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Archimedes and the Emergence of Pi Evonne Pankowski Edward Knote Archimedes of Syracuse (287 – 212 B.C.) o 2-3 decades separated Euclid & Archimedes o He pushed well beyond Euclid’s findings o Historical info is known, but validity is questioned 2000 years had past before the math world would see anyone like Archimedes… WHY?! o •Greece was taken over by the Romans •They were devoted to different values •Church played a role in limiting the focus on education •Mathematical research was not prioritized •It is crucial to maintain and develop the value systems in our schools and education! o His mathematic works & commentary have survived o They give us a picture of this revered, eccentric genius who dominated the math landscape o Birthplace: Syracuse (the island of Sicily) o His father was thought to be an astronomer & Archimedes developed a life-long interest in the study of the heavens o Archimedes spent time in Egypt, studying at the great Library of Alexandria (Euclid’s base of operation) o He was trained in the Euclidean tradition, which was apparent in his writings o He left Alexandria & returned to Syracuse, but kept correspondence with the Greek world & scholars of Alexandria o During his time in the Nile Valley, he invented the Archimedean screw o This was used to transfer low-lying water into irrigation ditches o Archimedes’ invention “testifies to the dual nature of his genius: he could concern himself with practical, down-to- earth matters, or could delve into the most abstract, ethereal realms” o He had such an intense focus on his math works o Mundane concerns of life were often ignored o Plutarch (Greek Philosopher & Writer) wrote that Archimedes would: “…forget his food and neglect his person, to that degree that when he was occasionally carried by absolute violence to bathe or have his body anointed, he used to trace geometrical figures in the ashes of the fire, and diagrams in the oil on his body, being in a state of entire preoccupation, and, in the truest sense, divine possession with his love and delight in science.” o Famous ‘story’ portraying absent-minded Archimedes o Crown of King Hiero II of Syracuse was suspected to be substituted for a lesser alloy instead of gold o King challenged Archimedes to determine it’s composition o He solved it one day on a rare occasion he was bathing & ran running through the streets crying Eureka! Noted Accomplishments o Discovered the fundamental principles of hydristatics, with a treatise called On Floating Bodies, which developed these ideas o Advanced the science of optics o Pioneering work in mechanics-water pump, levers, pulleys, & compound pulleys o Created an array of weapons when Romeunder Marcellus’ leadership-attacked Syracuse Claw of Archimedes - “iron hand” o Archimedes helped Syracuse defeat the Romans o Marcellus’ grew great admiration for him & his work, Plutarch wrote about this o Archimedes died like he lived-”lost in thought with his beloved mathematics” What is true about all circles? o Modern mathematicians refer to this ratio as pi (Greeks did not yet use the symbol in this context) C = p or C = p d d o When Archimedes arrived on the math scene, the constant ratio of Circumference to Diameter of a Circle was known C1 C2 2p r = = =p d1 d2 2r Proposition 3 The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71. • In this proposition, he used inscribed and circumscribed polygons, beginning with a hexagon. • He did this because each side of the hexagon equaled the circle’s radius. • The initial Perimeter of the inscribed polygon was 3, which was a very rough estimate for pi. • How did he approach the Perimeter for the hexagon without √3 ? • Yes, this is simple for us know, but back then it was very difficult o He repeated the bisection process for a 24-gon, 48-gon, & finally a 96-gon o The inscribed 96-gon got him to 3 10/71 (lower bound) o The circumscribed 96-gon got him to 3 1/7 (upper bound) o During this whole process, it was necessary for Archimedes to approximate square roots and it still puzzles people how it managed to do it o With their decimal equivalents, Archimedes had pi nailed down to two decimal place accuracy, 3.14 3 1 < p < 310 7 71 Archimedes began