Dr. Dobb's Math Power Newsletter - April 2004

Level: Calculus

The assertion has been made that if a rocket simply accelerates long enough it will exceed the velocity of light. For example, the argument goes, arocket accelerating at one gravity (980 cm/sec²) for one year will have a velocity of 980•60•60•24•365=3.05X10¹⁰ cm/sec exceeding the velocity of light, 3.00X10¹⁰ cm/sec.

To see if this is true one needs to solve for the equations of motion of the rocket and determine its velocity using the principles of the theory of relativity.

The problem has been solved by several authors. Misner, Thorne and Wheeler, Gravitation, 1973, pp. 166-167 give the following solution. A detailed solution of the problem is available in a separate paper entitled "Solution of the Constant Acceleration Problem." [Contact the author for that paper...Ed.]

The equations of motion of a rocket accelerating at a constant one Earth gravity in the x direction in the fixed frame, the unaccelerated frame in which it began to accelerate, are

        t =(1/g)Sinh(g<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)                             

        x =(1/g)Cosh(g<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)                               (1)

        y=z=0                                        

where t, x, y and z are the coordinates of the rocket in a system at rest relative to the rocket's starting point, g is the acceleration of the rocket as measured in a frame moving with the rocket and the parameter is the time measured by a clock in the rocket. The units used in these equations are "natural coordinates" in which the velocity of light, c, is taken to be one.

Then to get the time coordinate, the 1/g factor in the equation for t must be multiplied by c to get c/g which does indeed have units of seconds; and to get the x-coordinate, the 1/g factor in the equation for x must be multiplied by c² to get c²/g which does have units of centimeters. These equations result:

        t = (c/g)Sinh(g<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">/c)                          

        x = (c

<sup><div align="left">2</div></sup><div align="left">/g)Cosh(g<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">/c)                           (2)

        y=z=0                                        

</div>

Now let g=980 cm/sec² and c=3.00X10¹⁰ cm/sec, and make use of the fact that a year is 60•60•24•365=3.1536X10⁷ s and a light year is 60•60•24•3X10¹⁰ which equals 9.4608X10¹⁷ cm to write

        t = 0.971 Sinh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)                       

        x = 0.971 Cosh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)                         (3)

        y=z=0                                        

where now t and are in years and x is in light years.

Now we are in a position to ask "How fast is the rocket moving?" Simply differentiate x with respect to t to obtain the velocity v of the rocket in the fixed frame:

                                       dx     d 

                                       --     -- [0.971 Cosh(1.039<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)]

                    dx d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">   dx   1     d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">     d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif"> 

        v = dx/dt = -- -- = -- ---- = ---- = -------------------------- 

                    d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif"> dt   d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">  dt     dt     d 

                                --     --     -- [0.971 Sinh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)]

                                dd<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif"><img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">     d<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif"> 

            0.971•1.030 Sinh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">) 

          = ------------------------ = Tanh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">)                    (4)

            0.971•1.030 Cosh(1.030<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">) 

This indicates that the fastest the rocket can go, as measured in the fixed frame, is one light year per year, or just the velocity of light, as the limit of Tanh is 1.0 as its argument approaches infinity. But the navigator in the rocket after one year of rocket clock time thinks he should have exceeded the velocity of light because he accelerated for one year at one gravity. Has he? Does he think he has done so? The answer is "No." A fixed observer will see the rocket traveling with the velocity given by eq. (4) so he will say that the rocket does not exceed the velocity of light. What about the navigator in the rocket? He will see the observer at the start point receding at the same velocity.

So the navigator will be very puzzled because he should be traveling at the velocity of light but is not. What is wrong?

Let's conduct a "thought experiment" to see if we can figure out what's going on. Let's say that after the rocket has accelerated for some months it has reached a velocity of half the speed of light. The navigator on the rocket drops a probe with a radar at that point and programs the probe to measure its speed by measuring the distance to passing asteroids and comets. Since it's a thought experiment we can assume the necessary asteroids and comets are there. Also assume that the rocket's home base has a radar with which it can measure the probe's speed. Call the fixed home base observer O and the probe dropped at the half speed of light observer O'. The speed of the probe must be the speed of the rocket at the time the probe was dropped. The reports of speed from the probe O' and home base O come in and, as expected, both say their relative velcities are c/2. So far so good.

