March 07, 2002
Dr. Dobb's Math Power Newsletter - March 2002
Math Power Vol. 8, No. 3, March 2002 - ISSN 1087-2035
21st Century Algebra: Sensing Color
Traditionalists correctly point out that the retinal rod receptors cannot be one of the three receptors in a trichromatic retina; but experiments show that rods do participate in sensing color, suggesting that the human retina is not trichromatic in the Youngsian sense. A Helmholtzian model may be assumed instead, based on the color variables brightness, hue, and saturation.[1]
In the Dec. 2001 issue, "The great color-mixture caper," we considered the CIE color mixture diagram and how it is constructed. That diagram is the result of measurements taken in thousands of experiments by hundreds of experimenters over several centuries and it is part of the CIE empirical model of color. Now we consider hue, the color variable that tells whether a stimulus favors red, orange, yellow, green, cyan, blue, violet or some other color of the spectrum.
Exercise 1. Graph exp(-x2) for -5<x<5. This is the simplest form of the Gaussian function which is characteristic of the threshold spectral response functions of the retinal photoreceptors. Note that its log is a parabola. In the photopic (daylight) range of light intensities, the retinal photoreceptor responses are very nearly logarithmic, and the difference in rod (r) & cone (c) responses is hue; thus, letting x be the wavelength of a monochromatic light stimulus, the function h(x)=Ln[r(x)]-Ln[c(x)] gives the hue value of an object in daylight; and with x specified in nanometers (nm), we find that r(x) = exp{-(1/2)[(x-500)/40]2} and c(x) = exp{-(1/2)[(x-560)/40]2} very nearly. This simplified version of the Helmholtzian model applies over the entire range of unique hues from the deepest violet to the deepest red; i.e., for 406<x<682.
Exercise 2. Simplify h(x). A linear function should be obtained. Use the resulting function in the following exercises. Hint: Recall that Ln[exp(u)] = u. [Ans: h(x) = -0.0375x + 19.875 ]
Exercise 3. The violet hue corresponds to 380<x<429. Calculate h(420), the nominal hue value of a violet. (4.125)
Exercise 4. The turquoise hue corresponds to 499<x<513. Calculate h(505), the nominal hue value of a turquoise. (0.9375)
Exercise 5. The lime hue corresponds to 528<x<546. Calculate h(540), the nominal hue value of a lime. (-0.375)
Exercise 6. The lemon hue corresponds to 575<x<587. Calculate h(580), the nominal hue value of a lemon. (-1.875)
Ex.7. The orange hue corresponds to 599<x<610. Calculate h(600), the nominal hue value of an orange. (-2.625)
Ex.8. The tangerine hue corresponds to 610<x<622. Calculate h(615), the nominal hue value of a tangerine. (-3.1875)
Ex.9. The ruby hue corresponds to 622<x<636. Calculate h(630), the nominal hue value of a ruby. (-3.75) Reference - [1] See H. B. Tilton, "A history of color vision and the modern Helhmholtzian brightness-hue-saturation model," ATTI DELLA FONDAZIONE GIORGIO RONCHI, Italy, Vol. LVI, No. 3, May-June 2001, ISSN 0391-2051; pp 487-513. Gravity control
It is nearly 300 years since Newton theorized about gravity—he saw it as a universal force, Einstein saw it as a geometry. But in spite of all that, a complete understanding of gravity eludes. How can I say that? The absence of antigravity machines says it all—machines that fly without using buoyancy, aerodynamics, magnetic levitation, or rockets. We do not have them. In all cases our flying machines remain suspended in mid-air by using a force other than (anti)gravity to counteract the gravity of the Earth. Satellites harness gravity the way the moons and planets do.
We are looking for a landmark invention here. But before there can be an invention, there must be malleable theory—one that can be molded into an invention. What the theorist does when developing a theory is try to reverse- engineer a portion of the universe. He begins by bringing all his knowledge to bear. Two possible bases for a malleable theory of gravity are:
(1) We can assume a natural gravitational charge exists which is analogous to
electric charge, except that like charges attract instead of repel;
Under the second basis there is no doubt that such neutral electric dipoles will rotate rapidly. Physicists refer to a parameter they call spin in gravitation theory, the notion of rotation being contained in that term.
