Statistical estimation of ␲ using random vectors S. C. Blocha) Physics Department, University of South Florida, Tampa, Florida 33620-5700 R. Dressler 2611 Bayshore Boulevard, Tampa, Florida 33629-7343 ͑Received 16 July 1998; accepted 13 August 1998͒ We present a simple application of a spreadsheet to estimate ␲ using random vector ensembles. The worksheet, combining elements of the works of , Liu Hui, and Buffon, can be used as a brief introductory tutorial on the Monte Carlo method, random number generators, and the logical IF function. Complete instructions are given for composing the worksheet. © 1999 American Association of Physics Teachers.

I. INTRODUCTION It is well known that ␲, and our other favorite transcen- dental number e, are associated with random processes ͑see The objective of this paper is to provide a brief, simple Appendix B͒. It has been said, but not proved, that there is spreadsheet tutorial on the Monte Carlo method and random no randomness, only uncertainty; ␲ appears throughout number generators, using random vector ensembles to esti- physics and we can never know ␲ exactly, so ␲ is a funda- mate ␲. At present there is intense worldwide activity in the mental source of uncertainty. In his Nobel Lecture,8 Max development of specialized random number generators for practical applications. These applications include Monte asked ‘‘Is there any sense—and I mean any physical Carlo methods in second-order phase transitions of magneti- sense, not metaphysical sense—in which one can speak of zation, solution of the transport equation in semi- absolute data? Is one justified in saying that the coordinate conductors, low-observability radar, computer testing, and xϭ␲ cm where ␲ϭ3.1415... is the familiar transcendental cryptography ͑see Sec. IV͒. This tutorial is related to one of number that determines the ratio of the circumference of a the oldest problems in geometric probability,1 ‘‘Buffon’s circle to its diameter? As a mathematical tool the concept of needle,’’ but the spreadsheet version is easier to understand a real number represented by a non-terminating decimal frac- and calculus is not required. tion is exceptionally important and fruitful. As the measure Georges Louis Leclerc Comte de Buffon reported his ex- of a physical quantity it is nonsense.’’ periment in 1777. Buffon probably raised some people’s eye- The Monte Carlo9–11 method was invented by Fermi, von brows during the experiment, because he stood on a tiled Neumann, and Ulam to calculate neutron transport in solids, floor and threw loaves of French bread over his shoul- but it has now become a general tool in diverse applications. ders. Counting the number of bread loaves that touched or A difficulty with the Monte Carlo method is that nonphysical crossed lines on the floor, and counting the total number of results may be obtained when a random number generator loaves, provided a means of estimating ␲. ͑A less expensive, ͑RNG͒ is not sufficiently random; it is more accurate to re- but less exciting, version of the experiment can be done with 12 parallel lines on a table, with one needle dropped repeatedly name RNG as PRNG ͑pseudo-random number generator ͒. to prevent interference among multiple needles. The problem Inadequacies of theoretical tests for PRNGs led to the use of is simpler if the line spacing equals the needle length. See physical tests with the Ising model ͑see Sec. IV͒. Compari- Sec. IV.͒ Buffon is credited with the introduction of calculus son of the exact Ising model ͑a result of Onsager’s heroic into probability theory; his memory is honored as the name analytical calculations͒ with Monte Carlo computations is of a street ͑Rue Buffon͒ in the Fifth Arrondissement in Paris2 used as a test of a PRNG. and as the name of a lunar crater ͑Crater Buffon͒. Inadequacies of computer-produced PRNGs led to the use The ubiquitous transcendental number ␲ has a fascinating, of physical devices to produce random sequences. For ex- almost mystical, history spanning several thousand years.3–7 ample, semiconductor diodes, thermionic vacuum diodes, During the last 100 years, and particularly in the last 2 years, and radioactive decay are used at relatively low frequencies, more was learned about ␲ than had been learned in the pre- and plasma tubes in waveguide are used at microwave fre- vious 5000 years. In ancient times, Archimedes and Liu Hui quencies. Silicon Graphics, Inc., recently applied for a patent estimated ␲ using circles inscribed in polygons, and circum- for the ‘‘lavarand,’’ a device using six Lava Lite® lamps and scribed outside of polygons, to obtain lower and upper limits digital video in conjunction with the Secure Hash Algorithm on its value. Supercomputers are now used in the calculation ͑SHS or SHA-1͒ of the National Institute of Standards and of ␲ to billions of decimal places ͑see Sec. II͒. Despite the Technology. The moving blobs in a Lava Lite, a sort of importance of ␲ in pure mathematics, numerical analysis, physics, and engineering, it may appear that billions of digits macroscopic Brownian motion, can be viewed on the Inter- are excessive because 39 decimal places are sufficient to net at http://lavarand.sgi.com. New blocks of random num- compute the circumference of the visible universe to a pre- bers are generated every minute at the web site, and are cision comparable to the size of a hydrogen atom. Neverthe- available for public use in octal and base-10 number sys- less, even multibillion decimal precision is not enough for tems. some current applications in which security is thought to be For more information on ‘‘Buffon’s needle’’ and ␲, achievable through exhaustion of an adversary’s computer access the Math Forum at Swarthmore College, http:// resources. forum.swarthmore.edu/dr.math/.