The navigator decides to accelerate for another month when he should have added about 0.1c to his speed, drop another probe, O", and again measure speeds. He does so and gets the speed reports from O' and O". As expected they report their relative velocities as 0.1c. Then the navigator compares the reports from O and O". They report their relative velocities are 0.571c not 0.6c as expected. What happened? The problem is that velocities do not simply add at relativistic speeds. Rather one must use this formula (from Bergman, Introduction to the Theory of Relativity, 1942, p. 43):

                v'+v" 

        v = --------------                              (5)

             1 + v'v"/c

<sup><div align="left">2</div></sup>

where v' is the velocity of O' with respect to O and v" is the velocity of O" with respect to O'. Using eq. (4) we get that the rocket goes 0.5c when is 0.5493 year. The rocket will accelerate to 0.1c in about 0.0971 year. So for =0.5493+0.0971=0.6403 eq. (4) gives a velocity of 0.565 in close agreement with eq. (5) which gives 0.659. (The disagreement comes because we applied eq. (5) to the 0.1c increment as one increment rather than breaking it into many smaller increments.) So no matter how long the rocket accelerates, no matter how fast the navigator thinks he should be going the rocket can never exceed the velocity of light. Each increment of added speed is less and less effective in increasing the rocket's speed as it nears light velocity. There is no need to cite any rules about increasing mass or infinite force being needed to accelerate the rocket. Even if infinite energy is available it still will not cause the rocket to exceed the velocity of light. The limiting velocity of material objects is the speed of light. [Emphasis added...Ed.]

A final comment: The limit on the velocity of the rocket does not come from eq. (5), the formula for the relativistic composition of velocities. Equation (5) is used above simply to explain what is happening. The limit comes from solving for equations of motion of the rocket and computing its velocity. The result follows simply from the application of the principles of relativity that Lorentz transformations relate observations between observers in inertial frames, that observations by various observers in inertial frames of the same events must agree (general covariance) and the relativistic equations of motion which follow from them.

A final, final comment: The argument has been made that the mathematics are correct, but that the "apparent" velocities are just an illusion like the apparent bending of a straight stick extending into a pool of water. This is false. The predictions of relativity are real and have been experimentally verified (see ch. 38 in Misner, Thorne and Wheeler for a detailed discussion of experimental results). Verification of the composition-of-velocities formula near the speed of light is also provided by particle accelerators which accelerate particles near, but never beyond the velocity of light. [Emphasis added, two places...Ed.] Time dilation has been verified by measuring the lifetimes of unstable particles in cosmic ray debris and particle accelerator experiments. Experiment is the final arbiter in science, and experiment verifies that the velocity of light is the speed limit.

The theory of relativity is incomplete, since it is not compatible with Quantum Mechanics. At some future time relativity will be replaced by a more complete theory, just as Newtonian mechanics was replaced by relativity, but whatever replaces relativity must still agree with the predictions of relativity in those cases where experiment has shown relativity to be correct.

-- The End --

The author of "Begin the Adventure" responds -

My friend Jim Malmberg severely misses the point. (See pp. 1-3.) The theory of special relativity is not in contention; the contest is between Einstein's reality view and the gathering return of the pre-Einsteinian kinetic-perspective view; it has nothing to do with the composition-of-velocities formula, and that is not in contention.

Like a true disciple of Einstein's reality view, Jim declares these things:

• In connection with his opening statement "The assertion has been made that if a rocket simply accelerates...",

Jim declares: "To see if this is true one needs to solve for the equations of motion of the rocket..." which he then apparently and curiously takes to be his equations (1) thru (4), culminating in eq. (5), the composition of velocities formula.

In taking the approach he has, that of applying special relativity to a problem of general relativity, Jim has fallen into a logical trap called "an alternate reality." Massimo Piattelli-Palmarini (pages 142-3):* "Few of us suffer from any illusion that the summary paths taken by our intuitions and approximations would lead us to exactly the same point to which reason and exact calculation might have brought us. But we do delude ourselves into thinking that we are thereby brought to a neighboring area, one that is close enough." One cannot analyze the real motion of an accelerating rocket by applying the theory of special relativity since that deals only with inertial systems and the accelerating rocket is not an inertial system. If we apply it anyway (as Jim has) that may or may not take us to a neighboring area; but, under the kinetic perspective view, it would be an appearance only.

