The following sections are intended as a concise but solid introductions to current scientific understanding of the physical universe. We do not assume any prior knowledge of physics, but familiarity with Multivectors as later described will be subsequently assumed. Multivectors enable a clearer and cleaner picture than many standard treatments.
    In this section we lay some basic groundwork in terms of units and scales, not all of which will make immediate sense to the physics novice, who should skim over any terms, phrases, and equations he does not recognise. The intent is to provide a programmers' perspective of what we might call the numerical playing field, inpartcular the notion that every physical quantity can be measured in terms of a a reference length unit such as the meter.  

Sierpinski (<1K) Physical Units


    Because length, time, mass, energy, and temperature are all related, we can measure all of them in various powers of the same unit. We will favour the meter length unit m here since it is a particularly human-comprehendable measure.
    Lightspeed c = 299 792 458    m s-1 » 228.16 m s-1 » 3×108 m s-1 provides 1 s = 299792458 m » 228.16 m, where = denotes "unit equivalence", in the sense that light travels that far in a second. We can write this as 1 s = c m but must be careful in that this c is the m s-1 value without the units, for strictly speaking c m has units m2 s-1 rather than s. We use = rather than = to remind us that the units should be "stripped" and just the (S.I.) values used for variable symbols like c and h; and that the equivalence is predicated on the precise values used for these constants.
    Our time unit becomes c-1 m, the time taken for light to travel one meter, and we can effectively measure time in meters. Speed becomes unitless in the sense that we can measure all speeds as fractions of c and can choose our units so that c=1. This is not the same as dimensionless. A value is dimensionless if it has the same numeric value under every system of units.

    The energy of a photon of wavelength l is given by E=hcl-1 so we can measure energy in m-1 (the spacial frequency of a photon with the desired energy) with 1 J = (hc)-1 m-1 = 5.0327126 × 1024 m-1 where 1 J = 1 N m = 1 kg m2 s-2 is the S.I. Joule unit; and 1 m = (hc)-1 J-1 = 5.0327126 × 1024 J-1 .
    [  h = 6.626068 × 10-34 m2 kg s-1  is known as the Planck constant. The Dirac-Planck Constant (aka. h-bar) h º (2p)-1 h can be regarded as the fundamental quanta of angular momentum. ]

    Via Einstein's E=Mc2 we have 1 kg = c2 J   = ch-1 m-1 so we can measure mass in energy units and so in inverse length units.
    Temperature is merely a measure of energy and via E=kBT where kB is Boltzman's constant we have 1 K = 1.38065×10-23 J so we can measure temperature in energy as the energy per molecule of a reference ideal gas at the given temprature.

    Thus we can measure mass, energy, temperature, frequency, and inverted length in a single S.I. unit such as m-1 or J or K [   Without reference to the universal gravtiational constant G = 6.6742×10-11 m3kg-1 =0×102147483647 m4 ]provided only that we have accurate values for c, h, and the Boltzmann temeperature-energy ratio kB. It is worth noting that kB » 8×5½×10-16h½c (within 0.014 %).

S.I. Units Conversion Table based on c=2.99792×108 ms-1 ; h=6.62607×10-34 Js ; kB=1.38065×10-23 JK-1
m1 mc-1 = 2-28.1594
=3.33564×10-9 s
ch-1 = 2138.377
=4.52444×1041 kg-1
c-1h-1 = 282.058
=5.03412×1024 J-1
c-1h-1kB = 26.11902
=6.95036×101 K-1
sc = 228.1594
=2.99792×108 m
1 sc2h-1 = 2166.536
=1.35639×1050 kg-1
h-1 = 2110.217
=1.50919×1033 J-1
h-1kB = 234.2784
=2.08366×1010 K-1
kgch-1 = 2138.377
=4.52444×1041 m-1
c2h-1 = 2166.536
=1.35639×1050 s-1
1 kgc2 = 256.3188
=8.98755×1016 J
c2kB-1 = 2132.258
=6.50965×1039 K
Jc-1h-1 = 282.058
=5.03412×1024 m-1
h-1 = 2110.217
=1.50919×1033 s-1
c-2 = 2-56.3188
=1.11265×10-17 kg
1 JkB-1 = 275.939
=7.24296×1022 K
Kc-1h-1kB = 26.11902
=6.95036×101 m-1
h-1kB = 234.2784
=2.08366×1010 s-1
c-2kB = 2-132.258
=1.53618×10-40 kg
kB = 2-75.939
=1.38065×10-23 J
1 K

