1/23/2024 0 Comments Motion in a 4d sphere![]() In 2003, Oleg Musin proved the kissing number for n = 4 to be 24. As in the three-dimensional case, there is a lot of space left over - even more, in fact, than for n = 3 - so the situation was even less clear. It is straightforward to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled 24-cell centered at the origin). ![]() ![]() In four dimensions, it was known for some time that the answer was either 24 or 25. A coordination number of 12 is found in a cubic close-packed or a hexagonal close-packed structure. The twelve neighbors of the central sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size (as in a chemical element). Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by Reinhold Hoppe, but the first correct proof (according to Brass, Moser, and Pach) did not appear until 1953. Newton correctly thought that the limit was 12 Gregory thought that a 13th could fit. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. This leaves slightly more than 0.1 of the radius between two nearby spheres. A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a regular icosahedron. Therefore, the circles 1 and 2 intersect – a contradiction. Therefore, the triangle C C 1 C 2 is isosceles, and its third side – C 1 C 2 – has a side length of less than 2 r. The segments C C i have the same length – 2 r – for all i. Then at least two adjacent rays, say C C 1 and C C 2, are separated by an angle of less than 60°. These rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°.Īssume by contradiction that there are more than six touching circles. Proof: Consider a circle with center C that is touched by circles with centers C 1, C 2. In two dimensions, the kissing number is 6: In one dimension, the kissing number is 2: Known greatest kissing numbers One dimension For others investigations have determined upper and lower bounds, but not exact solutions. Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century. ![]() Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.įinding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in ( n + 1)-dimensional Euclidean space. Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere.
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