- polarizability (1,-1,1) (t,d,m)

angular mass linear capacitance density density inertia m density gradient (0,2,1) (0,0,1) (0,-1,1) (0,-2,1) (0,-3,1) (0,-4,1)

angular momentum electric dynamic conductance electrical acoustic momentum charge viscosity conductivity impedance (-1,2,1) (-1,1,1) (-1,0,1) (-1,-1,1) (-1,-2,1) (-1,-3,1) (-1,-4,1) atomic stopping energy force electric pressure pressure stopping cross current gradient power section (-2,2,1) (-2,1,1) (-2,0,1) (-2,-1,1) (-2,-2,1) (-2,4,1) (-2,3,1) heat power radiance source power (-3,2,1) (-3,0,1) (-3,-1,1)

insulation thermal time inverse temporal efficiency resistivity t velocity density (1,2,0) (1,1,0) (1,0,0) (1,-1,0) (1,-3,0) moment of volume area distance number wave fuel volume section d N number efficiency concentration (0,4,0) (0,3,0) (0,2,0) (0,1,0) (0,0,0) (0,-1,0) (0,-2,0) (0,-3,0) volume electric velocity frequency thermal heat volume transfer flow potential conductivity activity coefficient (-1,3,0) (-1,2,0) (-1,1,0) (-1,0,0) (-1,-1,0) (-1,-2,0) (-1,-3,0) mass specific acceleration specific stopping angular energy acceleration pressure power (-2,2,0) (-2,1,0) (-2,0,0) (-2,-1,0) (-2,4,0) specific jerk angular power jerk (-3,2,0) (-3,1,0) (-3,0,0) snap

inductance permeability (-4,1,0) thermal expansion (2,2,-1) (2,1,-1) (2,-2,-1) thermo- resistivity resistance fluidity transport electric diffusion power (1,3,-1) (1,2,-1) (1,1,-1) (1,-1,-1) (1,0,-1) specific specific specific mass volume area length concentration ( 0,3,-1) (0,2,-1) (0,1,-1) (0,0,-1) specific activity (-1,0,-1)

FIG. 9: The Periodic Table of Dimensions. Three space-time coordinate planes, graduated from top to bottom as the m, N, and m−1 planes, respectively, give an ordered presentation of every measurement that can be derived from the prime dimensions of mass m, space d, and time t. 17

TABLE III: A representative summary of the Catalog of Synonymous Dimensions. Sample Number of Fundamental SI SP Synonymous measurement citations quantity unit dimensions dimensions dozen 1 number mole n 213 yard 3 distance meter d 209 gallon 3 volume cubic meter d3 203 pound-mass 4 mass kilogram m 131 acre 3 area square meter d2 105 torque 4 energy newton-meter md2t−2 96 half-life 4 time second t 80 bar 5 pressure pascal md−1t−2 72 rpm 2 frequency hertz t−1 42 luminous intensity 6 power candela md2t−3 32 electric chargea 5 mass flow coulomb mt−1 13 surface tension 11 surface tension newton per meter mt−2 13 thermal conductivity 30 thermal conductivity watt per meter-kelvin d−1t−1 1 aA discussion on the dimensions of electric charge follows.

