1. Chapter 2 Greek Geometry
The Deductive Method
The Regular Polyhedra
Ruler and Compass Construction
Conic Sections
Higher-degree curves
Biographical Notes: Euclid

2. 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)

3. Euclid’s “Elements” ( ≈ 300 BCE)
Postulates
Common Notions
Axioms
Principles of logic

4. 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

5. 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

6. 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

7. 2.2 The Regular Polyhedra
Definition A polyhedron which is bounded by a number of congruent polygonal faces, so that the same number of faces meet at each vertex, and in each face all the sides and angles are equal (i.e. faces are regular polygons) is called the regular polyhedron
Regular polygon: any number n > 2 of sides
Regular polyhedron: only five!

8. 5 Regular Polyhedra (The Platonic Solids)

9. Four Elements

10. Proof that there are only five
Consider polygons that can occur as faces
The sum of angles with common vertex must be 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 5
3π / 5
6 (and more)
≥ 2π / 3
Impossible

11. Construction of Icosahedron (by Luca Pacioli, 1509)
3 golden rectangles (with sides 1 and (1+√5) /2 ) A
AD = (1+√5) /2 )
BC = 1 B C
AB = AC = BC =1 D

12. Icosahedron and Dodecahedron

13. 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)

14. 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

15. Kepler’s diagram of the polyhedra

16. 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 √

17. Three famous problems V=2 V=1 Duplication of the cube
The impossibility of solving 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
α/3 α
A=π
A=π

18. 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)

19. 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))

20. 2.4 Conic Sections

21. Ellipse Parabola Hyperbola
In general, any second-degree equation represents a conic section or a pair of straight lines – Descartes (1637)

22. 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)

23. Drawing of conic sections
Generalized compass described by the Arab mathematician al-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

24. 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)

25. The Cissoid of Diocles ( ≈ 100 BCE)
Cubic curve with Cartesian equation
y2 (1+x) = (1-x)3
Diocles showed that the cissoid could be used to duplicate the cube

26. 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

28. The Epicycles of Ptolemy
Almagest – astronomical work of Claudius Ptolemy
Epicycles were candidates for the planetary orbits P

29. 2.6 Biographical Notes: Euclid
Taught in Alexandria (around 300 BCE)
“There is no royal road in geometry”
“Elements”
Euclid