While Bohr's model of the atom could account for spectral lines, it still could not account for why electrons had quantacized angular momentum in atoms and why electrons could be in orbit in atoms, which would involve acceleration around the nucleus, without radiating away all their energy, which accelerated electrical charges do. The atom could simply not be a little solar system, based on charge rather than gravity, since electrons, on classical principles, would lose energy and fall into the nucleus.
An answer to these questions was offered by Prince Louis de Broglie in 1923 with the theory that, as Einstein had introduced the idea that light could behave like both waves and particles, perhaps particles of matter could also behave like waves. Thus, electrons in an atom were not moving in orbits but filled an orbit as standing waves. Familiar electromagnetic radiation, like light, exists as traveling waves, moving through space at the velocity of light. A standing wave does not move, but vibrates between fixed points, like the string on a violin. The sine wave at right represents a whole wavelength. It has a portion with a positive magnitude, a portion with a negative magnitude, and a node, which has zero magnitude. The ends of the wave, which also have zero magnitude, are usually not considered to be nodes. Half a wavelength would have no nodes; one and a half wavelengths, two nodes; and two whole wavelengths, three nodes. A one dimensional wave has nodes that are points, and it vibrates into two dimensions. Similarly, a two dimensional wave, like a wave of water on the ocean, has nodes that are lines, and it vibrates into three dimensions. A three dimensional wave, which is what electrons in an atom would be, has nodes that are surfaces. Such surfaces can be planes, cones, or spheres. By analogy, we might want to say that a three dimensional wave would vibrate into four dimensions, but this aspect of the matter does not seem have been much discussed or explored. In electron waves, each non-spherical node represents a quantum of angular momentum. Thus, a half wavelength, with no non-spherical nodes, is 0 angular momentum; a full wavelength, with one non-spherical node, is angular momentum; a wavelength and a half, with two non-spherical nodes, is 2 angular momentum; etc.