−1 60,61,64 of the velocity element dPtP is found to be light speed c, remove the unwanted factors of π from the results. the derived magnitude of the angular momentum element The SI and SP systems are rationalized systems in that 2 −1 mPdPtP is found to be Planck’s reduced constant ~, and µo is defined with a factor of 4π, o is defined as a function −1 3 −2 of µ , and their proportion is determined by the speed of the element mP dPtP , which had no corresponding mea- o surements in the survey, is found to be the gravitational light in the relation constant G. Still other elements are derivations of these 2 constants. The derived magnitude of the mass flow ele- µooc = 1. (88) −1 3 ment mPtP is c /G. Since the Planck units themselves It is in the dimensions of q that SI and SP depart, are derived from the constants c, G, and ~ alone, it can and in SP q is derived from the fundamental electro- be expected that all further derivations will be functions magnetic relationship given in Eq. (88). In SI q/t is a of some order of each. fundamental dimension, and the unit of inductance, the Of all the dimensional resolutions examined in this sur- henry, has the dimensions4,61 md2t−2(q/t)−2. Permeabil- vey, electric charge q stands out along with the prime ity has the dimensions of henry per meter, which resolve dimensions as essentially unresolved. This seems to re- to mdt−2(q/t)−2, in which q is the only dimensionally sult from an arbitrary emphasis on electric current q/t ambiguous factor in SP terms. as a fundamental unit.60 However, charge is not without In SP, the derivation of q follows from the SI definition its derivations as a product of prime dimensions. The of the fundamental unit of electric current: CRC Handbook discusses three variants of the cgs system: the electrostatic, the electromagnetic, and the Gaussian The ampere is that constant current which, systems.61 In the electrostatic system, q has the dimen- if maintained in two straight parallel conduc- sions m1/2d3/2t−1; in the electromagnetic system q/t is tors of infinite length, of negligible circular defined with the dimensions m1/2d1/2t−1 so that q has cross-section, and placed 1 meter apart in the dimensions m1/2d1/2t0 and the product qc has the vacuum, would produce between these conduc- −7 dimensions m1/2d3/2t−1; the Gaussian system is a hy- tors a force equal to 2×10 newtons per me- brid of these two. These dimensions of q are confirmed ter of length. (9th CPGM [1948], Resolutions by Allen as m1/2d3/2t−11/2, where the permittivity  is 2 and 7). 64 dimensionless. When the SP dimensions of , which are The measurable qualities in this definition of electric cur- 3 m/d , synonymous with density, are taken into account, rent are force per distance. Resolved to the fundamental 1/2 3/2 −1 1/2 m d t  becomes m/t. components of the DCS axes, force per distance becomes In order to derive the m1/2d3/2t−11/2 dimensions, two mt−2. All other electric quantities can be derived from fundamental electromagnetic constants are left dimen- electric current. In SI units, the ampere-second com- sionless: the permeability µo and permittivity o of free prises the fundamental unit of charge, the coulomb. In space. Furthermore, factors of π appear where they are DCS terms, the geometric description of q becomes a line not expected or wanted in equations using these systems. with slope m/t. In the Periodic Table of Dimensions, this This is an example of a nonrationalized system. A ratio- makes q synonymous with mass flow, as opposed to the nalized system includes factors of π in the constants to electrostatic system definition with fractional exponents. 18

Given m/t as the dimensions of q, the dimensions of 2 -1 µo become dt /m, the reciprocal of pressure P . From -1 N m N t Eq. (88), and with c and µo defined dimensionally, o has N m N t F = × N = N F = × N = N N a f dimensions synonymous with density ρ. Rearrangement N a f (a) (b) 1/2 (a) (b) of Eq. (88) gives the relation c = (1/µoo) . Since µo 1/2 is synonymous with the reciprocal of P and o is syn- 1/2 N 2 N N onymous with ρ, the speed of light is then qualitatively N 2 N N = N v = 1/2 = N v = S/ρ Vq v2 defined by the relation c ∝ (P/ρ) , which is synony- S/ρ Vq v2 (c) (d) mous with the theoretical formula for the speed of sound (c) (d) 66,67 1/2 in air, vsound = (1.4Pair/ρair) . Furthermore, the electric field E is defined60,68 as the force per unit charge, FIG. 10: Dimensional arithmetic. Each cube represents an E = F/q. This has the SP dimensions of velocity. Analo- element in The Periodic Table of Dimensions: (a) Force F is gous to E is the gravitational field G, derived as the force derived from the multiplication of mass m and acceleration a, (b) frequency f is the inversion of time t, (c) mass stopping per unit mass, G = F/m. This has the SP dimensions power S/ρ is derived by squaring electric potential Vq, and of acceleration. This gives the analogous result in both (d) velocity v is derived as the square root of specific energy systems that the force is a product of the source and the v2. field:

F (G) = ma, (89a) gives P = nRT/V , where n and R are dimensionless so that their product is the origin N. Then nR times F (E) = qv. (89b) the temperature T is synonymous with energy E, whose position is 2 forward, 2 left, 1 up; volume V is 3 left. The qualitative similarity between field equations and For multiplication, movement is away from the origin; boundary conditions in different physical contexts and for division, movement is toward the origin. Then for the SI definition of electric current are strong grounds T/V , the coordinates of V are subtracted (move 3 right for favoring the use of m/t over m1/2d3/2t−1 as the di- instead of left) from those of E (2 forward, 2 left, 1 up) mensions of q. to arrive at P (2 forward, 1 right, 1 up). Arithmetically this is accomplished as

