Statistical estimation of p 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 1998! We present a simple application of a spreadsheet to estimate p using random vector ensembles. The worksheet, combining elements of the works of Archimedes, 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 p, 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 B!. It has been said, but not proved, that there is spreadsheet tutorial on the Monte Carlo method and random no randomness, only uncertainty; p appears throughout number generators, using random vector ensembles to esti- physics and we can never know p exactly, so p is a funda- mate p. 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 Born 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 Boltzmann transport equation in semi- absolute data? Is one justified in saying that the coordinate conductors, low-observability radar, computer testing, and x5p cm where p53.1415... is the familiar transcendental cryptography ~see Sec. IV!. 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 back 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 p. ~A less expensive, ~RNG! 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. IV!. 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 calculations! with Monte Carlo computations is of a street ~Rue Buffon! in the Fifth Arrondissement in Paris2 used as a test of a PRNG. and as the name of a lunar crater ~Crater Buffon!. Inadequacies of computer-produced PRNGs led to the use The ubiquitous transcendental number p 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 p 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 p 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-1! of the National Institute of Standards and of p to billions of decimal places ~see Sec. II!. Despite the Technology. The moving blobs in a Lava Lite, a sort of importance of p 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 p, 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 ~4!, 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. Hurben!. II. PRESENT STATE OF p 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.html!, and M. McDonagh ~http://www.tvgen.com/ tions and hidden periodic structures are eventually discov- movies/mopic/pictures/40/40269.htm!. 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- p, but p 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 Tokyo! calculated p to 5.153 96310 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 531010 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 rences! and number 3 is the least frequent ~4 999 914 405 more realistic than a uniform distribution. Abramowitz and occurrences!. Stegun20 provide several simple PRNGs, based on the nor- As an example of a modern use of p, we showed how mal distribution, the bivariate normal distribution, and the decimal digits of p 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, p 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 2eip51, 2eip2eip52 `. ~1! 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, (2e) and a purely imaginary transcendental number (ip) http://www.ccsf.caltech.edu/;roy/pi.formulas.html, dis- combine to produce all of the real, positive integers, the cusses computation of p 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.