Chapter 2 Greek Geometry

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Chapter 2Greek Geometry

• The Deductive Method
• The Regular Polyhedra
• Ruler and Compass Construction
• Conic Sections
• Higher-degree curves
• Biographical Notes Euclid

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2.1 The Deductive Method

• The most evident possible statementsAxioms
• New statements (theorems, propositions etc.) can
  be derived from axioms and statements already
  established statements using evident principles
  of logic

Originator of the method Thales (624 547 BCE)
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Euclids Elements( 300 BCE)
Postulates
Common Notions
Axioms
Principles of logic
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Postulates

• To draw a straight line from any point to any
  point
• To produce a finite straight line continuously in
  a straight line
• To describe a circle with any centre and distance
• That all right angles are equal to one another
• That if a straight line falling on two straight
  lines make the interior angles on the same side
  less than two right angles, the two straight
  lines, if produced indefinitely, meet on that
  side on which are the angles less than the two
  right angles

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Common Notions

• Things which are equal to the same thing are also
  equal to one another
• If equals be added to equals, the wholes are
  equal
• If equals be subtracted from equals, the
  remainders are equal
• Things which coincide with one another are equal
  to one another
• The whole is grater than the part

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Remarks

• The intention was to deduce geometric
  propositions from visually evident statements
  (postulates) using evident principles of logic
  (the common notions)
• Euclid often made use of visually plausible
  assumptions that are not among the postulates
• 17th century development of analytic geometry
  (Descartes)
• 5th postulate the parallel axiom
• 19th century non-Euclidean geometries

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2.2 The Regular Polyhedra
Definition A polyhedron which is bounded bya
number of congruent polygonal faces,so that the
same number of faces meet at eachvertex, and in
each face all the sides and anglesare equal
(i.e. faces are regular polygons) is calledthe
regular polyhedron
Regular polygon any number n gt 2 of sides
Regular polyhedron only five!
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5 Regular Polyhedra (The Platonic Solids)
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Four Elements
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Proof
Consider polygons that can occur as faces The sum
of angles with common vertex mustbe less than 2
p 360o
Number of sides n Angle Max. number of faces
3 p / 3 5(3,4, or 5)
4 p / 2 3
5 3p / 5 3
6(and more) 2p / 3 Impossible
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Construction of Icosahedron(by Luca Pacioli,
1509)
3 golden rectangles (with sides 1 and (1v5) /2 )
AD (1v5) /2 )
A
BC 1
B
C
AB AC BC 1
D
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Icosahedron and Dodecahedron
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Spheres

• For every regular polyhedron there exist the
  sphere which passes through all its vertices (its
  radius is called circumradius) and the sphere
  which touches all its faces (its radius is called
  inradius)

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Keplers diagram of the polyhedra

• Johannes Kepler (1571 1630) great astronomer
• Three Laws of planetary motion
• Keplers theory of planetary distances (based on
  regular polyhedra)
• The theory was ruined when Uranus was discovered
  in 1781

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Keplers diagram of the polyhedra
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2.3 Ruler and Compass Constructions

• The ruler and compass elementary operations
• given two points, construct the line through them
• given two points, construct the circle centered
  at one point passing through the other point
• given two lines, two circles, or a line and a
  circle, construct their intersection points
• A point P is called constructible from points P1,
  P2, , Pn, if P can be obtained from these
  points with a finite sequence of elementary
  operations
• One can show that the points constructible from
  P1, P2, , Pn are precisely the points which
  have coordinates in the set of numbers generated
  from the coordinates of P1, P2, , Pn by the
  operations , -, , / , and v

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Three famous problems
V2
V1

• The impossibility ofsolving of these problems by
  ruler and compass constructions was proved only
  in 19th century
• Wantzel (1837) (impossiblity of the duplication
  of the cube and trisection of the angle)
• Lindemann (1882) (impossibility of squaring of
  the circle)

• Duplication of the cube
• Trisection of the angle
• Squaring the circle

Ap
Ap
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Equivalent form

• Starting from the unit length, it is impossible
  to construct
• 3v2 (duplication of the cube)
• p (squaring the circle)
• sin 20o (trisection of the angle a 60o)

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Open problem

• Which regular n-gons are constructible?
• Equivalent problem circle division
• Gauss (1796) 17-gon is constructible and a
  regular n-gon is constructible if and only ifn
  2mp1p2pk, where each pi is a prime of the form
  22h 1 (Fermat prime)
• What are these primes?
• Are there infinitely many of these primes? (the
  only known are for h0,1,2,3,4 (3, 5, 17, 257,
  and 65537))

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2.4 Conic Sections
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Hyperbola
Parabola
Ellipse
In general, any second-degree equation represents
a conicsection or a pair of straight lines
Descartes (1637)
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Menaechmus (4th century BCE)

• Invention of conic sections
• Used conic sections to solve the problem of
  duplication of the cube(finding the intersection
  of parabola with hyperbola)

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Drawing of conic sections
Generalized compass described by the Arab
mathematicianal-Kuji (around 1000 CE)
Application of conic sections

• Kepler (1609) the orbits of planets are
  ellipses
• Newton (1687) explained this fact by his law of
  gravitation

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2.5 Higher-degree Curves

• There was no systematic theory of higher-degree
  curves in Greek mathematics
• Greeks studied many interesting special cases
• The Cissoid of Diocles ( 100 BCE)
• The Spiric Sections of Perseus ( 150 BCE)
• The Epicycles of Ptolemy ( 140 CE)

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The Cissoid of Diocles ( 100 BCE)
Cubic curve with Cartesian equation y2 (1x)
(1-x)3
Diocles showed that the cissoid could be usedto
duplicate the cube
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The Spiric Sections of Perseus

• Torus (or Spira) one of the surfaces studied by
  Greeks
• Spiric section the section of torus by plane
  parallel to the axis of the torus
• Curves of degree 4

Cassini oval
Lemniscate of Bernoulli
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The Epicycles of Ptolemy

• Almagest astronomical work of Claudius Ptolemy
• Epicycles were candidates for the planetary
  orbits

P
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2.6 Biographical Notes Euclid

• Taught in Alexandria(around 300 BCE)
• There is no royal road in geometry
• Elements

Euclid