Squaring the Circle. A Classical Problem

Transcription

1 Squaring the Circle A Classical Problem

2 The Rules Find a square which has the same area as a circle Limited to using only a ruler and compass Only a finite amount of steps may be used Image from OneMomsBattle.com, Lucy K. Wright: Squaring the Circle : Adjusting to Life after My Father Drove Away

3 Rhind Papyrus Image from Mathground.net, Rhind Mathematical Papyrus

4 Rhind Papyrus Egyptian papyrus dated to 1850 BC Perhaps based on a work going as far back as 3400 BC Used 8/9 the length of the circle s diameter as the square s base Using r=1, A= Approximated this as About 0.6% different

5 History - Greece A very popular (but never solved) Greek problem. One of three geometric problems of antiquity Claims that Anaxagoras was first to attempt Antiphon the Sophist attempted inscribing regular polygons within the circle take a limit fills the circle with infinite sided polygon Creation of inscribed octagon from inscribed square Image from The Ancient Tradition of Geometric Problems (p. 27)

6 History - Greece (cont.) Lune of Hippocrates The shaded region has the same area as triangle ABC Like many other attempts, only gave false hope for solving the issue of squaring the circle Archimedes solved, but used spirals Not constructible with ruler and compass Image from Wikimedia Commons File:Hipocrat_arcs.svg

7 History - (Much) Later Attempts James Gregory: false proof of its impossibility (i.e. falsely proved pi is transcendental) De Morgan: written in a book on paradoxes He suggested St. Vitus to be the patron saint of circle squarers, referencing St. Vitus s dance which was wild and leaping, leading to mass hysteria

8 Impossible? Algebraic solution New problem: How to get π? Close approximations have been viable for a long time Although close, none are exact Is it possible with Euclidean methods? No, because π is transcendental Image from Wikipedia Commons, File:Squaring_the_circle.svg

9 Constructible numbers A real number c is constructible iff a line segment of length c can be constructed from a line segment of length 1 Only addition, subtraction, multiplication, division, and square-rooting are allowed All rational numbers are constructible If a is constructible, then a ½ is constructible

10 Constructible number (cont.) Image from MTL.Math.UIUC.edu, Step 5 Constructible Real Numbers

11 Transcendental Numbers A non-constructible, non-algebraic number A number that cannot be made via: addition/subtraction multiplication/division exponents/roots e is another transcendental number If π is transcendental then it is non-constructible Proved transcendental by Ferdinand von Lindemann

12 Proof of π is Transcendental Frederick von Lindemann s proof Proof by contradiction e has been proven to be transcendental Euler s identity e πi = -1 is transcendental (false) i is algebraic so π must be transcendental Implies impossibility of exactly squaring the circle Still possible to provide arbitrarily close approximations to values of π

13 Ramanujan s Approximations 1913: Used 355/113 for π Differs in 7 th decimal place). Note.- If the area of the circle be 140,000 square miles, then [the side of the square] is greater than the true length by about an inch. (O Connor) 1914: used ( /22) ¼ Differs in the 9 th decimal place. Accuracy: For a circle of diameter 8000 miles about 50 million), the side is a fraction of an inch off.

14 Bibliography O'Connor, J., & Robertson, E. (1999, April 1). Squaring the circle. Retrieved March 8, 2015, from html Weisstein, Eric W. "Constructible Number." From MathWorld--A Wolfram Web Resource. "Step 5: Constructible Real Numbers." Step 5: Constructible Real Numbers. University of Illinois at Urbana-Champaign. Web. 8 Mar < edu/node/49>. Knorr, W. (1993). The ancient tradition of geometric problems. New York: Dover. Retrieved March 8, 2015 from files/knorr_ch_2.pdf Pampena, Simon [Numberphile]. Transcendental Numbers [Video file] Retrieved February 28, 2015, from