298 Am. J. Phys. 67 ͑4͒, April 1999 © 1999 American Association of Physics Teachers 298 Fig. 1. ‘‘Buffon’s needle’’ experiment, implemented as a Java applet ͑com- posed by Dr. M. J. Hurben͒.

II. PRESENT STATE OF ␲ tival in 1998. The movie received excellent reviews by film critics G. Siskel and R. Ebert on TV and the Internet Pascal observed that, although random events cannot be ͑http://www.tvplex.com/BuenaVista/SiskelAndEbert/͒,J. predicted, statistical patterns emerge. This is one of the dif- Beradinelli ͑http://www.movie-reviews.colossus.net/movies/ ficulties with PRNGs, in which distinctive distribution func- p/pi.html͒, and M. McDonagh ͑http://www.tvgen.com/ tions and hidden periodic structures are eventually discov- movies/mopic/pictures/40/40269.htm͒. The central character, ered. Nevertheless, it is often very difficult to distinguish Max, believes mathematics is the language of nature, every- 13,14 15 between random and deterministic processes; Stewart thing can be represented and understood through numbers, clarified several misconceptions about random numbers. and there are patterns in nature. This simple premise evolves So far, no pattern has been found in the decimal digits of into a complex multilevel convoluted plot that, in Beradinel- ␲, but ␲ cannot be considered purely random because we li’s opinion, ‘‘is eccentric even for a science fiction effort.’’ can compute the same digits time after time, even though we cannot compute a particular digit without computing all pre- vious digits. In July 1997, Y. Kanada16 and his colleagues III. OTHER RANDOM NUMBER GENERATORS 10 ͑University of Tokyo͒ calculated ␲ to 5.153 96ϫ10 deci- The search continues for useful PRNGs, but what is good mal digits, using 1024 processors. To check accuracy, the in one application may be not good in another. Collins calculation was performed by two methods. The slower et al.19 presented a nonlinear, aperiodic random generator method required 37 h of computer time; the faster method based on the logit transform of the logistic variable, studied required only 29 h. In the first 5ϫ1010 digits, Kanada found earlier by Ulam and von Neumann. In some applications, a that number 8 is the most frequent ͑5 000 117 637 occur- Gaussian or other distribution of random numbers may be rences͒ and number 3 is the least frequent ͑4 999 914 405 more realistic than a uniform distribution. Abramowitz and occurrences͒. Stegun20 provide several simple PRNGs, based on the nor- As an example of a modern use of ␲, we showed how mal distribution, the bivariate normal distribution, and the decimal digits of ␲ generate a spread-spectrum wavelet by exponential distribution; these can implemented in a spread- the direct-sequence method.17 The noise-like wavelet was sheet. encoded with the information AAPT, compressed, decom- The Internet is an excellent resource for random number pressed, and demodulated to recover AAPT. information, and a good place to start is http:// In the Euler–De Moivre theorem, ␲ appears as one of the random.mat.sbg.ac.at/ at the University of Salzburg. This ‘‘quarks of mathematics,’’ web site contains tests for random numbers, RNGs, the latest Ϫei␲ϭ1, Ϫei␲Ϫei␲ϭ2 ϱ. ͑1͒ news in the field, a virtual library of relevant literature ¯ concerning random numbers and Monte Carlo methods, In this enigma, a purely negative transcendental number and links to other web sites. An interesting URL, (Ϫe) and a purely imaginary transcendental number (i␲) http://www.ccsf.caltech.edu/ϳroy/pi.formulas.html, dis- combine to produce all of the real, positive integers, the cusses computation of ␲ to billions of decimal places by the things we can count on our fingers and toes. Feynman ex- arctangent method. pressed his admiration for this as ‘‘...this most amazing jew- ‘‘Buffon’s needle’’ is at http://www.angelfire.com/wa/ el...the most remarkable formula in mathematics.’’ Kasner hurben/buff.html in an interactive Java applet ͑see Fig. 1͒.A and Newman6 wrote ‘‘It appeals equally to the mystic, the two-dimensional Ising model of magnetization can be found scientist, the philosopher, the mathematician.’’ at http://planck.uni-muenster.de:8080/java/ising.html in an Benjamin Pierce, an eminent mathematician at Harvard interactive Java applet. University in the nineteenth century, after deriving the The Cornell Theory Center library includes a set of rou- Euler–De Moivre theorem in a lecture, said, ‘‘Gentlemen, tines for generating and using random numbers within that is surely true, it is absolutely paradoxical; we cannot parallel programs. The URL is http://www.tc.cornell.edu/ understand it, and we don’t know what it means, but we have UserDoc/Software/PTools/prng/. This web site gives the user proved it, and therefore, we know it must be the truth.’’ access to more than a thousand independent PRNGs. Later, Hilbert18 said ‘‘We must know and we shall know.’’ The EXCEL spreadsheet provides seven PRNG distribu- A science fiction movie named Pi, written and directed tions: uniform, normal, , binomial, Poisson, pat- by D. Aronofsky, was introduced at the Sundance Film Fes- terned, and discrete ͑consult spreadsheet Help for more in-