In 1913 Niels Bohr proposed that the lines in the spectrum of Hydrogen could be explained if electrons could only assume certain energy states in the atom, states which corresponded to quantacized values of angular momentum. The classical definition of angular momentum is mass times velocity times distance from the center of circular motion (kg*m/s*m = m2*kg/s). This turns out to be the units of Planck's Constant (h), which can also be expressed in units of energy times time (J*s).
Thus, each electron will only have angular momentum (l) values that are an integer times Planck's Constant divided by 2 (the number of radians in a circle; h/2 is often expressed as the "reduced Planck's Constant," -- a symbol used by Paul Dirac). The spectral lines of hydrogen result when an electron drops from one angular momentum state to a lower one and releases energy. The energy is then emitted as a photon, a quantum of electro-magnetic radiation, as explained by Albert Einstein in 1905 (the "photo-electric effect"), at a frequency and wavelength proportional to its energy, according to Planck's equation, E = h = hc/ , where c is the velocity of light, is the frequency (1/s, Hz), is the wavelength (m), and c = . Bohr's equation for hydrogen is as follows, where n and n' are integer values for the level that the electron is leaving (n) and the level to which the electron is falling (n').
The additional constants are the electrostatic force constant (k) and the mass (me) and charge (e) of the electron. Evaluating the equation simply gets us a quantity of 912 times the factor , with the quantum integers, in units of length as Angstroms ( = 10-10 m).
.
When electrons drop down to the second lowest quantum level (n' = 2), they emit photons in the part of the electromagnetic spectrum that we perceive as visible light. This is the "Balmer" series of spectral lines. The lowest energy jump (3 to 2), produces a photon with the wavelength of 6560 , which is a bright red line in the spectrum of Hydrogen, usually called the "hydrogen alpha" (H ) line. This is one of the conspicuous colors of the universe, since atoms of hydrogen glow with this red color when excited. That contributes a bright red line to the spectra of most stars; and "bright" nebulae in galaxies tend to be red from the excited hydrogen, blue from scattered starlight, or purple from a combination of the two.
While Bohr's model of the atom could account for spectral lines, it still could not account for why electrons had quantacized angular momentum in atoms and why electrons could be in orbit in atoms, which would involve acceleration around the nucleus, without radiating away all their energy, which accelerated electrical charges do. The atom could simply not be a little solar system, based on charge rather than gravity, since electrons, on classical principles, would lose energy and fall into the nucleus.
An answer to these questions was offered by Prince Louis de Broglie in 1923 with the theory that, as Einstein had introduced the idea that light could behave like both waves and particles, perhaps particles of matter could also behave like waves. Thus, electrons in an atom were not moving in orbits but filled an orbit as standing waves. Familiar electromagnetic radiation, like light, exists as traveling waves, moving through space at the velocity of light. A standing wave does not move, but vibrates between fixed points, like the string on a violin. The sine wave at right represents a whole wavelength. It has a portion with a positive magnitude, a portion with a negative magnitude, and a node, which has zero magnitude. The ends of the wave, which also have zero magnitude, are usually not considered to be nodes. Half a wavelength would have no nodes; one and a half wavelengths, two nodes; and two whole wavelengths, three nodes. A one dimensional wave has nodes that are points, and it vibrates into two dimensions. Similarly, a two dimensional wave, like a wave of water on the ocean, has nodes that are lines, and it vibrates into three dimensions. A three dimensional wave, which is what electrons in an atom would be, has nodes that are surfaces. Such surfaces can be planes, cones, or spheres. By analogy, we might want to say that a three dimensional wave would vibrate into four dimensions, but this aspect of the matter does not seem have been much discussed or explored. In electron waves, each non-spherical node represents a quantum of angular momentum. Thus, a half wavelength, with no non-spherical nodes, is 0 angular momentum; a full wavelength, with one non-spherical node, is angular momentum; a wavelength and a half, with two non-spherical nodes, is 2 angular momentum; etc.
At it happens, the atom turned out to be a bit more complicated than Bohr's original atom of 1913. Each level of energy contains, not only a new integer quantum of angular momentum, but all the smaller quanta of angular momentum as well. Each energy level in an atom is distinguished, however, by the absolute number of nodes, spherical and otherwise. Thus, the first energy level, with no nodes, has only one form, with 0 angular momentum. The second energy level has one node. This can be either spherical, which means a spherical standing wave with a hidden spherical node inside (another S orbital), or a plane. A plane node gives us an angular momentum of . The plane node cuts the orbital in two, separating a side of positive magnitude from a side of negative magnitude. Such a wave is then called a P orbital. Now we get a further complication. An asymmetrical node produces an angular momentum vector. In classical physics, that vector can assume any orientation; but, as we might suspect, this doesn't happen the same way in quantum mechanics. The vector is quantacized and can only assume certain orientations: 2l + 1 , which is all integer values from +l to -l. These different vectors make a physical difference when an atom is placed in a magnetic field. The orbiting electrons produce a magnetic field, where the angular momentum vector produces a magnetic vector, which then assumes different orientations in an ambient field. The different vector orientations are thus called "magnetic substates" of angular momentum. In a P orbital, the magnetic substates are +1, 0, and -1. In the 0 substate, the vector is conventionally regarded as perpendicular to the z axis, and the node is thus the xy plane, symmetrical around the z axis. In the +1 and -1 substates, the nodes are the xz and yz planes. Since each substate can contain two electrons, the P orbital can contain 6 electrons overall. The second energy level thus has one S and one P orbital, and can hold 8 electrons. This row in the Periodic Table starts with Lithium and ends with Neon, which has an atomic number of 10. That is then the second of the "magic numbers," the atomic numbers where the energy levels have filled up, producing the particular chemical stability that we see in the Inert Gases.

The third energy level is characterized by 2 nodes. This can occur as an S orbital with two spherical nodes, a P orbital with one plane and one spherical node, and a new kind of orbital, the D. The two nodes of the D orbital occur in different ways. To be symmetrical around the z axis in the 0 magnetic substate, two cones are necessary, one below the xy plane, one above. We can also understand these cones to be produced by the rotation, around the z axis, of two lines that pass through the origin. This divides the wave into three lobes, one above the xy plane, one below the xy plane, and then a ring that is wrapped around the z axis in the xy plane. The +1 and -l magnetic substates are then characterized by one plane node that is symmetrical around the z axis in the xy plane and another plane that occupies the xz or the yz planes. The +2 and -2 magnetic substates then have two plane nodes, which intersect each other along the z axis, dividing the wave into four different parts. The first D orbital (3d), which can contain 10 electrons, however, turns out to have an energy comparable to the S (4s) and P (4p) orbitals in the fourth energy level. It fills up, then, only in row four of the Periodic Table, which goes from Potassium to Bromine (where the magic number is 36). This delay in filling the "higher" orbitals is characteristic of the Periodic Table, but it does occur fairly regularly. The D orbitals correspond to the "transition metals" in the Periodic Table.
Only one more kind of orbital occurs with electrons in atoms, the F orbital, which has an angular momentum of 3 , allows for 7 magnetic substates, and so will hold 14 electrons. The nodes are all combinations of cones and planes, analogous to the D orbital described above. An F orbital (4f) does not begin to fill until row 6 of the Period Table, giving us the first Rare Earth series. By the time we get to the second Rare Earth series (5f), the elements are so unstable that most only exist artificially. Their chemistry is mostly not really a practical question.