• "The limiting velocity of material objects is the speed of light." As justification (it seems) for that statement he invokes the composition-of-velocities formula again. But, must I repeat, an accelerating rocket operates in a different realm -- the realm of general relativity -- because it is accelerating; it is not an inertial frame. The formula Jim uses [eq. (5)] was presented long before 1973 (the date of his reference) and is not in contention. [It appeared in A. Einstein, Relativity, 1916 (English translation, 1931), chapter XIII, "THEOREM OF THE ADDITION OF VELOCITIES, THE EXPERIMENT OF FIZEAU." It is eq. (B) on page 46.

• ...In connection with the stick-stuck-in-water simile: "This is false." (It is not clear what he means by "this," specifically.) ...Following that with another declaration: "The predictions of relativity are real..." then a true but unconnected statement: "[They] have been experimentally verified."

His argument is unconvincing. The bent-stick effect, too, has been "experimentally verified" but that does not make it real. (Consider that a nonzero angle of bend can be measured even on a photo of the "bent" stick.) If Jim's point of view is to be supported, he must present a better rationale for the opposite realnesses he assigns to the bent stick and special relativity. What is at issue: When is the observed speed limit a real limit, and when it is not? Even a truly bent stick stuck in water may appear bent, but not all such appearing sticks are really bent.

* Massimo Piattelli-Palmarini, Inevitable Illusions: How Mistakes of Reason Rule Our minds, John Wiley & Sons, 1994, ISBN 0-471-58126-7; Dr.Massimo Piattelli-Palmarini is Principal Research Associate of the Center for Cognitive Science at MIT and Director of the Department of Cognitive Science at the Institute of San Raffaele in Milan, Italy.

Jim's formulas are correct but his paper misses the larger point. Thus when he writes "The fastest the rocket can go, as measured in the fixed frame, is one light year per year" [following eq. (4)] he is, of course, correct. But the key phrase there is, "as measured in the fixed frame"; such a measurement is only a measurement not necessarily indicative of the reality, just as the measurement of the straightness of a stick stuck in water is not necessarily indicative of the reality.

• "Verification of the composition-of-velocities formula near the speed of light is also provided by particle accelerators which accelerate particles near, but never beyond the velocity of light"

But Jim, the composition-of-velocities formula needs no such verification and is not in contention. And again, Jim, I hope you would re-read Part 1 of "Begin the Adventure" where the lightspeed limit on such particles is explained in terms of true rationality, not as an alternative rationality as you have done. Massimo Piattelli-Palmarini: "The age of 'alternative' solutions is fortunately over."

The gathering kinetic perspective view of special relativity runs counter to the reality view held by Einstein & Malmberg. (But you are in famous company, Jim!) How could Einstein have been fooled into thinking that the effects of special relativity are real if they are not?

Einstein may well have been the victim of a cognitive illusion. Massimo Piattelli-Palmarini defines cognitive illusions as "mental eyeshades...biases, tunnels, or blind spots." From pages 139-41:

That passage appears in Chapter 8 entitled, "How to Emerge from the Tunnel of Pessimism." The dead-end "tunnel" (the light barrier) which Einstein led us into a century ago is truly one of pessimism; it is time we emerged from it.

If one might think that Massimo Piattelli-Palmarini was simply engaged in some kind of put-down of some particular acknowledged expert or genius when he wrote that book, note that no mention of any particular expert, by name, could be found anywhere in the book; nor could "relativity." "Relativist" appears (on page 4); but that is clearly a reference to one who engages in relativism, a particular way of thinking: "The romanticists and relativists (of both yesterday and today)..."

-- End --

Henry Margenau:* If sense data alone were recruits for reality, its domain would be ill-defined. ... It would be inadequate to ascribe reality to nothing but momentary surges of perception.

