verify the Euler's formula for zircon crystal with full explnation
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Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways. Examples of this include the existence of infinitely many prime numbers, the evaluation of zeta, the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a formula for testing whether an arithmetic progression contains a square) and the Pythagorean theorem (which according to Wells has at least 367 proofs). This also sometimes happens for unimportant theorems, such as the fact that in any rectangle dissected into smaller rectangles, if each smaller rectangle has integer width or height, so does the large one.
This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V−E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2.
A version of the formula dates over 100 years earlier than Euler, to Descartes in 1630. Descartes gives a discrete form of the Gauss-Bonnet theorem, stating that the sum of the face angles of a polyhedron is 2π(V−2), from which he infers that the number of plane angles is 2F+2V-4. The number of plane angles is always twice the number of edges, so this is equivalent to Euler's formula, but later authors such as Lakatos, Malkevitch, and Polya disagree, feeling that the distinction between face angles and edges is too large for this to be viewed as the same formula. The formula V−E+F=2 was (re)discovered by Euler; he wrote about it twice in 1750, and in 1752 published the result, with a faulty proof by induction for triangulated polyhedra based on removing a vertex and retriangulating the hole formed by its removal. The retriangulation step does not necessarily preserve the convexity or planarity of the resulting shape, so the induction does not go through. Another early attempt at a proof, by Meister in 1784, is essentially the triangle removal proof given here, but without justifying the existence of a triangle to remove. In 1794, Legendre provided a complete proof, using spherical angles. Cauchy got into the act in 1811, citing Legendre and adding incomplete proofs based on triangle removal, ear decomposition, and tetrahedron removal from a tetrahedralization of a partition of the polyhedron into smaller polyhedra. Hilton and Pederson provide more references as well as entertaining speculation on Euler's discovery of the formula. Confusingly, other equations such as ei pi = -1 and aphi(n) = 1 (mod n) also go by the name of "Euler's formula"; Euler was a busy man.
The polyhedron formula, of course, can be generalized in many important ways, some using methods described below. One important generalization is to planar graphs. To form a planar graph from a polyhedron, place a light source near one face of the polyhedron, and a plane on the other side.
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