299 Am. J. Phys., Vol. 67, No. 4, April 1999 S. C. Bloch and R. Dressler 299 Fig. 2. Suggested format for work- sheet. The graph inset shows a typical ensemble. For graphical clarity, only the points corresponding to the ends of 103 vectors are shown. The estimate shown in cell I3 is based on 104 vec- tors. All vectors radiate from the origin ͑0,0͒. The cell editing box ͑below the icons͒ displays the formula in cell G4, indicated by the rectangle. The mouse pointer on the graph reads the compo- nents of a vector in the fourth quad- rant.

formation͒. In estimating ␲ we are only interested in the Ͼ7,1,0) will return the quantity 0 ͑used here as ‘‘false’’͒ uniform distribution. To access Random Number Generation because 6 is not greater than 7. For convenience in the work- in EXCEL, click on the Tools menu. ͑If you do not see Data sheet, we use cell addresses in the IF statement instead of Analysis in the Tools menu, then click on Add-Ins and check numbers like 6 and 7 ͑see the example in Table I͒. Cell the box adjacent to Analysis ToolPak. This is not a typo- addresses enable the spreadsheet to recalculate quickly when graphical error; ToolPak is the way it is spelled in EXCEL.͒ Function Key F9 is pressed to produce a new ensemble, and When you see Data Analysis appear on the Tools menu, the IF function provides efficient counting. Consult on- click it and choose Random Number Generation. Next, screen Help in your spreadsheet for additional information choose Uniform Distribution in the dialog box. ͑You may concerning IF. need to click on Update Add-Ins in the Tools menu.͒ Table I shows the details of the formulas used in compos- ing the worksheet in EXCEL format, and Fig. 2 illustrates a IV. WORKSHEET DESIGN suggested worksheet layout. The first two lines in Table I show the cell formulas used to generate the random compo- The primary spreadsheet functions used in the worksheet nents of a vector. Use the spreadsheet Copy operation to are RAND and IF. In our modification of ‘‘Buffon’s copy these cells in their respective columns to fill the desired needle,’’ we use an ensemble of random vectors, radiating number of vectors in the ensemble. For example, for 104 from the origin of coordinates, xϭ0, yϭ0. The random x,y vectors, copy cell A2 to all cells through A10001. Cell E2 components of each vector, generated by RAND, are in the computes the random vector magnitude associated with cells domain Ϯ1. Consequently, all vectors are contained in a A2 and C2. Cell G2 uses the expression ϭIF͑E2Ͻϭ1,1,0͒ to square of area 4 ͑see Fig. 2͒. Following the polygon method of Archimedes and Liu Hui, we can imagine a circle of ra- dius 1 and area ␲ inscribed in the square. Therefore, the ratio Table I. Worksheet details ͑Microsoft EXCEL format͒. of the circle’s area to the square’s area is ␲/4. If the vector ensemble is sufficiently random and large, we can obtain an Cell Formula Purpose estimate of the ratio /4 by counting the vectors that lie on ␲ A2 ϭRAND( )*2Ϫ1 Generate random x, Ϫ1рxрϩ1 and inside the circle, and counting the total number of vec- C2 ϭRAND( )*2Ϫ1 Generate random y, Ϫ1рyрϩ1 tors. In other words, the task of estimating ␲ is reduced to E2 ϭSQRT͑A2ˆ2ϩC2ˆ2͒ Compute random vector magnitude r counting. G2 ϭIF͑E2Ͻϭ1,1,0͒ Determine if r is not outside of unit 17,21 In this worksheet, the IF function is the key to count- circle ing the random vectors. The IF function is a logical function H2 ϭATAN2͑A2,C2͒ Compute random vector 4-quadrant analogous to the exclusive—or gate device used in digital polarangle͑rad͒. This column can be electronics; it returns a selected quantity if a condition is omitted true, and another selected quantity if that condition is false. G10 003 ϭSUM͑G2:G10001͒ Count number of points on, and inside For example, the expression ϭIF(6Ͻ7,1,0) will return the of, unit circle G10 004 ϭG10 003*4/10000 Estimate ␲ based on 10 000 random quantity 1 ͑used here as ‘‘true’’͒ because 6 is less than 7. In vectors words, you can think of the IF statement as ‘‘if 6 is less than G10 005 ϭPI͑͒ Display actual value of ␲ 7, then 1, otherwise 0.’’ Conversely, the expression ϭIF(6

300 Am. J. Phys., Vol. 67, No. 4, April 1999 S. C. Bloch and R. Dressler 300 Fig. 3. A gallery of random vector ensembles, showing the end points of the vectors. Each ensemble contains 103 vectors. Note the lack of spatial uniformity.

determine if the vector magnitude is not outside of the unit the two-dimensional random distribution changes in the circle or, in other words, not greater than one. graphical representation. Figure 3 shows four typical en- Column H contains the random vector phase angles; it can sembles. be omitted. We have not found a use for the phases, except As with ‘‘Buffon’s needle,’’ and the polygon methods of to produce histograms to examine the uniformity of the Archimedes and Liu Hui,22 the random vector method esti- phase distributions. Cell G10003 counts the number of vec- mates ␲ as a ratio of integers, and therefore it can never truly tor magnitudes on, and inside of, the unit circle. Cell G10004 compute the value of ␲, a transcendental number. A question estimates ␲ based on the ensemble. Cell G10005 displays the for the student: Should this worksheet be regarded as a actual value of ␲ to a selected number of decimal places. method of estimating ␲, or would it be appropriate to con- The spreadsheet has 65 536 cells in a column and en- sider this as a test of the RNG? A challenge for the student: sembles can be stored in multiple columns, so ensembles can For what values, if any, of real positive integer n is the ex- be large. Spreadsheet recalculation can be set to Manual or pression exp͓␲ͱn͔ an integer? ͑Try nϭ163 and nϭ652. Automatic using the Tools, Options, Calculation menu. The Perseverance is required to decide if these values of n recalculation time for an ensemble of 104 vectors is less than produce integers. The spreadsheet is not useful for this, 1 s, using EXCEL 97 in WINDOWS95, a 166-MHz Pentium pro- because the results exceed its 30-digit display ability.͒ cessor, and 64 Mbytes of RAM. Compare this with Ref. 11, Hint for the instructor: This simple expression is the subject which uses FORTRAN 77 and a 150-MHz Pentium processor of considerable scrutiny and discussion. For examples, to estimate transcendental number e by three Monte Carlo see http://www.ccsf.caltech.edu/ϳroy/epissqrtn.html and methods, with CPU times of 186, 1969, and 32 896 s. ftp://ftp.netcom.com/pub/so/somos/math/radix.html. If only Ramanujan were here!