    The charge of the electron is considered to be  Qe- º e = -1.60217646 × 10-19 C where C denotes a Coulomb, the S.I. (Systematised Idiocy) unit of charge. This is defined as one Amp Second A s where the Amp is defined as the current producing a force of 2×10-7 Newtons per meter between two infinite wires one meter apart , a disasterously misconcieved approach.
    The Coulomb force law F = k q1q2 r-2 for two charges distance r apart suggests measuring charge in f º N½ m = kg½ m3/2 s-1 = c-1 kg½ m½ = c-1 (ch-1 m-1)½ m½ = (ch) m0 = (ch) where ch is unit stripped, ie. a dimensionless (unitless) measure.   
    The saner C.G.S. ("centimeter-gramme-second") system has the equivalent Franklin or electro-static-unit defined by 1 esu º 1 g½ cm3/2 s-1 = 10-3/2 kg½ 10-3 m3/2 s-1 = 10-9/2 f . For the Lorentz-heaviside system we take k=(4p)½ but if we don't take this approach, k has to be given compensatory units and in S.I. we have k = (4pe0)-1 = 10-7 c2 kg m C-2 and e0 = (4p)-1 c-2 107 C2    kg-1 m-1 .
    It turns out that  1 C = 10c esu = 10-7/2 c f where c is the m s-1 lightspeed value (with the units stripped) so f = c-1 107/2 C   = (ch) and   
1 C = 10-7/2 (ch-1)½ m0 yielding
    Qe- = -1.60218×10-19 C = -4.8032×10-10 esu = -1.51891×10-14 f = -0.0340795   = -0.0340795 = -2-4.874952 = -(2pae--1) = -(2p) ae-½where dimensionless ae- » 137.036-1 » 2-7.0981 known as the fine structure constant provides a more traditional dimensionless measure of Qe-.
    The Coulomb does not appear in the above S.I. units conversion table because charge, having no units, is effectively measured in m0; though it is perhaps most natural to measure charge in multiples of the electron charge Qe-, or possibly of the downquark charge Qd = 3-1 Qe- = 0.0113598 = 2-6.4599188 serving to remind us of natures strong indication that charge is an integer variable.
    Benjamin Franklin's coin toss dictates that we consider the electron charge Qe- [  often denoted e ] to be negative and the positron and proton charges (assumed both equal to -Qe-) to be positive.
    We are free to postulate an electron charge of -1 and interpret ae- as a measure of nonunit e0=(4pae-)-1 , but physicists usually favour taking either e0 = ½ae--1 (hc)-1Qe-2 or 4pe0   = 2pae--1 (hc)-1 Qe-2 to be unity and so take Qe- to be either -(2ae-hc)½ or -(½p-1ae-hc)½ = -(ae-hc)½ , ie. Qe- = -(2ae-)½ or Qe- = -((2p)-1ae-)½ Regardless, permitivitties of other media are usually considered as proportions of e0.