When electrons are thought of as orbiting the atomic nucleus like planets do the sun, what the nucleus itself is doing is a question that may not even occur. However, when we realize that the electrons are not "orbiting" but only occupying energy levels as standing waves, and that electrons in S orbitals, which have no nodes passing through the geometrical origin, can thus be found in the nucleus, we should realize that the protons and neutrons in nuclei must occupy energy levels, and so orbitals comparable to the electrons, themselves. There are some complications with nuclear orbitals, however. Protons and neutrons are different particles and so occupy their own respective sets of orbitals. The nuclear force, by which protons and neutrons are attracted to each other (they are hadrons and baryons), observes "parity," which ends up meaning that each energy level can contain even or odd quantities of angular momentum, not both. The second energy level, therefore, with an angular momentum of , contains the familiar P orbital but no S orbital. This means that much higher level orbitals get filled up for the same number of elements by protons (up to I orbitals, with 6 angular momentum). This goes up even higher with neutrons, which accumulate faster than protons. In the heaviest known elements, neutrons are thus filling J orbitals, with 7 angular momentum. Nuclear orbitals, like the electrons in the atoms, also fill up at different levels than one would expect. This is complicated by two factors: One that it works differently for protons and neutrons; and the other that we have a phenomenon called the "spin-orbital interaction" by which particles with positive spin and particles with negative spin becomes separated from each other and fill at different times, producing very different "magic numbers" for stable nuclei than occur for electrons in nuclei.

The G, H, I, and J orbitals become increasingly complicated. Only the G orbitals, with an angular momentum of 4 , and 9 magnetic substates, are shown at right. These are all superimposed on each other, of course, in the atomic nucleus, as the electron orbitals occupy the volume of the atom.

Two further complications: The waves shown are the "real" part (using real numbers) of the wave function. There is also the imaginary part of the wave function, using imaginary numbers ( -1 = i). What the physical significance of this is is a good question. On the other hand, the physical significance of the real wave function is also a good question. Werner Heisenberg and, again, Niels Bohr regarded the wave function as a "probability cloud": The square of the wave function gives a distribution for the probability of finding the electron as a particle. The wave function collapses into an actual location for the particle when an attempt is made to observe the particle. The idea that the observation creates the reality is Bohr's classic "Copenhagen Interpretation" of quantum mechanics. However, it seems inescapable that the wave function is a real and physical thing, since only a wave phenomenon can account for the interference effects that can be observed with radiation and with particles. That kind of quantum mechanics, which still observes Bohr's principle of Complementarity, but allows for two different levels of reality, seems best accommodated by a Kantian dualism of phenomena and things-in-themselves. The standing wave electrons, protons, and neutrons thus occupy real space and account for the physical size of atoms and nuclei. As long as the atom or the nucleus maintains its integrity, the waves persist; but an experimental or observational intervention to locate individual particles collapses the waves and does produce discreet locations for the particles.

Units of Measurement

1024 yetta Y
1021 zetta Z
1018 exa E
1015 peta P
1012 tera T
109 giga G
106 mega M
104 myria my
103 kilo k
102 hecto h
101 deka da
dk
10-1 deci d
10-2 centi c
10-3 milli m
10-6 micro
10-9 nano n
10-12 pico p
10-15 femto f
10-18 atto a
10-21 zepto z
10-24 yocto y
The Metric System of 1795
length metre m
area are a (10 m)2
volume litre l 1 dm3
stere 1 m3
mass gramme g 1 cm3
water
The metric system as used by international agreement in science today is officially the "Système International d'Unités" (SI). Apart from the prefixes, shown at right, which have mostly simply been expanded from the original ones, the system of units is based on a fragment, indeed a version of a fragment, of the original metric system. Only one of the original units, the meter, is still part of the basic "official" system. The "stere" was a unit intended for firewood, and is not used at all, as far as I can tell. The "liter" and the "are" (usually the "hectare") are "unofficial" units that are simply based on the meter. The "gram" has now been demoted from a basic unit to a derived unit, derived from the "kilogram." But how a unit with a prefix becomes "basic" is curious.

