MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle...
Transcript of MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle...
![Page 2: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/2.jpg)
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
![Page 3: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/3.jpg)
Lunes of Hippocrates
Find the shaded area of the lunes.
![Page 4: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/4.jpg)
Geometrical Calculations
What is the relationship between a and x?
Can you construct this to verify your solution?
![Page 5: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/5.jpg)
![Page 6: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/6.jpg)
Squaring the
Circle and
Other Shapes
Kevin Lord
![Page 7: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/7.jpg)
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.
Greek GeometryBefore 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.
![Page 8: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/8.jpg)
Quadrature Quadrature (or squaring) of a plane shape is the
constructions – using only compass and straight-
edge – of a square with the same area as the
original figure.
![Page 9: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/9.jpg)
Compass and Straight-Edge
• Perpendicular line
• Dropping a perpendicular from a point
• Bisecting a line
• Bisecting an angle
• Marking equidistant points
• Calculating
![Page 10: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/10.jpg)
Quadrature of a rectangle• Construct (or draw ) an “arbitrary”,
say 9cm x 4cm, rectangle - labelled LMNO
• Extend the line MN
![Page 11: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/11.jpg)
• Use a compass to mark off segment NG equal in
length to ON
• Find midpoint MG – point H
![Page 12: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/12.jpg)
• 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
![Page 13: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/13.jpg)
Can you prove that the areas are equal?
Use the triangle HPN to show that
area of NPQR = area LMNO
![Page 14: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/14.jpg)
ProofLet a, b, c be the lengths of triangle
HP, HN and PN.
Pythagoras theorem a2 = b2 + c2
a
b
c
Now NG = ON = a - b and MN = a + b.
Area (rectangle LMNO) = MN x ON = (a + b)(a - b)
= a2 - b2
= c2 = Area (square NPQR)
a - b
![Page 15: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/15.jpg)
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
A B
C
![Page 16: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/16.jpg)
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
A B
C
D
![Page 17: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/17.jpg)
Quadrature of a triangle
• Construct perpendicular through midpoint of CD
• Construct perpendiculars to base through A and
B to complete the rectangle
A B
C
D
![Page 18: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/18.jpg)
Quadrature of a curved shapeOne of the famous problems from antiquity was
how to construct a square with the same area as a
circle.
“squaring the circle”
![Page 19: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/19.jpg)
Squaring the Circle
![Page 20: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/20.jpg)
Hippocrates of Chios c.470-410 BCE
Hippocrates’ investigated the
quadrature of lunes.
Lune
a plane shape
bounded by two
circular arcs.
![Page 21: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/21.jpg)
Lunes of HippocratesFind the shaded area of the lunes.
10
8
![Page 22: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/22.jpg)
Quadrature of a lune
In general, show that the area of the lunes is
equal to the area of triangle ABC.
![Page 23: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/23.jpg)
Lunes of Hippocrates
![Page 24: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/24.jpg)
= =+
+ =
Proof by pictures
![Page 25: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/25.jpg)
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
![Page 26: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/26.jpg)
Proof by
pictures
![Page 27: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/27.jpg)
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.”
![Page 28: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/28.jpg)
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.
![Page 29: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/29.jpg)
Geometrical Calculations• What is the relationship between a and x?
• Can you construct this to verify your solution?
![Page 30: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/30.jpg)
Geometrical Calculations
p q
![Page 31: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/31.jpg)
Geometrical CalculationsConstruction
• 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
![Page 32: MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle Hippocrates’ work with lunes offered some hope that there may be a generalisation of](https://reader033.fdocuments.in/reader033/viewer/2022060414/5f12d1820811d320df306292/html5/thumbnails/32.jpg)
About MEI
• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students
and work with employers to enhance
mathematical skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources