By Martin Aigner, Günter M. Ziegler, Karl H. Hofmann

This revised and enlarged 5th version positive aspects 4 new chapters, which include hugely unique and pleasant proofs for classics reminiscent of the spectral theorem from linear algebra, a few newer jewels just like the non-existence of the Borromean jewelry and different surprises.

**From the Reviews**

"... inside of PFTB (Proofs from The booklet) is certainly a glimpse of mathematical heaven, the place shrewdpermanent insights and gorgeous rules mix in extraordinary and wonderful methods. there's tremendous wealth inside of its pages, one gem after one other. ... Aigner and Ziegler... write: "... all we provide is the examples that we've got chosen, hoping that our readers will percentage our enthusiasm approximately impressive rules, smart insights and lovely observations." I do. ... "

*Notices of the AMS, August 1999*

"... This publication is a excitement to carry and to examine: considerable margins, great pictures, instructive photos and gorgeous drawings ... it's a excitement to learn besides: the fashion is obvious and unique, the extent is as regards to effortless, the mandatory heritage is given individually and the proofs are awesome. ..."

*LMS e-newsletter, January 1999*

"Martin Aigner and Günter Ziegler succeeded admirably in placing jointly a large number of theorems and their proofs that may definitely be within the ebook of Erdös. The theorems are so basic, their proofs so dependent and the rest open questio

ns so fascinating that each mathematician, despite speciality, can reap the benefits of analyzing this e-book. ... "

*SIGACT information, December 2011.*

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This revised and enlarged 5th version beneficial properties 4 new chapters, which include hugely unique and pleasant proofs for classics resembling the spectral theorem from linear algebra, a few newer jewels just like the non-existence of the Borromean earrings and different surprises. From the Reviews". .. inside of PFTB (Proofs from The ebook) is certainly a glimpse of mathematical heaven, the place shrewdpermanent insights and gorgeous principles mix in impressive and excellent methods.

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**Extra info for Proofs from THE BOOK**

**Sample text**

VAN DER PORTEN: A proof that Euler missed . Apery's proof of the irrationality of ((3). An informal report, Math. Intelligencer 1 (1979), 195203. [9] T. J. RANSFORD: An elementary proof of E;'" ~ Summer 1982, 3-5. = ~2, Eureka No. 42, [10] T. RIVOAL: La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs, Comptes Rendus de l' Academie des Sciences (Paris), Ser. I Mathematique, 331 (2000),267-270. " Hilbert's third problem: decomposing polyhedra Chapter 7 In his legendary address to the International Congress of Mathematicians at Paris in 1900 David Hilbert asked - as the third of his twenty-three problems - to specify "two tetrahedra oJ equal ba es alld equal altitudes which call ill 110 IVa)' be plit illfo congruent tetrahedra, alld which canl10t be combined II ith COli ruen!

N=3 3 lope n=6 6 lope 71=4 4 lope n=7 6 I pe r n=3 3 lope n= 4 lope n= lpe n=6 6 lope n=7 6 I pe After some attempts at finding configurations with fewer slopes you might conjecture - as Scott did in 1970 - the following theorem. The rem. lfn 2: 3 point ill the plalle do 1I0t lie on aile illgle line. tlleT! they derermille at lea r n - 1 differelll lope. where equality i po ible 0111 , iJn i odd alld n 2: 5. Our examples above - the drawings represent the first few configurations in two infinite sequences of examples - show that the theorem as stated is best possible: for any odd n 2: 5 there is a configuration with n points that determines exactly n - 1 different slopes, and for any other n 2: 3 we have a configuration with exactly n slopes.

652341 /: 625314 (4) A touching move is a move that reverses some string that is adjacent to the central barrier, but does not cross it. For example, ~ 1 A touching move 4 1 is a touching move. Geometrically, a touching move corresponds to the slope of a line of the configuration that has exactly m points on one side, and hence at most m - 2 points on the other side. Moves that are neither touching nor crossing will be called ordinary moves. For this 7fl = 213:546 --+ 213:564 = 7f2 ° is an example.