Squaring the Circle Sydney Ricks, Colin Byun, and Sekou Koram Introduction Proof: Rope Method Interesting facts • Given: A unit circle It is possible to square the circle with the addition of a rope • Goal: Construct a square • A square within a triangle within a larger circle began • Start with a circle with radius 1 with the same area 1 to be used in the 17th century to represent alchemy • Measure 4 of the circumference of the circle and the philosopher’s stone. Notice: We are attempting from the point T where T is a point on the to construct a square with circle. 1 휋 휋 퐶 = 2휋 4 2 side length 휋 • Since , of 2π is . So 2 is the length of the quarter of our circle from point T. • Often used in literature to denote the impossibility of Anaxagoras: something. Greek philosopher was the • Philosophically and spiritually, to understand the first mathematician who 휋 meaning of life, to be whole, complete, and free. • Extend the “rope” of the L = along the attempted to square the 2 circle in the diameter of the circle. (vertically) This 5th century BCE. [4] creates a new segment TU. [6] James Gregory: [1] A Scottish mathematician applied the • Construct a horizontal line, from U to a point Further Questions idea of sequences and on the edge of the circle, V. This new line UV convergence to prove must be perpendicular to TU. • Is it possible to Pentagon the Circle? Or any other there was no plane shapes? construction for squaring • What exactly are constructible numbers? [7] the circle. [4] • Connect points V and T to create a right Citation triangle. • Then construct a square from the new [1] Beyer, C. (n.d.). Squaring the circle is a geometric impossibility and an alchemy symbol. Retrieved April 29, Impossible? 2021, from https://www.learnreligions.com/squaring-the-circle-96039 segment VT. This segment VT = 휋 is the Lindemann-Weierstrass Theorem: [2] Bourne, Murray. “Squaring the Circle Rope Method.” Intmathcom RSS,Interactive Mathmatics, side of a square with an area equal to the 2017, www.intmath.com/plane-analytic-geometry/squaring-the- circle.php. • 휋 is a transcendental number and, circle. [3 ]HH, Patriarch Sir Godfrey Gregg D.Div. “SQUARING THE CIRCLE.” THE MYSTICAL COURT, THE MYSTICAL therefore, not a constructible COURT, 14 Oct. 2017, mysticalcourt.com/2017/10/14/squaring-the-circle/. number. [5] [4] O'Connor, J J, and E F Robertson. “Squaring the Circle.” Maths History, 1999, mathshistory.st- andrews.ac.uk/HistTopics/Squaring_the_circle/. • Its property of being irrational and • In the right triangle CUV we have: [5] Pierce, Rod. "Transcendental Numbers" Math Is Fun. Ed. Rod Pierce. 26 Nov 2020. 27 Apr 2021 not being a root to polynomials with 휋 휋