The Basic S.I. Units
length meter m
mass kilogram kg
time second s
electric
current ampere A C/s
temperature kelvin K oC +
273.15
amount of
substance mole mol
luminous
intensity candela cd
plane angle radian rad
solid angle steradian sr
One of the original problems of the metric system was the disparity in scale between the unit of length, the meter (m), and the unit of mass, the gram (g). Resolving this meant scaling down the meter or scaling up the gram. Both were done, and two rival systems emerged: The "CGS," system, for "centimeter, gram, second," and the "MKS" system, for "meter, kilogram, second." The "second," of course, was accepted by all as the basic unit of time. The CGS system was popular for a long time in American science. Some of its units, like the dyne and the erg, are still encountered. However, the MKS was probably destined to dominate from the beginning. In the first place, the centimeter and the gram are probably too small to be convenient for most purposes. Second, there are some MKS units, like the Volt and Ampere, that simply do not have corresponding CGS units. Thus, some CGS units, like the gauss, were defined in part through MKS units. So, to be as consistent as possible, the MKS system has become standard.
Consistency, however, is not always possible. Although the beauty of the metric system is its foundation on decimal values, which has sold it to every country in the world except the United States, and some other oddballs -- though even American customary units are officially defined in metric terms -- some customary units and strange usages have been retained or crept in for convenience. Most importantly, the systematization of decimal counting failed to anticipate the binary basis of modern computer technology. The powers of 2 now rival the powers of 10, and even metric prefixes have been corrupted. Thus, when the unit "kilobyte" ("kB" or just "K") is used, it usually does not really mean 1000 bytes of information. It means 1024 bytes, i.e. 210. A "megabyte" ("MB" or "Meg") is not 1,000,000 bytes, but 1,048,576 bytes, i.e. 1024 x 1024 or 220. This situation contains the potential for serious and dangerous confusions. I am now informed that new prefixes have been proposed for a binary system, infixing, in fact, "ibi" between the decimal prefix and the SI unit -- "KibiB" would now be the binary "kilobyte." This would clear up the current situation, but it also reveals that the decimal preference at the foundations of the metric system is not always appropriate to the material at hand.
Derived Units, S.I. & "Customary" Metric
length Angstrom 10-10 m
frequency hertz Hz 1/s
velocity m/s
acceleration m/s2
momentum m*kg/s
angular
momentum J*s
m2*kg/s
force newton N m*kg/s2
energy joule J N*m
m2*kg/s2
power watt W J/s
V*A
A2
m2*kg/s3
mass metric
ton t 1000 kg
area hectare ha (100 m)2
pressure pascal Pa N/m2
electric
charge coulomb C A*s
electric
current ampere A C/s
V/
electric
potential volt V J/C
A*
electric
resistance ohm V/A
W/A2
electric
capacitance farad F C/V
electric
conductance siemens S A/V
magnetic flux weber Wb V*s
magnetic flux
density tesla T Wb/m2
magnetic
inductance henry H Wb/A
luminous flux lumen lm cd*sr
illuminance lux lx lm/m2
radioactive
activity becquerel Bq 1/s
radioactive
dose gray Gy J/kg
This transformation of a uniform system in fact happened before. Both ancient Egyptian and ancient Babylonian systems of measures were based on a uniform basis of counting, decimal (base 10) for the Egyptians and sexagesimal (base 60) for the Babylonians. In fact, Babylonian 60's are still with us, unthreatened by the metric system, for seconds/minute and minutes/hour. The ancient duodecimal (base 12) reckoning of day and night (giving 24 hours in a day) is also still with us and unthreatened. The complications that eventually produced things like the 5280 foot mile came from historical adaptations and the introduction of what seemed like "appropriate" units for different purposes, a process that continues, and not just with binary computer language. Indeed, the duodecimal twelves that turn up in many ancient and customary units, like the 12 inch foot, are arguably better than decimals, since 12 can be evenly divided by 2, 3, 4, and 6, twice as many factors as 10 (evenly divisible only by 2 and 5). This is certainly why twelves started being used, and probably will again. Meanwhile, everyone must do something that was never supposed to happen with the metric system: remember that "kilo" means "1000" in one word and "1024" in another! -- a usage that may die hard even with the introduction of binary prefixes.
Finally, the table at left show a great many official derived S.I. units, and a couple of "customary" metric units, like the metric ton and the hectare. These units falls into four broads categories:
1. Basic physical units, from frequency to pressure;
2. Electric units, from charge to conductance;
3. Magnetic units, from flux to inductance; and,
4. Radiation units, from flux of light to a dose of radioactivity.
All these units all by themselves say a lot about the history of science and the structure of nature. The unit of force, named after Isaac Newton, is an artifact of Newton's equation F = ma, "force equals mass times acceleration." That volts and amperes can be multiplied together to give units of power, Watts, is something that everyone plugging things into an electrical outlet should know.
It is curious to reflect that while now in lingustics the value of customary usage reigns supreme, often resulting in the dimissal of educated, elevated, or traditionally grammatical speech as unnecessary, inauthentic, or the classist and oppressive tool of the capitalist patriarchy, just the opposite can be found in discussion of SI units, where even traditional metric units, like the convenient ngstrom (10-10m or 0.1 nm -- about the size of an atom), can be dismissed as archaic or reactionary. This difference is instructive. The variety of customary units resulted from the practice of those dealing with particular materials. Gold was thus weighed (and still is) in (troy) ounces rather than tons. It is not often that anyone is going to get a ton of gold together. Altough kitchen measures now are available in metric units, teaspoons, tablespoons, cups, etc. provided convenient, integer values for cooking. While 250ml is about a cup, the large integer betrays an origin foreign to the kitchen -- also now using a unit, the liter, that ironically is no longer an "official" SI unit. Those who despise customary units as mediaeval nonsense are thus in the position of the kind of grammatical martinet who tells people who say, "It's me," that they should say, "It is I" (so Louis XIV perhaps should have said, "L'état c'est je"?). What is awkward is when inappropriate units are imposed because of the uncompromising rationalistic cookie-cutter, as in the case of milliliters in the kitchen; but what is dangerous is losing a multi-million dollar Mars spacecraft because the engineers mixed up metric with customary units. The appropriate units for science are the SI ones, and it is as inexcusable (and more) that JPL engineers should be using feet or miles as it is that metric enthusiasts should be disparaging tablespoons or Fahrenheit temperatures in daily usage. Custom, even elevated grammatical usage, is the result of need and usage. The fundamental inspiration of the metric system, however, was rationalistic and dictatorial. In life there is in fact a place for both, and it is wisdom to know the difference.