* Henry Margenau, The Nature of Physical Reality, McGraw-Hill, 1950, page 291

Space is like a banjo string

The speed of light is like a resonance condition. With _o and _o being characteristic magnetic and electric constants of space (called permeability & permittivity respectively), James C. Maxwell saw that 1/(_oo) gives the speed of light. The constants _o and _o can be measured in the laboratory using inductive and capacitive circuit elements.[2] In this way theory shows how the speed of light can be determined without involving a light beam at all.

Similarly, with L and C being characteristic magnetic and electric constants (called inductance and capacitance respectively) of a tuned circuit, we have that 1/(LC) equals the resonant angular frequency of the circuit. Compare with1/(_oo) from above. Therein lies an important analogy well known to radio hams, another element of which is the intrinsic impedance of space, _o/_o 377 ohms. Radio antennas must match that impedance if we would accomplish best power transfer from the electrical circuit to space.

As with the LC circuit, the acoustic analog of the speed of light is not the speed of sound as is usually assumed, it is the resonant frequency of a tuning fork or banjo string. When a banjo string is twanged (plucked), it resonates at its characteristic frequency. Analogously, when empty space is twanged as by an electric spark, it resonates at its characteristic velocity. Electromagnetic energy is launched into space, moving at the resonant speed c relative to the receiver. The energy, E, in the twang is coded into the frequency, f, of the radiation according to f=E/h, h being Planck's constant.

The twang itself may be considered a quantum of energy, with

     <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/taud12.gif">
E = <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/integ.gif">  Pdt
     0

where P is the power profile and T the duration of the twang; but it was Planck's view that quanta (photons) do not comprise the electromagnetic beam. Even if the energy is transmitted as a wave, it is understandably detected at the receiving end in terms of quanta when the receptor elements are discrete nuggets of electricity such as electrons.

The wave theory of light had come into nearly full acceptance following Maxwell's work when Einstein resurrected the photon to explain the photo-electric effect,[3] and it seems everyone fell into lock step; indeed, a popular 1966 physics text, Halliday & Resnick (H&R), reported (p. 1181): "Three major features of the photoelectric effect cannot be explained in terms of the wave theory of light" (underline added). A strong indictment indeed. But why should the detection process say anything about the transmission mode? The key to understanding the photoelectric effect in terms of the wave theory of light may lie simply in recognizing that the analog of the speed of light is not the speed of sound but the frequency of a banjo string. That's a statement you won't find in every physics text!

Notes for "Space is like a banjo string" -

[1] Although c is the measured speed with which a packet of light moves across an observer's bow, the speed with which a packet travels from a far-distant source to an observer seems not to be directly measureable.

[2] See for example eq. (36-5) & (30-7) in Halliday & Resnick, 1966 (H&R). The speed of light is 1/() in general, and when =_o & =_o that is c= 299 792 458 meters per second; but and will increase in gravitational & inertial fields and so the speed of light will be less there; and the resonant frequency of an LC circuit will drop when spun in a centrifuge at a high rotation rate -- not because of the motion but because of the large centrifugal inertial field.

[3] Einstein felt the need for corpuscular photons; Planck did not. Planck: "That he [Einstein] may sometimes have missed the target of his speculations, as for example in his hypothesis of light quanta, cannot really be held against him." [As reported in Optics News (a publication of the Optical Society of America), Winter 1979, p. 38.]

Doppler Light

Postulate 2 of special relativity suggests many questions, some of which are answered by the relativistic foreshortening (shrinking) and apparent mass increase & time distortions coming out of the equations of special relativity. Those effects are finely interwoven with the Doppler effect announced earlier.

The same speed is measured for light no matter what the relative speeds of the light source and the observer are. The measured frequency... will change... . Such frequency shifts are called Doppler shifts, after Johann Doppler (1803-1853), who first predicted them.