with Euclid’s Proposition 3: The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71. o Then using this, he inscribed a regular hexagon because the radius of the circle would have the same length as each side of the hexagon C (circle) Pi (hexagon) 6r p= > = =3 d (circle) d (circle) 2r o He also circumscribed the circle with a regular hexagon C (circle) Pc (hexagon) 4r 3 p= < = = 2 3 » 3.464 d (circle) d (circle) 2r o Euclid’s Proposition 4.6 instructed how to inscribe a square in a circle o Through bisection of the square’s sides, we can create an inscribed regular octagon o Along that same process, we can get a 16-gon, 32-gon, etc… o This is the essence of Eudoxus’ famous method of exhaustion Starting with a Triangle Starting with a square Proposition 1 The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle. r r C Circumference = C Area = A Area = T o Preliminaries for Circular Area o THEOREM The area of the regular polygon is o PROOF Area (regular polygon) 1 1 1 = s1a + s2 a +... + sn a 2 2 2 1 1 = a(s1 + s2 +... + sn ) = aP 2 2 1 aP 2 Archimedes approached the Area of a Circle indirectly, using double reductio ad absurdum (reduction to absurdity/proof by contradiction) Proposition 1 r r Case 1: Suppose A>T C Circumference = C Area = A Area = T A - Area (inscribed polygon) < A - T T < Area (inscribed polygon) 1 rC = T > Area (inscribed polygon) = 1 aP 2 2 Contradiction! Proposition 1 Case 2: Suppose A < T Area (circumscribed polygon) - A < T - A Area (circumscribed polygon) < T Area (circumscribed polygon) = 1 aP > 1 rC = T 2 2 Contradiction! If both cases are false, then A = T! Circle Area Estimation Circle Dissection o Euclid’s Proposition 12.2 proved that two circular areas are to each other as the squares on their diameters, there is a constant k such that A 2 = k or A = kd 2 d 2 pr p = = =k 2 (2r) 4 http://knote.pbworks.com/w/page/52 418631/NCTM Archimedes o Archimedes’ treatise Measurement of a Circle revealed 3 propositions, 2 of which we looked at…but what about Proposition 2?? o According to the book, as a result of bad editing, copying, or translating, the 2nd proposition is out of place and unsatisfactory. o Other noted works of Archimedes: Geometry of spirals, conoids, and spheroids Finding the areas under curves, including the parabola On the Sphere and Cylinder-finding volumes and surface areas of 3D solids Proposition 13 The surface of any right circular cylinder excluding the bases is equal to a circle whose radius is a mean proportional between the side of the cylinder and the diameter of the base. x h x = = x 2 = 2rh x 2r LSA (cylinder) = A (circle) = p x 2 = 2p rh Proposition 33 The surface of any sphere is equal to four times the greatest circle in it. o Archimedes proved this in the same manner he used for Area of a Circle o He proved it impossible for the spherical surface to be more than and less than four times the area of its greatest circle, therefore it must be equal Surface area (sphere) = 4pr 2 Proposition 34 Any sphere is equal to four times the cone which has its base equal to the greatest circle in the sphere and it’s height equal to the radius of the sphere. Volume of a Cone = 1 pr 2h = 1 pr 2 r = 1 pr 3 3 3 3 Proposition 34 Any sphere is equal to four times the cone which has its base equal to the greatest circle in the sphere and it’s height equal to the radius of the sphere. Volume of a Sphere = 4 Volume (cone) = 4 pr 3 3 Volume of a Cylinder = pr 2 (2r) = 2pr 3 4 3 Vsphere 3 Vcyl r .2r 2 2r 3 4 Vsphere Vcyl r 3 3 3 2r 4 6 A sphere 4r 2 A cyl 2r.2r 2r 2 6r 2 2 A sphere 3 A cyl 4r 2 6r 2 2 3 If a sphere is inscribed in a cylinder, then the sphere is 2/3 of the cylinder in both surface area and volume! o Euclid’s Proposition 12.18 had proven that the volumes of two spheres are to each other as the cubes of their diameters (volume constant). 4 pr 3 = Volume (sphere) = mD3 = m(2r) 3 = 8mr 3 3 4 pr 3 = 8mr 3 3 m=p 6 EPILOGUE Derivation of π - Archimedes’ Approach o Archimedes started by inscribing a square inside a circle of radius 1, and approximating the circumference of the circle by the perimeter of the square. o To improve the approximation he then doubled the number of sides in the inscribed regular polygon, and so on, actually stopping at n = 96.