V. RESULTS ACKNOWLEDGMENTS Press Function Key F9 to recalculate the spreadsheet. A We wish to thank Dr. Ismail Sakmar for calling our atten- new ensemble appears every time F9 is pressed. Observe tion to the works of Liu Hui, and Dr. Michael J. Hurben for how the estimate of ␲ changes with each new set, and how the use his Java applet of ‘‘Buffon’s needle.’’

301 Am. J. Phys., Vol. 67, No. 4, April 1999 S. C. Bloch and R. Dressler 301 Kasner and Newman6 wrote, ‘‘No matter how mathemat- ics is approached, ␲ forms an integral part. In his Budget of Paradoxes, Augustus De Morgan23 illustrated how little the usual definition of ␲ suggests its origin. He was explaining to an actuary what the chances were that, at the end of a given time a certain proportion of a group of people would be alive, and quoted the formula employed by actuaries which involves ␲. On explaining the geometric meaning of ␲, the actuary, who had been listening with interest, inter- rupted and exclaimed, ‘My dear friend, that must be a delu- sion. What can a circle have to do with the number of people alive at the end of a given time?’ ’’ Compare Wigner’s amusing tale.24 ‘‘There is a story about two friends, who were classmates in high school, talking Fig. 4. ‘‘Pi’’ on a sphere. Here, ‘‘pi’’ is the ratio of the circumference of a about their jobs. One of them became a statistician and was circle to its diameter, measured on the curved surface of the sphere. A great working on population trends. He showed a reprint to his circle, corresponding to polar angle 90 deg, has ‘‘pi’’ϭ2. As the polar former classmate. The reprint started, as usual, with the angle approaches zero, the limit of ‘‘pi’’ is ␲. Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His class- APPENDIX A mate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. ‘How can you know Students may ask, ‘‘Is ␲ a constant in relativity?’’ Pi is that?’ was his query. ‘And what is this symbol here?’ ‘Oh,’ constant if it is defined without reference to a circle. For said the statistician, ‘this is ␲.’ ‘What is that?’ ‘The ratio of examples, Leibniz showed that ␲ can be expressed as an the circumference of the circle to its diameter.’ ‘Well, now infinite series involving ratios of integers, you are pushing your joke too far,’ said the classmate, ‘surely the population has nothing to do with the circumfer- ␲ 1 1 1 1 1 ϭ Ϫ ϩ Ϫ ϩ Ϫ ͑A1͒ ence of the circle.’ ’’ 4 1 3 5 7 9 ¯ a͒Electronic mail: [email protected] and Wallis provided the product formula, 1L. Schroeder, ‘‘Buffon’s needle problem: An exciting application of many mathematical concepts,’’ Math. Teach. 67 ͑2͒, 183–186 ͑1974͒. ␲ 2 2 4 4 6 6 8 8 2 ϭ ϫ ϫ ϫ ϫ ϫ ϫ ϫ ϫ¯ . ͑A2͒ A cosmic convergence occurs about 300 m from the Arc de Triomphe, 2 1 3 3 5 5 7 7 9 where streets named for Newton, Galileo, and Euler come together. One of Equations A1 and A2 , and many other curious represen- the mysteries of Paris is that there is no Rue Fourier. ͑ ͒ ͑ ͒ 3P. Beckmann, A History of ␲ ͑Golem, Boulder, CO, 1982͒. tations, are given in Ref. 6. It would be naive to assume that 4D. Blatner, The Joy of Pi ͑Walker, New York, 1997͒. we now know all representations of ␲. 5J. M. Borwein and P. B. Borwein, ‘‘Ramanujan and Pi,’’ in The World In general relativity, space–time is a non-Euclidean geom- Treasury of Physics, Astronomy, and Mathematics, edited by T. Ferris etry. The ratio of the circumference to the diameter of a ͑Little, , Boston, 1991͒, pp. 647–659. 