...The [Doppler] frequency [f] heard by the observer is [with =u/v]

        f/f

<sub><div align="left">o</div></sub><div align="left">=1/(1+<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/beta12.gif">)                            (40-16)



</div>

observer fixed in medium, source receding from observer [at speed u]. In this equation fo is the frequency heard when the source is at rest and v is the speed of sound. ...

        f/f

<sub><div align="left">o</div></sub><div align="left">=1-<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/beta12.gif">                                (40-17)



</div>

source fixed in medium, observer receding from source... For light, as contrasted with sound...it has proved impossible to identify a medium of transmission relative to which the source and the observer are moving. This means that "source receding from observer" and "observer receding from source" are physically identical situations and [so the Doppler effect for light is only the geometric mean of eq. 16 & 17; namely f/f_o=((1-)/(1+)) which can be written]

        f/f

<sub><div align="left">o</div></sub><div align="left">=(1-<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/beta12.gif">)/<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif">(1-<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/beta12.gif"></div><sup><div align="left">2</div></sup><div align="left">,                     (40-18)



</div>

[or rationalizing the denominator, f/f_o=(1-²)/(1+). Here means u/c with] source and observer separating...

eq. (40-18) puts points at (,f/f_o) = (0,1),(1,0). Also

lim <i>Re</i>(f/f

<sub><div align="left">o</div></sub><div align="left">) = <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/infinity.gif">,0; 
<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/beta12.gif"><img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/arrow12.gif">1<img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/plusminus12.gif">
</div>

and for ²>1,

<i>Re</i>(f/f

<sub><div align="left">o</div></sub><div align="left">)=0.                                 

</div>

What does the above analysis mean for a superluminary rocketship? Two things:

(1) When the ship is moving away from the observer with =1-0², we have f/f_o=¦0¦/2 which is real and zero; and if =1+0², we have f/f_o=i¦0¦/2 which is imaginary and zero; and if > 1 then Re(f/f_o) is zero.

(2) When the ship is approaching the observer with =-1+0², then f/f_o=2/¦0¦ which is the ultimate in hard cosmic rays; and if =-1-0², we have f/f_o=-i2/¦0¦ which is imaginary with real part zero; and if < -1 then Re(f/f_o) is zero.

What that means is that the stars directly in front of and in back of the ship will vanish in a widening circle or cone as the speed increases past lightspeed. The stars still visible in front will have their light shifted to the blue and those in back will have their light shifted to the red.

What is the speed of light?

Since 1 Jan. 1990 the speed of light has been put at exactly 299 792 458 meters per second. "Exactly" because the meter is now defined in those terms. Wildi (p. 51):[1] "The meter (m) is [defined as the distance] travelled by light in a vacuum during a time interval of 1/299 792 458 of a second."

Ex.1. Given, one inch is exactly 2.54 cm. Find how many meters there are in a mile (5280 feet). Find an exact conversion.

Ex.2. Calculate the speed of light in miles per second. Is your answer exact?

The title question has a deeper meaning. When we say that a light signal propagates at "the speed of light," that statement needs to be clarified. By Postulate 2 of special relativity, that speed is independent of v, the speed of the source relative to the observer; but what if the source is receding at a speed vc?[4] What is the situation then? Doppler provides an answer.

According to Doppler, the frequency, f, of a light signal drops if the source is receding from the observer. The cosmic red shift is often given as an example. For a receding source (v positive) or an approaching source (v negative) the ratio of apparent frequency to intrinsic frequency is Re ((1- )/(1+)) with = v/c for all real <> -1, which is 0 for 1.

Even though the speed of an incoming light signal is independent of the source speed, the received energy is not.[5] Thus the incoming speed of a light signal from a source which is receding with a speed of v= 0.5c, 0.9c or 0.99c is always c; and one can even say it is c for vc but that is moot, for then no energy reaches the observer from that source.

Ex. 3. Given =¹/₂-Arcsec() rad. Show a complete proof that at a speed of =2 the Cerenkov shockwave cone contains an interior half angle of exactly =30^o.

Notes for "What is the speed of light? -

[1] Theodore Wildi, Metric Units and Conversion Charts, 2nd ed., IEEE Press, NY, 1995, ISBN 0-7803-1050-0; IEEE Order Number PP4044.

[2] From 1961 thru 1989 the meter was defined as 1 650 763.73 wavelengths of the orange-red light of krypton-86. Before that, the meter was defined as the distance between two scratches on a platinum-iridium bar kept at the International Bureau of Weights and Measures in Sèvres, France.

[3] The scientific second is presently defined in terms of the transition between the two hyperfine levels of the ground state of cesium-133.