The Fine Structure Constant

    As the most natural measure of the dimensionless electron charge, the fine structure constant ae- = (2e0hc)-1Qe-2 = (2p)Qe-2 is arguably the primary empirically derived parameter of our universe. Its value cannot currently be theorectically derived but must rather be directly or indirectly measured experimentally and then "fed" into theories such as the Standard Model. The best current estimates for ae- are the 2006 QED-based value a2006=137.035999710(96)-1  and the 2002 CODATA value a2002=137.03599911(46)-1 , with non-intersecting standard uncertainty ranges [137.035999614, 137.035999806] and [137.035998650, 137.035999570] respectively   .
    James Gilson suggests   ae-=137-1 cos(137-1p) Tan(n-1p) where integer n=3973=137×29 and Tan(x) º x-1 tan(x) is the cardinal tangent function.
    This lies within the a2006 uncertainty range but the n giving closest approximation to a2006 is n=3968=(25-1)×27 with 137-1 cos(137-1p) Tan(3968-1p) = 137.0359997146-1 . However n=3967 is almost as good with 137-1 cos(137-1p) Tan(3967-1p) = 137.035999700237-1 and 3967 is the (137×4)th prime (if 2 is considered the 0th prime) , which is either a remarkable coincidence or a strong indication that Gilson is on the right track.
fsc (3Kbyte)

    Setting HN º åk=1N  (-1)½k(k-1) k! we observe that H5 = 1!-2!-3!+4!+5!=137 and letting Pk denote the kth prime we thus have the equivalent conjectures ae- = HN -1 cos(HN -1 p) Tan(P(N-1)HN -1 p) and ae- = HN -1 cos(HN -1 p) Tan(P4HN -1 p) for N=5.

Composite Units
    The Newton unit of force 1 N = 1 kg m s-2 = ch-1 m-1 m (c m)-2 = (ch)-1 m-2 while the Pascal unit of pressure is 1 Pa = 1 Nm-2 = (ch)-1 m-4 . Density in kg m-3 = ch-1 m-4 thus has the same units as pressure.
    Dynamic viscosity has units Pa s = N m-2 s = kg m-1 s-1 = h-1 m-3, while fluidity has units Pa-1 s-1 = h m3 and kinematic viscosity has units m2s-1 = c m.
    Linear momentum has units kg m s-1 = h-1 m-1 while angular momentum in kg m2 s-1 = h-1 m0 is unitless.
    Power is measured in Watts with 1 W º 1 J s-1 = c m-2.
    Since Ñ = åieiÐei has units m-2 on account of the reciprocal basis vectors, and a potential fP is defined so that -ÑPf = F is a force we see that potentials have units N m2 = (ch)-1 m0. However,  physicists typically regard potential as the spacial integration of force and so measure it in joules , with electrical potential measured in Volts where 1 V = 1 J C-1 = (hc)-1 m-1 (10-7/2 (ch-1)½)-1 = 107/2 (hc-1)3/2 m-1 .
    Resistance is measured in Ohms with 1 W º 1 V A-1 = 1 J s C-2 = (hc-1) m-1 c-1 m 107 hc-1 = h2 c-3 m0 .   
    Capacitance is measured in Farads with 1 F º 1 C V-1 = C2 J-1 = 107 ch-1 hc m = 10-7 c2 m . The permitivitty of vacuum e0= (2hca)-1Qe-2= 8.85419×10-12 Fm-1 = (4p)-1 .
    Inductance is measured in Henrys with  1 H = 1 m2 kg s-2 A-2 = 1 m2 kg C-2 = 107 m .

Non-S.I. Units
    In atomic physics the electronvolt eV »1.60218×10-19 J » 1.78266×10-36 kg » 8.06554×105 m-1 is common, often as mega-electronvolt 1 MeV = 106 eV and giga-electronvolt 1 GeV = 109 eV . From 1 m = (2p)-1hc(Mc2)2   we have 1 GeV »   1.602×10-10 kg m2s-2 » 2-32.54 J = 5.0632911 × 1015 m-1   » 252.17 m-1 » 280.33 s-1 .