D. Dimensional arithmetic P = nR × T ÷ V (−2, −1, 1) = (0, 0, 0) + (−2, 2, 1) − (0, 3, 0). (91) The table not only has organizational properties, but also mathematical properties. In the Periodic Table of Inversion is realized as a symmetric translation across Dimensions (Fig. 9), mathematical operations on dimen- the origin and is illustrated in Fig. 10(b). Frequency f sions cause movement through the table in proportion with position 1 forward is the inversion of time t, 1 back; to the dimensions of the terms; all of these operations for 3 left it is 3 right. For permeability at 2 back, 1 left, are the result of standard exponential convention. Addi- 1 down it is pressure (2 forward, 1 right, 1 up). For those tion and subtraction are limited to like terms. The addi- who think better in coordinate notation than in shapes, tion or subtraction of like terms results in no change to just multiply the dimensional triad by −1, i.e., the dimensions of the terms and therefore no movement µ = P −1 through the table—no operation of addition or subtrac- tion gives rise to diversity. As an example, 5 meters (di- (2, 1, −1) = −1 × (−2, −1, 1). (92) mension d) can be added to 1 meter (dimension d) and Powers of any dimension are multiples of that position. the dimension of the result, 6 meters, is still d. For velocity v with position 1 forward, 1 left, the square is -1 Multiplication of any two dimensions proceeds from v2 at 2 forward, 2 left. Figure 10(c) shows that mass stop- N m N t F = × N = N one factor, by addition of the coordinates (the exponents) ping power S/ρ (2 forward, 4 left) is derived by squaring N a f of the second factor, to the product. Dimensional multi- (a) (b) electric potential Vq (1 forward, 2 left), or as an arith- plication is illustrated in Fig. 10(a). For Newton’s second metic operation law F = ma, force is arrived at by adding the position of 2 1/2 N N N m [1 up (from the origin)] to the position of a [2 forward, S/ρ = V 2 = N v = q S/ρ Vq v2 1 left (from the origin)]. On paper this is accomplished (−2, 4, 0) = 2 × (−1, 2, 0). (93) (c) (d) arithmetically by addition of the individual components of the dimensional triad, i.e., Roots of any dimension are fractions of that position; v at 1 forward, 1 left, is the square root of v2 (2 forward, F = m × a 2 left) as shown in Fig. 10(d), or arithmetically (−2, 1, 1) = (0, 0, 1) + (−2, 1, 0). (90) √ v = v2 Division is simply the inverse operation proceeding from 1 (−1, 1, 0) = × (−2, 2, 0). (94) the numerator. Rearranging the ideal gas law for pressure 2 19

The cube root of volume (3 left) is distance (1 left). The by the principles of geometry. By contracting the space geometric mean of dimensions is the root of their product. metric to the only kinematically relevant direction, that This is given as the geometric center of their positions of velocity, space and time can be expressed in equivalent in the table. For energy (2 forward, 2 left, 1 up) and terms. The relations of these terms can be operated on inductance L (2 back, 2 left, 1 down), the geometric mean by the space-time metric in any of its forms. As indepen- is area (2 left); 2 forward and 2 back cancel out, and 1 dent quantities expressed in like terms, space and time up and 1 down cancel out. In coordinate terms this is form an orthogonal plane. Detailed geometric analyses of kinematic functions in the space-time plane accurately (E × L)1/2 = A predict real physical phenomena. With the addition of 1 only one more measurable physical quantity, mass, or- [(−2, 2, 1) + (2, 2, −1)] = (0, 2, 0). (95) 2 thogonal to the space-time plane, a geometric represen- tation of over 1600 derived units reduced to 72 elements The geometric mean of the left hand side of Eq. (88) is brings to light a geometry of dimensional analysis. Each the origin. This can also be proven by constructing the of the elements is a measurable physical quantity derived 2 triangle µ, , and v , and noting the intersection of the from the three primary axial quantities mass, space, and lines originating from each apex and bisecting the side time; all physical law can be derived from these elements opposite. Question: Is this intersection the square root with geometric principles. or the cube root? And direction? No, it does not imply dimension. “Di- mensioned” is used as a verb when a technical drawing V. CONCLUSIONS is dimensioned in each direction. Under close scrutiny, directions are found to combine in a different proportion What is a dimension? A dimension is any measurable than do the primary physical qualities from which all physical quantity. From the simple space metric given physical law can be derived. When dimension is used as by the Pythagorean distance formula in Eq. (1), to the a noun to refer to measurable physical quantities, direc- ordered Periodic Table of Dimensions presented in Fig. tion does not measure up to space, time, or mass. Then 9 which demonstrates geometric relations between physi- in a formal discussion of direction versus dimension, it cal phenomena, this discussion has been guided primarily must be said that direction does not imply dimension.

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