 Saturday, January 3, 2004

MIND GAMES
A question of cricket

It doesn't take long for anyone to check how many coins of equal size can be placed around a central coin so that these touch the central coin and yet not overlap.

"Only six. To calculate how many coins would touch the one in the centre, think of circles of radius r. Think of another imaginary circle drawn by joining the diameters of all the circles around the central one. The radius of this imaginary circle is 2r, since the radius of each circle is r and the radius of the imaginary circle is got when we join the centres of two circles. Its circumference is 2.piX2r=4.pi.r. To know how many circles are around, we divide this circumference by the dia of each circle (i.e. 2r), since the imaginary circle is created by joining the dias of all circles. (4 X 22/7 X r) / 2r = 44/7 = 6.2857142`85 or 6, if we take the integer part only," says Vimal Jit Kaur.

 Halley believed that elliptical orbits existed. Using this theory, he calculated that the comet of 1682 (now called Halley's Comet) was periodic and was the same object as the comet of 1305, 1380, 1456, 1531 and 1607. In 1705, he predicted that it would return in 76 years and appear in December, 1758. It did`85 on December 25, 1758.  From Halley's biography

"However, if the spheres are placed in three dimensions, three spheres on each side can touch the central sphere, totalling the number of spheres to 12 (6+3+3), so that all these are touching the one in the centre."

Viney Yadav and Dr Tarsem Lal agree with her views, but in Suhail's view, the number of spheres which can be placed around a sphere is 14. Each sphere subtends an angle of pi(2 - 3^(0.5)) steradians, and since the total solid angle about a point is equal to 4pi steradians,14 spheres can be placed around one sphere.

After Halley had thrown them out of his house, Newton and Gregory resumed this debate and soon agreed that "six" was indeed the answer for coins. However, they still disagreed on the second point. Gregory stated that in three-dimensional space, the first layer surrounding a central ball would contain 13 spheres, while Newton was more in favour of 12: "Put one sphere on the bottom, then arrange five spheres in a pentagon around the central sphere, just below its equator; place another five spheres more or less in the interstices of the lower five spheres (slightly above the equator of the central sphere), and finally put the 12th sphere on the top." Only Imperial College, UK, has the Newton papers and even it is silent on the question of cricket. Not until the 1950s it could be proved that theoretically, there was space for nearly 15 spheres around the central ball. Practically though, only 12 have been achieved. Think. (Write at The Tribune or adityarishi99@yahoo.co.in)

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