The Electromagnetic Spectrum

The electromagnetic spectrum is the range of types of electromagnetic radiation from the weakest radio waves to the strongest gamma rays. As described by James Clerk Maxwell (1831-1879), electromagnetic waves are an electric field oscillating in one dimension, a magnetic field oscillating perpendicular to this, and then an electromagnetic wave traveling in the direction perpendicular to the other two at the velocity of light. The velocity of light, c, is now defined as exactly 299,792,458 m/s, where c = : is the frequency (1/s, Hz) and is the wavelength (m). By international agreement, the radio portion of the electromagnetic spectrum is divided into bands based on powers of ten. For this purpose, the velocity of light is taken to be 300,000 km/s, the frequency is used in multiples of 3, and the wavelength then simply becomes multiples of 10. The first band is thus 3Hz to 30 Hz in frequency, corresponding to wavelengths of 100,000 km to 10,000 km.
In the center of the radio spectrum are the LF, "low frequency" band, also called LW, "long wave," the MF, "middle frequency" band, also called MW, "middle wave," and the HF, "high frequency" band, also called SW, "short wave." AM radio in the MF band, short wave radio in the HF band, and FM radio in the VHF band are the most familiar forms of radio broadcasting. The UHF band contains some television broadcast frequencies, but at about 30 cm we get into microwaves, which are roughly classified into centimeter and millimeter wavelengths, corresponding to the SHF and EHF bands, respectively. Beyond microwaves, we get into infrared light, at micrometer wavelengths, and here there is no reason not to extend the radio wave system of classification by bands. This puts visible light right into the middle of band 15, with a fair amount of infrared and ultraviolet on both sides. Wavelengths of light are traditionally given in Angstroms ( ), but nanometers (nm) are becoming more common. Photographs can be made with infrared light up to wavelengths of about 1.3 micrometers. At longer wavelengths, infrared light given off simply by warm objects can be detected with electronic equipment. Ultraviolet wavelengths are now divided into three, A, B, and C, stretching very nearly to the end of band 16. There we pick up the X-rays, which go down to a hundredth of an Angstrom, or a picometer (pm). Shorter wavelengths are gamma radiation ( ), given in femtometers (fm) or "Fermis." There are shorter wavelengths of gamma radiation and cosmic rays beyond band 21, but it is there that we run out of metric prefixes for the frequency, offering a convenient place to stop.