6 circle in non-Euclidean geometry can be more or less than ␲. E. Kasner and J. Newman, Mathematics and the Imagination ͑Simon and As a familiar example, consider the curved surface of a Schuster, New York, 1943͒, Chap. 3, pp. 65–111. 7P. Dubreil, ‘‘The history of the mysterious numbers,’’ in Great Currents sphere. For every great circle on a sphere, the ratio of cir- of Mathematical Thought, edited by F. LeLionnais ͑Dover, New York, cumference to diameter has the value 2. For infinitesimal 1971͒, Vol. 1, pp. 92–114. This is a translation of Les Grandes Courants circles on a sphere, the ratio approaches the Euclidean value de la Pense´e Mathe´matique ͑Librairie Scientifique et Technique, Paris, ␲ ͑see Fig. 4͒. For the types of non-Euclidean geometry 1962͒. commonly used in physics, the ratio is very nearly ␲ over 8M. Born, ‘‘The statistical interpretation of quantum mechanics,’’ Nobel small distances so it may be difficult to detect the difference Lecture, 11 December 1954, in Nobel Lectures, Physics, 1942–1962 ͑Elsevier, Amsterdam, 1964͒, Vol. 3. in ordinary measurements far away from holes. The 9T. W. Ko¨rner, Fourier Analysis ͑Cambridge U.P., Cambridge, 1988͒, Laser Geodynamics Satellites ͑LAGEOS and LAGEOS II͒, Chap. 13. and the Gravity Probe-B experiment ͑to be launched in year 10R. H. Landau, and M. J. Pa´ez, Computational Physics, Problem Solving 2000͒, may have sufficient sensitivity for this, because they with Computers ͑Wiley, New York, 1997͒, Chaps. 6 and 7. may be able to detect the Lense–Thirring effect. If we define 11P. Mohazzabi, ‘‘Monte Carlo estimations of e,’’ Am. J. Phys. 66 ͑2͒, ␲ by Eq. ͑A1͒ or Eq. ͑A2͒, or in Euclidean geometry, not 138–140 ͑1998͒. A search of the AJP database produced 46 references to physical geometry, then is invariant but the ratio of cir- the Monte Carlo method in AJP and The Physics Teacher. The database is ␲ located at hhtp://www.amherst.edu/ϳajp/index.html. cumference to diameter in physical space may not be equal 12PRNG is also used to designate a parallel random number generator. See to that constant value. http://www.tc.cornell.edu/UserDoc/Software/PTools/prng/. 13Aristotle may not have been the first to identify this problem, but he illustrated it vividly. In the Ross translation of Nichomachean Ethics, APPENDIX B Book Two, Aristotle wrote ‘‘Even matters of chance seem most marvelous if there is an appearance of design as it were in them; as for instance the Events of low probability are rare but observable. For ex- statue of Mitys at Argos killed the author of Mitys’ death by falling down ample, the town of Wethersfield, in the state of Connecticut, on him when a looker-on at a public spectacle; for incidents like that we think to be not without a meaning.’’ Much later, Tom Robbins ͓Jitterbug has received more than its fair share of meteorites in recent Perfume ͑Bantam, New York, 1984͔͒ observed that ‘‘The division be- years. Also, consider the following examples involving ␲ tween what is random in Nature and what is purposeful is extremely dif- explicitly, and e implicitly, in the Gaussian distribution. ficult to determine.’’ In a 50-year study of the stock market ͑Ref. 9, p.