[4] It is assumed Einstein's light barrier is not a problem. See MATH POWER, Nov '03, p. 2, "Light sailing is not all there is."

[5] According to Planck's equation E=hf and in view of Doppler.

Secrets of the Old Masters

In a recent CBS "60 Minutes" a case was made that some portrait artists of the 17th century may have -- without saying so -- used a secret aid called camera obscura; that is a camera without film. This was in an age before there were cameras as we know them, but at a time when focusing mirrors and lenses were just coming into use. A case was made that some of the old masterpieces might have been made with the aid of an image projected onto a canvas by means of a concave mirror or lens. There would be a reversal, as in a mirror, and one argument of the proponents was that suddenly in that period, more than the expected number of left-handed people were depicted in paintings.

The making of a concave mirror is no easy task; but it was accomplished well before photographic film was manufactured. And anyway a lens or even a pinhole will also work. So the artist himself became the photographic film, simply tracing and coloring the optical image projected onto the canvas. What results is a proto-photo. Cheating? Well it fooled a lot of fools. Some art aficionados today, when told of the likely camera-obscura caper, may become nervous and highly defensive of their now-questionable views of such matters.

The point of this article is that seemingly difficult tasks may not be that hard if you use a little ingenuity. You can accomplish things to amaze others if you simply apply yourself to the task at hand with an intelligent eye to accomplishing it. Another example of working smarter not harder appears below (p. 6), at "Piano with a gearshift."

A special talent -

Idiot savants are celebrated for their ability to take any date they are given and supply its day of week by only thinking about it for a moment. The problem is, does anyone ever check their answers? Audiences ooh and aah when presented with such a performance, & seem willing to accept answers on faith. Even dates well into the future are analyzed in this manner; but again, it seems no one checks.

Am I missing something? Can someone shed light on the phenomenon? Is it genuine? Has anybody ever run scientific tests concerning it?

Speaking of exceptional powers -

The other day someone asked this reporter why someone doesn't check for possible telepathic powers in a group of subjects. What could it hurt?

Well someone has checked. Researchers at Duke University performed banks of experiments in the 1950s concerning just that over a period of several years. Results came up negative.

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. -HBT

The hard copy version of MATH POWER is published as a shareletter; that means you are permitted to make not-for-profit copies of 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. Editorial Assistant, Jo Taylor. All material is copyright 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.

What the pedestrian ear may hear

A bugle or trumpet with neither slide nor keys still lets the bugler play a variety of notes depending on the vibration mode of the air within the instrument which he controls with his blowing. Notes may include C, G, >C, >E, >G. (The > sign means "up an octave" from the keynote.) Those are related by frequency ratios of C:G= 2/3, C:>C=1/2, C:>E= 2/5, and C:>G=1/3 in Pythagorean tuning. This GENERIC PC MATH Smart BASIC miniprogram sounds the first 18 notes of "Reveille" using the 12-tone just scale:

 Echo Play"G>CEC<g>ECEC<g>ECEC<g>CEC"|ok                                (S01)

Compare that melody with the first 15 notes using Pythagorean tuning, as with a bugle, in this brute-force GENERIC PC MATH two-command suite:

 Set n=2,4:sound 261.6*

 Echo sound 261.6*3/%n%%n%5/%n%%n%3/%n%5/%n%%n%5/%n%%n%3/%n5/%n%%n%5/%n%

      %n%3/2,4:beep |ok                                                 (S02)

Ex. 1. For Pythagorean tuning there is this more-elegant suite than (S02):

    <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif"> 

 SET pyth=p(i) <i>[Symbolically]</i> 

 ECHO For i=1to 18:Sound 261.6*%pyth%/2,4:next:beep |ok                 (S03)

     <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif">   <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif">     <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif">        <img src="http://twimgs.com/ddj/ddj/images/ddj0404mp/sqrt.gif">   < --  --  mandatory spaces 

where p(i) is the 18-term sequence 3,4,5,4,3,5,4,5,4,3,5,4,5,4,3,4,5,4.