To zero in on the part of the spectrum of most interest to us, visible light in Band 15, we can take advantage of a couple of coincidences of nature:
1. Visible light occupies slightly less than one octave of the electromagnetic spectrum, i.e. the longest wavelengths (c. 7600 ) of visible light are slightly less than twice the length of the shortest wavelengths (c. 4000 ); and,
2. The musical note A below middle C is set by convention at a frequency of 440.0 Hz; and 440.0 terahertz (THz) is near the end of the red frequencies of visible light. (My thanks to the dear departed and sorely missed Issac Asimov for his discussion of the musical frequencies.)
Thus, the frequencies of light can be matched up in terahertz with musical notes in hertz. This can then be compared with the visible colors. As it happens, the most familiar colors of the visible spectrum -- red, orange, yellow, green, blue, and violet -- number 6, which is the number of white keys on a piano (all the natural notes, " ") between G and G. They look like the keys of the G Major chord: G, A, B, C, D, E, F#, & G (without the G's).
The table at right, with frequencies and wavelengths, is upside down in comparison to the table of electromagnetic frequencies above. It covers two full octaves above and below middle C. At the top it starts in the Ultra-Violet. Indeed, it includes the entire UV-A part of the spectrum. At the bottom, it is well into the Infra-Red, including the boundary between Band 15 and Band 14. These infra-red frequencies can all be photographed. As a list of music notes, this gives us some sense of how little we actually can see of the electromagnetic spectrum. What is significant about this part of the spectrum, however, is that the particular spectrum of radiation emitted by the Sun peaks right in the yellow wavelengths of visible light. Of all the colors of light, yellow seems to us to be the closest in bightness and transparency to white light itself. This is not a coincidence. The Sun is a yellow star.

Note

A femtometer (fm), 10-15 m, can also be called a "Fermi," since that scale, the approximate size of a proton, is useful for nuclear dimensions, as the Angstrom ( = 10-10 m) is for atomic dimensions. Indeed, the radius of an atomic nucleus is roughly equal to 1.2*A1/3 fm, where "A" is the atomic mass number, the number of protons and neutrons in the nucleus. Mass numbers (A or B) can be found in the Period Table of the Elements. The Angstrom unfortunately is now passing out of usage, since it is not part of the SI. Nanometers (10-9 m) are coming to be used instead.
I am intrigued that the "electrostatic force constant," which plays the same role in Coulomb's Law for electrostatic force that the gravitational constant does in Newton's equation for gravity, and which is given in Physics, The Foundation of Modern Science [by Jerry B. Marion, John Wiley & Sons, Inc., 1973], did not seem to be given in 62nd edition of the Handbook of Chemistry and Physics [edited by Robert C. Weast and Melvin J. Astle, CRC Press, 1981]. I thought, "Isn't there some use for Coulomb's Law anymore?" Well, apparently there is, but the law gets written differently and a different constant is used: the "permittivity constant." It is in different units, farads per meter (F/m), but this turns out to be equivalent to C2/N*m2, the reciprocal of the units of the electrostatic force constant (times 4 ). How that is used, and the rest of the constants, can be seen in Historic Equations in Physics and Astronomy. I now find the "permittivity" constant in the 83rd edition of the Handbook of Chemistry and Physics [edited by David R. Lide, CRC Press, 2002] called simply the "electric constant."
The "Planck" units are of significance (1) as "natural" units of measurement, based on the Planck Constant, the velocity of light, and the Gravitational Constant (with Boltzmann's Constant used to convert the Planck Energy into the Planck Temperature), and (2) for the recent and promising physics of Strings and Super-Strings. The Planck Length is regarded as the smallest physically significant distance, below which is a quantum chaos. It is also regarded as the length of the strings in String theory. The Planck Length is the scale, and the Planck Time the age of the universe, at which gravity is thought to act like all the other forces of nature. Note that while the Planck Length and Planck Time are very small, other units, like the Planck Energy, the Planck Temperature, and the Planck Force, are rather large. A 100 Watt light bulb would expend the Planck Energy in 568 days. What this may mean is that at very small scales, high energies are needed (experimentally) to reveal the structures. The Planck Temperature may have some connection to the temperature of the Universe itself when, shortly after the Big Bang, its size was at the Planck Length.

