, and by NASA Research Grant No The authors are thankful to Maria Beltran for valuable comments.

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1 WAS STONEHENGE AN OBSERVATORY? by Gilbert Castillo and Vladik Kreinovich 1 Department of Computer Science University of Texas at El Paso, El Paso, TX s {castillo,vladik}@cs.utep.edu Abstract. In 1965, researchers found that the direction formed by two stones from Stonehenge, a mysterious ancient structure, pointed exactly to the direction of sunrise at solstice. Several similar coincidences were discovered, which lead many people to believe that Stonehenge was an ancient observatory. In this paper, we analyze this problem from a geometric viewpoint and show that similar coincidences can happen (with a probability) for randomly placed stones. Thus, Stonehenge s coincidences cannot be viewed as proof that this place was an observatory. 1. HAWKINS: IT WAS In 1965, G. S. Hawkins and J. B. White published a book (Hawkins et al 1965) in which they argued that Stonehenge, a mysterious ancient three-circle stone structure in England, was actually an ancient observatory. For example, they discovered that a direction formed by two of 60 stones of the outer circle points (with a high accuracy, less than 1 ) was the direction in which the sun rises during the solar solstice (actually, to the direction in which sun rose 4, 000 years ago, when Stonehenge was built). If we simply place two stones at random, then the probability that these two stones point in this direction is minuscule. To convince those who were not convinced by this coincidence, the authors used a computer to check if any other directions between the stones could be interpreted in reasonable astronomical terms, and they indeed found many such directions. 1 This work was partially supported by NSF grant No. EEC , and by NASA Research Grant No The authors are thankful to Maria Beltran for valuable comments.

2 This book seemed to be very convincing and many encyclopedias still claim that Stonehenge was an observatory. 2. ARCHAEOLOGISTS AND STATISTICIANS DOUBT Archaeologists doubt. Many archaeologists were not convinced by Hawkins and White because accurate astronomical observations were not very consistent with the primitive character of their everyday tools such as knives, etc. Archaeologists doubts do not sound very convincing to astronomers. This inconsistency may have been very convincing for archaeologists but not for astronomers. To astronomers, the contradiction between a seemingly valid statistical conclusion of Hawkins et al. and vague non-statistical ideas of archaeologists were rather an indication that archaeologists must change their ideas. Statisticians doubt. From the astronomers viewpoint, a more serious objection to Hawkins et al. was raised by statisticians who re-calculated the probabilities of coincidences to occur for randomly placed stones using more accurate statistical techniques. Statisticians got new values for probabilities of random coincidence; these values were still small but much less convincing than the ones given by Hawkins et al (for a detailed statistical analysis and detailed bibliography, see (Heggie 1981)). Statisticians doubts are not very convincing to archaeologists. We have already mentioned that astronomers did not buy into archaeological objections. Well, archaeologists did not buy into statisticians objections either. The main reason was that both the original idea of Hawkins et al. and the archeological objections were easily explainable to a layperson, while the statisticians new estimates were quite sophisticated and non-intuitive at all. So, maybe, Stonehenge was not an observatory. These doubts did not disprove the idea that Stonehenge (or other ancient stone sites that were similarly analyzed) was actually an observatory: they just made this conclusion less convincing.

3 3. IT IS TIME FOR A GEOMBINATORIC ANALYSIS This is a geombinatoric problem, after all. We have mentioned two groups of people who doubted Hawkins conclusions: archaeologists who doubted these conclusions on archaeological grounds and statisticians who doubted these conclusions on statistical grounds. Surprisingly, what is missing is geometrical combinatoric (geombinatoric) analysis, because, after all, we are talking about the geometric properties of the points located on a circle. What we are planning to do. In this paper, we will show that such an analysis can lead to easy-to-understand probability estimates and, most probably, to a conclusion that there is no proof at all that Stonehenge was an observatory. Our main (heuristic) idea of geometric probability estimates. From the geometric viewpoint, N stones are simply N points on a plane. From N points, we can form N(N 1)/2 different pairs, and each of these pairs determines a straight line. For the generic orientation of the stones, we thus have N(N 1)/2 different straight lines on the plane. Each line defines two opposite directions. Each direction can be characterized by the angle between this direction and the West-East line. One of the two opposite directions has an angle from the interval [0, 180 ], and the other one from the interval [180, 360 ]. So, the directions defined by N stones form N(N 1)/2 angles on the interval [0, 180 ]. In particular, for N = 60, we have 60 59/2 1, 800 different directions. If these directions were equally spaced on these intervals, then the space between the two consequent directions would be equal to 60 /1800 = 2. Thus, any angle d (in particular, the angle corresponding to the solstice) is in the 2 angular minute wide interval between two stone-based directions. The stone-based direction that is the closest to the given one is at most 1 away. Thus, in this equally spaced case, we have a stone-based direction within one angular minute from any given direction! We will show that a similar conclusion can be made for randomly placed stones.

4 If stones are located at random, then, for a rough estimate, it is natural to assume that angles corresponding to these N(N 1)/2 directions can be represented by independent random variables uniformly distributed on the interval [0, 180 ]. Let d be the angle that we want to approximate by stone-based directions; let us compute, for a given accuracy δ > 0, the probability P (δ) that one of N(N 1)/2 directions is δ close to d, i.e., belongs to the interval [d δ, d + δ]. Since the direction is uniformly distributed on the interval [0, 180] of width 180, the probability P 1 of this direction to be within the smaller interval [d δ, d + δ] of width 2δ is equal to the ratio of the lengths of these intervals: P 1 = 2δ/180 = δ/90. Hence, the probability Q 1 that this direction is not δ close to d is equal to Q 1 = 1 P 1 = 1 δ/90. Since the directions are assumed to be independent, the probability Q(δ) that neither of N(N 1)/2 directions is δ close to d is equal to the product of the N(N 1)/2 probabilities that the first, second, etc., direction are not δ close, i.e.: Q(δ) = Q N(N 1)/2 1 = (1 δ/90) N(N 1)/2. The desired probability P (δ) that one of the directions is δ close to d is thus equal to P (δ) = 1 Q(δ) = 1 (1 δ/90) N(N 1)/2. This probability is 90%(0.9) iff (1 δ/90) N(N 1)/ Let us take natural logarithms of both sides of this inequality. We get N(N 1)/2 ln(1 δ/90) 2.3; for small δ, ln(1 δ/90) δ/90, so this inequality is equivalent to N(N 1) δ ,

5 i.e., to the inequality δ = = Hence, with probability 90%, for any direction d, be it a direction to the solstice or to the White House lawn, there is a stone-based direction that is 7 angular minutes close. Since this is true for any direction, Stonehenge s coincidences in no way prove that Stonehenge was an observatory. 4. CONCLUSIONS AND AN OPEN PROBLEM Our main conclusion is that geombinatoric analysis seems to indicate that Stonehenge was not an observatory. This conclusion raises a natural open problem: to make accurate estimates of the probability that Stonehenge s coincidences are accidental, and thus, to finally solve the question of whether Stonehenge was an observatory. REFERENCES Hawkins, G. S., and White, J. B. Stonehenge decoded, Doubleday, N.Y., Heggie, D. C. Megalithic science, Thames and Hudson, N.Y., 1981.