MEI Conference Squaring the Circle and Other Shapes

Transcription

1 MEI Conference 2017 Squaring the Circle and Other Shapes Kevin Lord

2 Can you prove that the area of the square and the rectangle are equal? Use the triangle HPN to show that area of NPQR = area LMNO

3 Lunes of Hippocrates Find the shaded area of the lunes.

4 Geometrical Calculations What is the relationship between a and x? Can you construct this to verify your solution?

5

6 Squaring the Circle and Other Shapes Kevin Lord

7 Greek Geometry Before the Ancient Greeks, Babylonian and Egyptian mathematicians were able to calculate the areas of various plane shapes. These calculations had practical applications in working out land usage etc. and required measurement. The Greek geometers were more interested in finding a unifying system for finding the area of any plane shape. Area was considered as a property of the shape. A square figure, the most fundamental shape, was both equal to its area and could be defined by its area.

8 Quadrature Quadrature (or squaring) of a plane shape is the constructions using only compass and straightedge of a square with the same area as the original figure.

9 Compass and Straight-Edge Perpendicular line Dropping a perpendicular from a point Bisecting a line Bisecting an angle Marking equidistant points Calculating

10 Quadrature of a rectangle Construct (or draw ) an arbitrary, say 9cm x 4cm, rectangle - labelled LMNO Extend the line MN

11 Use a compass to mark off segment NG equal in length to ON Find midpoint MG point H

12 Using H as the centre, draw an arc through M and G. Extend line ON to intersect the arc at P NP is one side of the square

13 Can you prove that the areas are equal? Use the triangle HPN to show that area of NPQR = area LMNO

14 Proof Let a, b, c be the lengths of triangle HP, HN and PN. a c Pythagoras theorem a 2 = b 2 + c 2 b Now NG = ON = a - b and MN = a + b. a - b Area (rectangle LMNO) = MN x ON = (a + b)(a - b) = a 2 - b 2 = c 2 = Area (square NPQR)

15 Quadrature of a triangle How could you use the method for a rectangle to construct the square of equal area to the triangle? Describe the steps C A B

16 Quadrature of a triangle Construct (or draw) an arbitrary triangle, ABC Drop a perpendicular line from C to the base Find the midpoint of CD C A D B

17 Quadrature of a triangle Construct perpendicular through midpoint of CD Construct perpendiculars to base through A and B to complete the rectangle C A D B

18 Quadrature of a curved shape One of the famous problems from antiquity was how to construct a square with the same area as a circle. squaring the circle

19 Squaring the Circle

20 Hippocrates of Chios c BCE Hippocrates investigated the quadrature of lunes. Lune a plane shape bounded by two circular arcs.

21 Lunes of Hippocrates Find the shaded area of the lunes. 8 10

22 Quadrature of a lune In general, show that the area of the lunes is equal to the area of triangle ABC.

23 Lunes of Hippocrates

24 Proof by pictures + = = + =

25 Lunes and the Regular Hexagon If a regular hexagon is inscribed in a circle and six semicircles constructed on its sides, then the area of the hexagon equals the area of the six lunes plus the area of a circle whose diameter is equal in length to one of the sides of the hexagon. Hippocrates of Chios, ca. 440 B.C.E

26 Proof by pictures

27 Quadrature of the circle Hippocrates work with lunes offered some hope that there may be a generalisation of his method leading to squaring the circle. In 1882, the German mathematician Ferdinand Lindemann proved that the quadrature of the circle is impossible by proving that π is a transcendental number.

28 Indiana Pi Bill Indiana 1897 Edwin Goodwin proposed a bill to the State Assembly which included a solution to the problem of squaring the circle. The bill would have defined π = 3. 2 in Indiana. It was eventually thrown out.

29 Geometrical Calculations What is the relationship between a and x? Can you construct this to verify your solution?

30 Geometrical Calculations p q

31 Geometrical Calculations Construction Draw horizontal line across lower part of the page Mark points A, B and C (AB= 12cm, BC = 3cm) Construct perpendicular line through B Construct bisector for AC (Mark it O) Draw a semi-circle, centre O, radius AO Measure X

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