A function to implement that sequence in Smart BASIC is p(i) =

 (CINT(9/8*cos((i-8)*0.8*5/3.14)-1/8+4)+

 2*(i\6)*(6\i)+2*(i\11)*(11\i)+(i\16)*(16\i)+(i\17)*(17\i)-(i\18)*(18\i))

It was found by fitting a cosine function to 13 of the 18 data points (to ¹/₂) and adjusting the remaining five points using the Kronecker delta, _mn, which is (m\n)*(n\m) =1 when m=n else =0 with m and n being natural numbers. The Kronecker delta subs for the DATA statement in miniprograms. Constants 9/8 (amplitude), 0.8*5/3.14 (frequency), and -1/8+4 (dc offset) have not been reduced so as not to obscure their makeup.

Ex. 2. If there is any doubt that the PLAY statement gives a just scale, this two-command suite might be used:

 SET just=j(i) <i>[Symbolically]</i>

 ECHO For i=1to 18:Sound 261.6*2^(%just%/12),4:next:beep |ok            (S04)

where j(i) is the sequence 7,12,15,12,7,15,12,15,12,7,15,12,15,12,7,12,15,12.

Find an expression for j(i). Hint: Study Ex. 1 for a clue.

Word of the month -

key¹ - The lever by which the sounds of a pianoforte, organ, or harmonium are produced. [88 such levers for the standard piano.]

key² - An arrangement by which certain holes are opened and closed in flutes, oboes, and other wind instruments.

key³ - A scale, or series of notes progressing diatonically, in a certain order of tones and semi-tones, the first [lowest-frequency] note of that scale being called the Key-note.

...Elson's Pocket Music Dictionary, Oliver Ditson Co, 1909.

Piano with a gearshift

You knew that a slide trombone doesn't need tuning the way a violin does. What's more, with the slide trombone it is no special problem to transpose a song from one key³ to another. What does it mean to "transpose" a song?

With a properly tuned piano as the standard, adjacent keys¹ or notes are said to be related as just semitones. "Just" means the ratio of frequencies for adjacent keys¹ (whether black-white, white-black, or white-white) is constant, numerically equal to 2^(1/12), or 1.0594631 to eight figures. [There are twelve semitones in an octave, as from Middle C to the C above (or below) Middle C.] That means a song played with one set of fingerings will sound the same, patternwise, when the fingerings are moved or transposed one or more semitones up or down. ...Easier said than done because not all adjacent white keys¹ are separated by a black key¹.

Although that "syncopated" layout of keys¹ developed accidentally, it is advantageous for keeping visual and tactile track of finger registration as one is playing along, but it makes transposition difficult. That syncopated key¹ layout is carried over to the saxophone keys² to facilitate playing in a so-called major key³ with either instrument, if I have my facts straight. Such keys³ relate to the diatonic scale (the white keys¹) developed first in Europe hence "major."

It has been reported that Stan Kenton, popular jazz pianist of the mid 20th century, had a piano with 24 just keys¹ per octave. It has also been reported that Irving Berlin, famed popular song writer of the early 20th century, had his piano fitted with a "gearshift" lever with which he could change the registration of the piano keys¹ to the strings nearly on the fly, that is in the middle of a rendition, to affect a transposition. Can a similar thing be done with, say a saxophone?

Many wind instruments have keys²; each bringing forth a particular note or selection of notes. So the tuning of, say a saxophone, relates to the configuration of the instrument, unlike a trombone. Turns out it is advantageous to manufacture differently tuned saxophones. From Schmidt (copy permission requested):*

And here all this time you marveled at the ability of musicians to play in any key³! Of course when a musician modulates, or transposes keys³ on the fly without changing instruments, that is an accomplishment of note. (Ouch!)

* Stanley Schmidt, "Scarce Skills and Scattered Substitutes," editorial in Analog Science Fiction and Fact, ISSN 1059-2113, Oct. 2003, pp. 4-7

The power of batch/BASIC collaboration in computer math

Only Smart BASICTM (GWBASIC) is required; it remains behind the scenes like a second command file. Batch programs and commands that do math have appeared in these pages whether for calculating or graphing. Presented here is some background.

