Chapter 2

Greek Geometry

• The Deductive Method

• The Regular Polyhedra

• Ruler and Compass Construction

• Conic Sections

• Higher-degree curves

• Biographical Notes: Euclid

2.1 The Deductive Method

• The most evident possible statements:Axioms

• 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)

Euclid’s “Elements”( ≈ 300 BCE)

Postulates Common Notions

Axioms Principles of logic

Postulates

1. To draw a straight line from any point to any point

2. To produce a finite straight line continuously in a straight line

3. To describe a circle with any centre and distance

4. That all right angles are equal to one another

5. 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

Common Notions

1. Things which are equal to the same thing are also equal to one another

2. If equals be added to equals, the wholes are equal

3. If equals be subtracted from equals, the remainders are equal

4. Things which coincide with one another are equal to one another

5. The whole is grater than the part

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

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 > 2 of sides

Regular polyhedron: only five!

5 Regular Polyhedra (The Platonic Solids)

Four Elements

ProofConsider polygons that can occur as faces

The sum of angles with common vertex mustbe less than 2 π = 360o

Number of sides n

Angle Max. number of faces

3 π / 3 5(3,4, or 5)

4 π / 2 3

5 3π / 5 3

6(and more)

≥ 2π / 3 Impossible

Construction of Icosahedron(by Luca Pacioli, 1509)

3 golden rectangles (with sides 1 and (1+√5) /2 )

B C

A

D

AB = AC = BC =1

AD = (1+√5) /2 )

BC = 1

Icosahedron and Dodecahedron

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)

Kepler’s diagram of the polyhedra

• Johannes Kepler (1571 – 1630) – great astronomer

• Three Laws of planetary motion

• Kepler’s theory of planetary distances (based on regular polyhedra)

• The theory was ruined when Uranus was discovered in 1781

Kepler’s diagram of the polyhedra

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 √

Three famous problems

• Duplication of the cube

• Trisection of the angle

• Squaring the circle

• 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)

V=1 V=2

α α/3

A=π A=π

Equivalent form

• Starting from the unit length, it is impossible to construct– 3√2 (duplication of the cube)– π (squaring the circle)– sin 20o (trisection of the angle α = 60o)

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 if

n = 2mp1p2…pk, 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 h=0,1,2,3,4 (3, 5, 17, 257, and 65537))

2.4 Conic Sections

12

2

2

2

b

y

a

x1

2

2

2

2

b

y

a

x2axy

Ellipse Parabola Hyperbola

In general, any second-degree equation represents a conicsection or a pair of straight lines – Descartes (1637)

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)

212

1

:1 hyperbola with 2

1 parabola ofon intersecti heConsider t

32

2

xxx

xy

xy

Generalized compass described by the Arab mathematicianal-Kuji (around 1000 CE)

Drawing of conic sections

Application of conic sections

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

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)

The Cissoid of Diocles ( ≈ 100 BCE)

Cubic curve with Cartesian equation

y2 (1+x) = (1-x)3

Diocles showed that the cissoid could be usedto duplicate the cube

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

The Epicycles of Ptolemy

• Almagest – astronomical work of Claudius Ptolemy

• Epicycles were candidates for the planetary orbits

P

2.6 Biographical Notes: Euclid

• Taught in Alexandria(around 300 BCE)

• “There is no royal road in geometry”

• “Elements”

Euclid