By Y. Eliashberg

This publication provides the 1st steps of a thought of confoliations designed to hyperlink geometry and topology of 3-dimensional touch buildings with the geometry and topology of codimension-one foliations on 3-dimensional manifolds. constructing virtually independently, those theories at the beginning look belonged to 2 assorted worlds: the idea of foliations is a part of topology and dynamical platforms, whereas touch geometry is the odd-dimensional 'brother' of symplectic geometry. despite the fact that, either theories have built a couple of impressive similarities. Confoliations - which interpolate among touch constructions and codimension-one foliations - might be useful us to appreciate higher hyperlinks among the 2 theories. those hyperlinks supply instruments for transporting effects from one box to the other.It's gains contain: a unified method of the topology of codimension-one foliations and phone geometry; perception at the geometric nature of integrability; and, new effects, particularly at the perturbation of confoliations into touch buildings

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**Example text**

A; B; C /. A divides BC but is relatively prime to B. A; B; C /. A divides BC but does not divide C . B; A; D/. B A/ C A it must also be relatively prime to B A. B A/C D BC AC but not C . 1): If BC D AD and A is relatively prime to B, then B divides AD and is relatively prime to A. 2): If BC D AD but A does not divide C , then B divides AD but does not divide D. A1 ; B1 ; C1 /, that minimizes A C B C C . Then by P2 , A1 ¤ 1, so by P1 , A1 ¤ B1 . A1 ; B1 ; C1 /. The only remaining possibility is B1 > A1 .

A C b/2 D a2 C b 2 C 2ab, so given the latter, only the diagram on the left is needed to carry out the proof. 8 below, taken from Fig. 7 A simple dissection proof 6 For a proof, see Boltyanskii (1963). 1 Two dissection proofs 33 Fig. 8 Sine of the sum of two angles Nelsen 2000, p. 40, where the figure is credited to Volker Priebe and Edgar A. Ramos. ©The Mathematical Association of America 2013. 7 is used to illustrate proposition II,4 of Euclid’s Elements. So the question again arises, could not Euclid have proved proposition I,47 more simply by using those diagrams?

Q1 p1 /q2 qs used for that purpose in Zermelo’s proof. Euclid’s Lemma may also be proved by reductio. Indeed, Gauss did so (for the contrapositive statement) in his Disquisitiones Arithmeticae. His proof, presented next below, is actually a double reductio that invokes the division algorithm thrice. Eighth proof: Gauss first showed by reductio that no prime p can divide a product of two smaller positive integers. For suppose to the contrary that p is a prime that divides such a product, and let r < p be the least positive integer for which there exists a positive integer s < p such that p divides rs.