The physics of our malleable theory of gravity must be simple, because for Nature it "came naturally"; it's just our math that makes things seem difficult, and it may be productive to develop a new math which makes gravitation theory easy. Efforts are afoot to do that in various quarters. The Gij notation of general relativity is an effort to do just that; but it may be too obscure. Surely a greater effort by more individuals would be helpful in this regard.
There have been those who've claimed to have invented an antigravity machine; some of those efforts have been discussed in our sister publication, LIGHT WORK, now defunct. Problem is, those inventions could not do what was claimed for them and the theory behind them was obscure.
Whoever the real future inventor of the first antigravity machine may be, he must have intimate knowledge of present-day theory—that is to say, of its mathematics. A good starting point is Einstein's general theory of relativity and Maxwell's electromagnetic theory. A union of those two theories may let us forge the key to unlock the kingdom of antigravity.
by Norman Nerdnick
One of the boss's favorite publications, Satellite TV Week, performs a magic trick involving a hitch in time that would surely have fascinated Albert Einstein himself. You know how the bulk of the nation "springs forward" in Spring and "falls back" in the Fall? The World Almanac: "Daylight Saving Time in the U.S. begins each year at 2 a.m. on the first Sunday in April and ends at 2 a.m. on the last Sunday in October." That means that at the Fall switch, the hour 1-2 a.m. occurs twice; then at the Spring switch the hour from 2-3 a.m. is missing. So in the Fall the hitch is a pleat and in the Spring it is a tear and a gap. And those hitches move in a rippling way across the nation from east to west because they occur at 2 a.m. in each time zone.
Now here's the magic part: As this is being written we just got our copy of STVW for the week Oct. 28-Nov. 3, 2001 which contains the last Sunday in October I'm looking at it now; and guess what—there is no hitch in the timeline in the program grids. On a hunch I checked the previous week's issue to see if maybe the hitch was there, at the end of the listing for Saturday. It wasn't. There is no hint of a hitch anywhere near the 2 a.m. columns. How do STVW's editors manage to make that hitch in the civil timeline vanish? It must be magic. Surely they wouldn't arbitrarily smooth it over out of expediency!?
Editorial
This is a call to each student to get serious about your math studies and contribute to a fuller understanding of the Universe in which we live.
When I was very young I was told that the purpose of life was to appreciate God. For me, that translated into a need to fully understand the Physical Uni- verse. I've had a lifelong fascination with many areas of physics, but as this is being written I am 75 and that will be ending soon. While I developed a new mathematics of relaxation oscillations to help the analog electronic circuit designer in his craft (because I needed it and no one else had done it), and I developed a new theory of human color vision (because the vested interests in Young's brave but premature theory had led everyone to fall prey to the "petrified knowledge" monster), chances are I will not be around long enough to find a full understanding of gravity.
I appeal to all who dream of going to the stars. Control of gravity is needed for that. Continue the quest! ... HBT
Dear Physicist - There exists a club on "Smareandache Non-Euclidean
Geometries" at http://clubs.yahoo.com/clubs/smarandachegeometries Ed. note: Send your comments directly to mikeantholy@yahoo.ca Division by zero is not a dead issue -
Under a new paradigm division by zero is not only undefined it is moot.