Graphing and calculator commands are processed in GENERIC PC MATH^TM by COMMAND.COM before being sent on to Smart BASIC, resulting in the following characteristics and advantages -- whether in a batch/BASIC program, or with no program at all:

1) All user input is via the DOS command line. Even if you decide to write a batch/BASIC program you will use the DOS EDIT text editor -- what Kay Y. Nelson calls "the Editor" (Voodo DOS, Ch. 7), never the BASIC editor. And if you labor under the illusion that DOS does not exist in Windows 95, 98, and 2000, know that it does. In W95 it is DOS 7.00, in W98 it is DOS 7.10, and in W2000 it is a variation of DOS 5. Linux can be configured to emulate DOS, also.

2) SET definitions can be used in commands. These are the %...% forms. This enables the use of multiple command suites, providing useful versatility. See p. 5 this issue for example. Examples also appear in Jan '04, Sep, Oct '03.

3) Nonkeyboard characters (ASCII 128-255) can be used in commands for annotation and BASIC will ignore them. SET commands won't ignore them. See all of the nonkeyboard characters along with their ASCII numbers with this miniprogram command:

 Echo For i=0to 127:j=4+i mod 20:k=9+8*(i\20):Locate j,k:u=i+128:?u;chr$(u):

 next:Locate 23,1|ok

4) SET definitions can be used in command suites for math functions or partial functions. DEF definitions can be used inside graphing programs but only for complete functions.

5) DATA statements inside graphing programs work as expected -- to transfer data to BASIC -- but not in graphing commands. DATA makes a fine introduction for comments to be placed anywhere inside a command, as opposed to REM which can be used only at the end of a command. For data transfer in a command, there is the Kronecker delta, (m\n)*(n\m), which is 1 for m=n and 0 otherwise where m and n are natural numbers. An example of its use for data transfer appears in the bugle command suite (Ex. 1) given above on page 10.

And strictly inside batch/BASIC programs:

6) Parameters %1, %2, etc. can be used inside BASIC code to carry command tails. COMMAND.COM replaces those definitions before BASIC ever sees the file, as with SET definitions. Very powerful.

Mail Matters

International mafia in science
Letter dated 16 Nov '03

Hi Homer,

On CNN 7 Aug '02 there was the following news item:

Debunking Einstein/ Aussie Researchers Prove Inconstant Speed of Light. That's it.

As I told you on the telephone, the theory of relativity became like a dogma in physics! Everything is being interpreted within its framework. Especially physicists in the USA are more conservative, while other physicists (in Europe or Asia) are more liberal.

If a phenomenon is observed at a greater speed than light, the researchers try to find a way of explaining it within the framework of the theory of relativity. Nobody is allowed to contradict it -- as if in a scientific totalitarianism. I feel that there is an international mafia; when a new theory emerges which doesn't comply with the theory of relativity, it is disregarded or blamed, or sometimes (in rarely happy cases) "unified" with the theory of relativity -- as for example, quantum theory. Relativity does not apply in quantum physics and, because everyone should accept the theory of relativity, the conservative scientists (Einstein's mafia) try to develop now a so-called string theory to "unify" the theory of relativity with quantum physics.

In a documentary on channel KNME in the last month called "Einstein's wife," it was said the papers he published on the theory of relativity were written together with his first wife; her name was on the manuscripts, but it disappeared in the published articles; nobody knows why, and the manuscripts were not found any longer.

...Florentin Smarandache/UNM/Gallup

PS: Did you receive a preprint of mine published in "Bulletin..." about quantum probability?

Dear Florentin:

PBSX on C-band satellite showed a one-hour presentation on Fri 6 Feb '04 called "Einstein's Wife." I watched it through and assume it is the same as your KNME reference. It might be considered alongside Denis Brian's 1996 book, Einstein: A Life, and I hereby put that task on my to-do list.

Your use of convervative (as opposed to liberal) I provisionally take to mean those who agree with Einstein's reality view.(?)

In view of Piattelli-Palmarini's recent book, Inevitable Illusions: How Mistakes of Reason Rule Our Minds, it seems we may have a tiger by the tail in the "no speed barrier in space" department. See the response starting on page 4, this issue, to Jim Malmberg's attack on our March paper, "Begin the Adventure."

Re your PS: yes, I believe I have it and may review and comment at some future time.

Your friend and colleague, ...HBT