We saw last week some of the properties of the “boring” number 42, and there is more. As Salva Fuster points out: “The number 42 is not only the second sphenic number, but the sum of the proper divisors of the first sphenic number (30) is precisely 42. It is also an oblong number, that is, the product of two consecutive natural numbers. (42 = 6 x 7)”.
Let us remember that the proper divisors of a number are all except the number itself, pardon the pun; in the case of 30: 1, 2, 3, 5, 6, 10 and 15. But the properties of this unfairly underestimated number do not end here:
42 is the sixth number of Catalan (no accent: he was Belgian). Let us remember that the Catalan numbers, which we have already dealt with on some occasion, represent, among other things, the different ways of dividing a polygon into triangles by means of diagonals that do not intersect or of inserting parentheses in a product of several factors. The first ten Catalan numbers are: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862 and 16796.
42 is the magic constant of a 3×3×3 magic cube, which contains in its cubicles the numbers from 1 to 27, so that the sum along any row, column or diagonal that passes through the center is the itself, called the magic constant (can you prove that this magic constant is 42?).
42 is the smallest dimension for which the “sausage conjecture” has been shown to be correct, which states that for five dimensions or more, the packing of spheres whose convex hull has minimum volume is always a row arrangement.
From the sieve (by Apollonius) to the tapestry (by Sierpinski)
But to talk about the Sausage Conjecture and other interesting aspects of sphere packing, a topic that we already covered, albeit far above, last year, it is convenient to go down one dimension and start talking about circle packing. In this case, the row (sausage) arrangement does not give rise to the minimum surface envelope, as is easy to see in the case of seven equal circles, which, as Gauss showed, can be packed as densely as possible with one of them surrounded by the other six (what saving of surrounding surface supposes the hexagonal arrangement with respect to the sausage?).
But the packed circles do not have to be the same, and as early as the 3rd century BC. C. the Greek mathematician Apollonius of Perga made important contributions to the study of circles of different sizes tangent to each other.
Apollonius discovered that, given any three circles tangent each to the other two, there are two other circles tangent to those three. If we repeat the process with the new triads of circumference to which the incorporation of these two gives rise and repeat the process indefinitely, we obtain a fractal called, in honor of the “great geometer” (as he was known in his time), the sieve of Apollonius, which was studied by Leibniz (which is why it is also known as “Leibniz packing”) and which is a clear precedent of the Sierpinski triangle and tapestry.
The Apollonian sieve can be constructed from different triples of generatrices. If the three initial circles have the same radius, as in the figure, we obtain the so-called symmetric Apollonian sieve. If one of the three generating circles has an infinite radius, that is, it is a line tangent to two circles tangent to each other, we obtain a family of Ford circles. But that is another article.
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