The good news is that it is not a dead issue; and as long as we do not
attempt to perform that operation, there is nothing wrong with indicating
it. Thus we break no law by simply writing 1/0; and indeed under the new
paradigm 1/0 may serve as a placeholder as in |Arctan(1/0)|= Recall that the ideal signum function, Isgn, is like SGN except it is not 0 for an argument of 0; instead it is +1 or -1 there, depending on direction. Now prove that the identity Isgn(x)/|x|=1/x holds even for x=0. To do that, first see that 1/Isgn(x)=Isgn(x) and Isgn(x)|x|=x hold even for x=0; then write Isgn(x)/|x| = 1/[Isgn(x)|x|] = 1/x, QED; and you've broken no law. Helmholtz said it, and other respected researches have said it more recently. Note that with the Helmholtzian omnichromatic model it is not helpful to assume a set of "primary colors" because all spectral colors have equal footing as Hermann von Helmholtz recognized in the mid 19th century: "To assert that there are simple colours in the spectrum...would not be correct." So, why has the idea of primary colors in human vision persisted for 200 years and counting? The "primary" reason is that it mostly works. But so did Ptolemy's geocentric model of the universe. Plotting three colors on his color-mixture diagram, Newton found 300 years ago that mixtures of three colored lights are able to produce any color lying within the triangle so defined. Further, there is one set of three (nominally red, green and blue) that maximizes the accessible color gamut. The use of those three in an analyzing camera or synthesizing display gives the widest range of colors possible with such a three-primary system, but not all colors can be produced that way. Even so that is what we've been doing all along with color photography, color television and color computer displays. We've erected a triangular fence in a curved barnyard,excluding part of the herd. One reason that idea has persisted is that no alternative was heretofore apparent. But it is now clear that cannot be the way the human eye does it. So isn't it time to access the entire color gamut? How to do that in a simple, evolution-friendly way is revealed in the lead article and its reference. ERRATUM: The February 2002 issue contained a simple error in arithmetic, not affecting the end result. See if you can find it. Also in that issue... As a follow-up to "Infinitely satisfying" where it was found that x=±  ,
it is an interesting exercise to let u=1/x in the original problem then
solve to get u=0. Even though we would normally throw out such a proof because
at some point division by zero is indicated, since we do not actually need
to perform division by 0, this deserves a second look. Final proofs must
of course be backed up by limit processes. Such borderline situations often
play a role in nonstandard analysis and special calculus.
Things to come - "The Stroud Project": Developing Gaussian (exponential) Function Module GCM-8805, and generating a Gaussian pulse train. An "Emerging Technologies" article in the May, 2001 issue of Photonics Spectra, p. 58, reports: Two [teams], working separately, have slowed light from a crawl to a stop, and then restarted its propagation...with the [same] phase and amplitude information of the initial signal... Group velocity, as opposed to phase velocity, was affected. One team was headed by Prof. Lene Vestergaard Hau of the Rowland Institute for Science and Harvard University in Cambridge, Mass. Mikhail Lukin was "a lead researcher" on the other team located at the Harvard-Smithsonian Center for Astrophysics, also in Cambridge. The latter team's work was announced in Physical Review Letters, Jan. 29, 2001 according to PhS. The full theoretical implications of this development are still obscure, at least to this writer. For example, when the light beam is stopped can the power source then be removed and the beam remain? Highly doubtful; so just what does all this mean? What are the implications beyond the mundane appli- cations to optical switching and optical computing in small packages? I mean, how must our understanding of basic physics be modified in the light of this development? Who knows; maybe next year someone will cause light not only to stop but to move backwards, retracing its steps! A PARADOX!
The cube root of -1 is -1, a real number; but if the radical index is
not exactly 3, it is complex. Investigate f(x) =(-1)^[1/(x+3)]
at and around x=0 where ^ indicates exponentiation. Hint: Write f(x)=exp[(1/(x+3)log(-1)].
Procedure: Note that log(-1)=i
f(0) = cos( Written reader comments are invited on all material. Those intended for my attention must be submitted by US Mail to my Tucson address. All such comments are subject to being published unless requested otherwise. They may also be subject to editing. ... Homer Tilton, Editor The hard copy version of MATH POWER is published as a shareletter; that means you are permitted to make not-for-profit copies from it for distribution to your colleagues and students. MATH POWER is published monthly. It is published and edited by Homer B. Tilton under the auspices of Pima Community College, East Campus, 8181 E. Irvington Rd. Tucson AZ 85709-4000. All material is copyrighted by Homer B. Tilton unless otherwise noted. A limited number of copies may be made at educational institutions for internal use of faculty and students. For more extended copying or to request additional copies contact the Editor at the above address. Letters and editorial material are welcome. All submitted material may be published in MATH POWER, and edited, unless specifically requested otherwise.
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/2,
meaning "in the limit as x in |Arctan(x)| goes to ±
0.5 + 0.866i ...but f(0) must equal -1... 