Duodecimal Counting

The prime factors of 12, however, are 2 and 3, which means that the base is evenly divisible by 2, 3, 4, and 6. The base 10 is only evenly divisible by its prime factors. The practical effect for duodecimal counting is especially to be seen in the fractions, at right and left (where, by analogy with hexadecimal counting, d10 = ddA & d11 = ddB). Common fractions like 1/2, 1/3, 1/4, 2/3, and 3/4 are all easily expressed as single digit duodecimals, without the repeating decimals [the repeating groups are here shown underlined] that plague the student who simply wants to deal with a third or two-thirds quantities. We get double digit duodecimals for 1/8 and 1/9. On the other hand, the simple decimals, 1/5 (d0.2), and 1/10 (d0.1), do get us repeating duodecimals, as 1/5 (dd0.2497) and 1/A (dd0.12497); but it is a good question how much more frequently the student or other calculator needs to express a one-fifth quantity rather than a third or a quarter. The frequency of fifths and tenths that we do have may largely be an artifact of the use of the decimal system itself. It may, indeed, be easier to remember the repeating group for 1/3 in decimal (d0.3) than for 1/5 in duodecimal (dd0.2497), but the same group also works for 1/B (dd0.12497).
The Babylonians, and the Sumerians before them, formulated their counting with a sexagesimal (base 60) system, which of course did not occur naturally in the Sumerian or Babylonian languages, precisely because of the large number of factors by which the base could be divided. Indeed, the only difference between sexagesimal and duodecimal counting is the factor 5 (5 x Consequently, the duodecimal system is somewhat more suited to the artifacts of Babylonian counting that we still use, like the 12 (= dd10) or 24 hour (= dd20) day, the 60 (= dd50) divisions of the minute or the hour, and the division of the circle into 360 (= dd260) degrees.
"Factorials" multiply together successive integers. Since more integers are multiplied in duodecimal than in decimal counting, we get rounder numbers for factorials in duodecimal than in decimal, as shown at right.
Indeed, it is very unlikely that duodecimal counting will ever replace decimal counting. It would be hell on those who, like me, still have to use their fingers occasionally -- though those with the genetic trait of six fingers on a hand would certainly feel vindicated. It would also be necessary to memorize a larger multiplication table, though I understand that students are no longer even expected to remember the multiplication table. This sort of nonsense, of course, is part of the program of the bureaucratic and political elites that control public education to promote their own power and foster the dependency of citizens, who can't even balance their own check books, much less do their own taxes, on government.
A compomise with sexagesimal counting might be a base thirty, whose factors are still 2, 3, and 5. All this loses is one of the 2's from 60, which was not that important anyway. Still, the requirement to have thirty symbols, and not just ten, twelve, or even sixteen (for the hexadecimal), would still be seriously cumbersome. But the number thirty does remind me of another historical counting system, that the Mayans used the base twenty. This is unique in the world, and does accompany an unusual interest in mathematics and numbers on the part of the Mayans. It does not, however, confer the sort of advantages that Babylonian counting did. The prime factors of 20 are still only 2 and 5, so Mayan numbers would still have as many repeating decimals as decimal counting. Thirty is the smallest number that is evenly divisible by the three smallest primes.

This is the third of four lectures on a rather difficult subject -- the theory of quantum electrodynamics -- and since there are obviously more people here tonight than there were before, some of you haven't heard the other two lectures and will find this lecture incomprehensible. Those of you who have heard the other two lectures will also find this lecture incomprehensible, but you know that that's all right: as it is explained in the first lecture, the way we have to describe Nature is generally incomprehensible to us.