302 Am. J. Phys., Vol. 67, No. 4, April 1999 S. C. Bloch and R. Dressler 302 584͒, the eminent statistician M. G. Kendall concluded that ‘‘... what looks 19J. J. Collins, M. Fanciulli, R. G. Hohlfield, D. C. Finch, G. v. H. Sandri, like a purposive movement over a long period is merely a kind of eco- and E. S. Shtatland, ‘‘A random number generator based on the logit nomic Brownian motion.’’ James Burke wrote that the essayist M. E. de transform of the logistic variable,’’ Comput. Phys. 6 ͑6͒, 630–632 ͑1992͒. Montaigne transcribed ‘‘The only thing that is certain is that nothing is 20M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, certain’’ on the ceiling of his study. Applied Mathematics Series Vol. 55 ͑USGPO, Washington, DC, 1964͒, 14A. Compagner, ‘‘Definitions of randomness,’’ Am. J. Phys. 59 ͑8͒, 700– pp. 952–953. Use the Ninth Printing, November 1970 or later, to avoid 705 ͑1991͒. In Fig. 1, p. 702, note the ͑seemingly͒ nonrandom features in earlier errors, one of which appeared on p. 953. the first 103 decimal digits of ␲, including the famous desert of fours 21S. C. Bloch, Introduction to Classical and Quantum Harmonic Oscillators where number 4 is absent in a sequence of 77 consecutive digits. ͑Wiley, New York, 1997͒, pp. 180–181, 299. 15I. Stewart, ‘‘Repealing the law of averages,’’ Sci. Am. ͑Int. Ed.͒ 278 ͑4͒, 22P. D. Straffin, Jr., ‘‘Liu Hui and the first golden age of Chinese mathemat- 102–104 ͑1998͒. ics,’’ Math. Mag. 71 ͑3͒, 163–181 ͑1998͒; A. Volkov, ‘‘Calculation of ␲ 16I. Peterson, ‘‘An enormous chunk of pi,’’ Sci. News ͑Washington, D.C.͒ in ancient China, from Liu Hui to Zu Chongzhi,’’ Historia Scientiarum 4, 152 ͑6͒,92͑1997͒. 139–157 ͑1994͒. 17S. C. Bloch, ‘‘Compression of wavelets,’’ Am. J. Phys. 61 ͑9͒, 789–798 23A. De Morgan, Budget of Paradoxes ͑Open Court, Chicago, 1940͒. ͑1993͒. See Figs. 10 and 11, p. 797. Quoted in Ref. 6. 18J. Dieudonne`, ‘‘David Hilbert ͑1862–1943͒,’’ in Ref. 7, p. 309. Hilbert’s 24E. P. Wigner, ‘‘The unreasonable effectiveness of mathematics in the optimistic statement is from one of his last lectures. natural sciences,’’ in Ref. 5, pp. 526–540.

CONDENSERS AND VACUUM TUBES I think we have to envisage two different purposes in the teaching of physics. If the student is planning to be what I may call a technician—perhaps the word is too specialized for the purpose I have in mind, but it will serve in lieu of something better—if he is planning to be a technician, then I think it is safe to say that the facts assume for him a primary importance, and he needs the ideas only to the extent necessary for the manipulation of the facts. It will be very useful for him to know of all of the different kinds of radio tubes which exist and of their characteristics. It will be useful for him to know what condensers leak and what condensers are well insulated. But if the student is studying physics as a mental discipline, then I will venture the heresy that ideas are all important and that facts are necessary only to the extent sufficient to provide material for the manipulation of the ideas.

W. F. G. Swann, ‘‘The Teaching of Physics,’’ Am. J. Phys. 19͑3͒, 182–187 ͑1951͒.

303 Am. J. Phys., Vol. 67, No. 4, April 1999 S. C. Bloch and R. Dressler 303