    1 eV = 1.602 × 10-19 J is considered a unit of energy. 1 GeV = 1.602 × 10-10 J and 1 J = 0.6242 × 1010 GeV.
    1 eV c-1 » 5.36×10-28 kg m s-1 can be thought of as a unit of momentum.
    1 eV c-2 » 1.78266175 × 10-36 kg can be thought of as a unit of mass. [   The c-2 is frequently ommitted in the literature, so that, for example, erroneous references to an "electron mass" of 0.511 MeV rather than 0.511 MeVc-2 are woefully common ]

    Other notable units are the mile (» 1609 m) ; the nautical mile (1852 m); the year (31 556 926 s » 25106 s ); the light-year (» 9.4605284×1015 m » 1016 m); and the parsec » 254.78 m .

Planck System
    In Planck's system, one choses the basic length unit so as to unify the squared-length gravitational constant Gc-2 » 6.674×10-11 m3s-2 kg-1 (2.988×108 m s-1)-2 » 2-33.80 (228.16)-2 m kg-1 » 2-90.12 m kg-1 = 2-231.14 m2 and our fundamental unit of length becomes the Planck length lP = (hGc-3)½ = 1.61624×10-35 m = 2-115.57484 m, with associated Planck time c-1lP = 2-143.73423 s.

    Note that G= 6.673×10-11 N m2 kg-2= 6.673×10-11 ch-1 (hc-1)2 m2= 1.47488×10-52 m2.

    The Planck mass ((2p)-1 h G-1)½ » 2.176 45(16) 10-8 kg is the mass of a black hole with Schwarchild radius as the Compton wavelength. . Multiplying by (8p) gives the reduced Planck mass.

Notable lengths
    Human physicists tend to work with (integer) powers of ten, ultimately because that is the number of fingers possessed by the majority of them. It is easier and   more natural to work with (fractional) powers of two, since these can be more readily multiplied.
      Based on the circumference of the Earth (the quarter latitude through Paris) the "arbitary" meter length unit actually has much to commend it numerically. We note here (novelly to the best knowledge of the author) that the electron Yukawa potential damping factor   Me- ch-1  =  6×262  m-1 to within 0.041%,  a consequence of the bizarrely suggestive dimensionless electroterran identity Me- R, h-1c » 1.00147 × 24×262×107 where R, denotes the radius of the Earth and Me- denotes the mass of the electron!
Illustrative Lengths
Length ( m)     Example
2-115.575Plank Length c-1(hGc-1)½
2-48.9192Heisenburg distance quanta 2Qe-2(3Me-c2)-1
2-48.3343Compton Radius (Electron) (4pe0)-1 Qe-2(Me-c)-1
2-42Gamma ray wavelength
2-38.5844Compton Wavelength (Electron) h(Me-c)-1
2-34.1375Bohr Radius h(Me-ca)-1
2-35.2193Hydrogen atom
2-34.9089Helium atom
2-33.21931 Angstrom (10-10 m)
2-31.4119C60 'buckyball' radius
2-29.8974DNA helix radius
2-21.2535Visible light wavelength (violet)
2-20.4461Visible light wavelength (red)
2-17.1242Human blood cell
2-16Silk fibre
2-12Dust mite
20Human child
Length ( m)     Example
21-Human adult
25-Blue whale
27Radio wavelength
29-Tallest human buildings
213.1105Mount Everest
214.6096Olympus Mons (Mars)
222.6047Earth radius
226.0913Jupiter radius
229.3735Solar radius
237.1223Earth orbit
242.032Neptune orbit
255.2051Distance to Alpha Centuri
268.5285Radius of Milky Way Galaxy
274.3243Distance to Andromeda Galaxy
278.7028Distance to Virgo Cluster
280.6463Radius of Local Supercluster
286.6939Distance to Abell 1835 IR1916 Galaxy
289Human guesstimated radius of universe

