Q.E.D.

Which famous proof did Archimedes inscribe on his tombstone How and why do knots make perfect pentagons Have you ever seen a proof so completely that it is just obvious In this delicious little book top mathemagician Dr Polster presents many of the most visually intuitive and exciting proofs from the dusty annuals of mathematical history. Test your ability to follow the logic, leap into mathemagnosis and experience Eurekamoment after Eurekamoment.

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• 3 First published 2006 AD This edition Wooden Books Ltd 2006 AD Published by Wooden Books Ltd. 8A M
• 4 With love to Anu who understands it all ... I am indebted to the many mathematicians of the past a
• 1 IntroductIon There are some mathematical objects whose beauty everyone is able to appreciate. The
• 2 3 treacherous truth what proofs are all about In mathematics, as in the physical sciences, we may
• 4 5 Pythagoras theorem a proof by dissection The Theorem of Pythagoras ca. 569 475 B.C.E. states th
• 6 7 Plane and sImPle your basic theorem toolbox The Elements of Euclid ca. 325 265 B.C.E. long ago
• 8 9 From PIe to PI mysteries of the circle Eratosthenes 276 194 B.C.E. is famous for his pisza pie
• 10 11 cavalIerIs PrIncIPle a proof by approximation in slices There are two versions of the celebrat
• 12 13 cavalIer cone carvIng serious dissection in action Cones come in all sorts of shapes and sises
• 14 15 a FrustratIng Frustum horses and moat walls Many ancient manuscripts contain algorithms for ca
• 16 17 archImedes theorem mysteries of the sphere Archimedes proved that the volume of a sphere is tw
• 18 19 InsIde out two proofs in wedges Archimedes demonstrated how to mathematically kill two birds w
• 20 21 mathematIcal domInos proofs by induction Setting up a number of dominos in a row, one domino f
• 22 23 the InFInIte staIrcase a proof by regrouping A classical paradox involves a number of identica
• 24 25 cIrclIng the cycloId a proof by dissection Start with a regular polygon under a line, mark one
• 26 27 slIcIng cones Dandelins sphere trick What kind of curve do you get when you slice a circular c
• 28 29 FoldIng conIcs burning mirrors whispering walls Mark a dot on a circular piece of paper, fold
• 30 31 KnottIng Polygons a proof by paper folding It is very easy to construct equilateral triangles,
• 32 33 cuttIng squares a fresh look at an old pattern Beautiful theorems are often arrived at by comi
• 34 35 Power sums proofs by double counting The marvellous Pythagorean oneglance dissection proof bel
• 36 37 neverendIng PrImes a proof by contradiction Just as every object of the real world can be spli
• 38 39 the nature oF numbers another proof by contradiction On the number line below, every point rep
• 40 41 the golden ratIo natures favourite number What does a rectangle look like that is not too slim
• 42 43 the numbers oF nature the geometry of growth A spiral of squares growing around a unit square
• 44 45 eulers Formula a proof by pruning A cut diamond is a solid without indentations all of whose f
• 46 47 PossIble ImPossIbIlItIes doubling, squaring and trisecting Socrates 469 399 B.C.E. once used
• 48 49 Ptolemys theorem says that for a quadrilateral inscribed in a circle a a b b d d. This redu
• 50 51 Theorem 2 The nth fraction is f n1fn where f n is the nth Fibonacci number. Proof The proof is
• 52 53 Some of the proofs in this collection, especially the dissection ones and those involving infi
• 54 55 Deep results are not usually discovered ataglance but rather as a result of a process of ever
• 56 57 This book started with the regular two and threedimensional polytopes, the regular polygons an
• 58 Visualisation Take one of the Platonic solids, choose a point outside but very close to the cent