    It is important to stress that one observer's radio wave is another's gamma ray in traditional relativity. The faster an observer travels "with" the wave, the higher the percieved frequency. The faster he travels "perpendicular" to the wave, the lower the percieved frequency. That said, visible light has a wavelength around 2-20 m (the size of a single biological cell) and a frequency of around 3×1014 Hz » 248 Hz . Gamma rays have a wavelength of order 10-12 m » 2-42 m with a frequncy of order 270 Hz. Radio waves have wavelengths of order 27m and frequency of order 231 Hz .
    Atomic radii (orbital radius of electron) range from 2-49 to 2-47 m .
    Current estimates are approximately 2226 atoms in our galaxy and 240 galaxies in the universe giving something in the order of 2266 atoms in the universe, which is presently considered by Big Bangers to be 15 × 109 years » 8 × 1014 s » 249 seconds old and roughly 296 m across.

    The unit of atomic mass is approx 2-89 kg » 252 m-1 while the mass of the electron is roughly 0.511 MeV c-2 » 2-10.93 GeV c-2 » 2-99.79 kg = 241.23 m-1 .

Time Quantisation
    Some authors (eg. Hotson) suggest time is quantised in units of t   =   2Qe-2(3Me-c3)-1 » 6.26642×10-24 s = 2-77.078633 s with associated Heisenberg length 2Qe-2(3Me-c2)-1 =1.87863×10-15 m = 2-48.91924 m . This is proportion 4Qe-2(3hc)-1 » 9/925 of the electron zitterbewegung period (2Me-c2)-1h = 6.44044×10-22 s = 2-70.395259 s .

    The universe is considered to be about 13.7×108 years = 258.5849 s » 2135.664 t and the furthest known galaxy is approx 2135.613 c-1t away.


    Via c and h (with temperature also requiring kB), charge, angular momentum, speed, resistance, permitivity and pemeability can all be asigned unitless values; length, time, inductance, capacitance, and kinematic viscosity can be measured in length units such as the meter m ; mass, energy, temperature, electric potential, current, and linear momentum can then be measured in m-1 ; power can be measured in m-2; dynamic viscosity in m-3; pressure and density in m-4.
    More generally, any scalar physical quantity may be expressed in the form   a mb where a and b are real numbers, b usually rational and typically an integer, Any reference length unit suffices and the implications for simple and robust programming and data type strategums are profound. However, even though, say,  momentum and temperature can both be measdured in m-1 they are not the same thing and it far from clear that they can be anymore meaningfully added when both in m-1 than when expressed in kg m s-1 and K. Typically the units of a value are implicitly implied by the program context rather than explicitly stored with the value but a natural way to represent a composite unit is as a reference length power and the powers of c, _placnk, and kB respectively so that momentum would be (-1; 0, -1, 0) and temperature (-1;-1,-1,1) . Length is (1;0,0,0) and charge (0;-½,-½,0). These powers will typically be small integer multiples of ½ so we might allocate four bits to each and represent physical units with 16-bit words.

    If we represent lengths in a fixed point meter value with fourteen fractional binary bits then a 32 bit intger can hold lengths in the range ±217 m » 131 km » 2-11 light-seconds to within 2-14 m » .06 mm. This is adequate for a human city but not for the Earth, let alone the solar system so if we are to used fixed point it is natural to use 64-bit integers taking us to ±249 m to the same .06 mm accuracy; which gets us out past Pluto but not to Alpha Centuri. For a fixed point Galactic simulation 96 bits are adequate but if we wish to go down to Compton radius accuracy we approach 118 bit. The visible universe at Compton resolution requires 138 bits and if we wish to work in Planck lengths we are looking at 205.
    All this suggests that a 256-bit (32-byte) integer data type would amply suffice for individual properties of particles and for a particle "ennumeration index" when simulating an isolated galaxy without recourse to hierachical coordinate systems, but that two or more further bytes would be needed to ennumerate particles in a multigalactic simulation. This exludes photons, however. A 60W lightbulb emmits 266 photons per second, the sun roughly 2148 .

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Copyright (c) Ian C G Bell 2006
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Latest Edit: 20 Mar 2007. Counter