Download Modern geometry 2012.8.27 - 9. 5 Introduction to Geometry Ancient

Modern geometry
2012.8.27 - 9. 5
Introduction to Geometry
Ancient
geometry
- Greek geometry ; Pythagoras, Euclid, Archimedes
(Long) period in Arab
17, 18C
19C
classical curves, Projective geometry
Gauss,
Non-Euclidean geometry
19C Differential geometry , Algebraic geometry,
Topology
Euclid's plane geometry
Euclid
edited
than Elements.
Elements – except Bible, no book was more used, published or studied
More than 1000 editions published. Old
longtime textbooks of
geometry.
Euclid’s geometry consists of undefined terms, 5 axioms and 5 postulates from which
he deduced 465 statements.
5 Axioms
An axiom
is
an assumption
common to all studies;
a postulate is particular to a
study.
1. Things that equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things that coincide with one another equal one another.
5. The whole is greater than the part.
The five postulates of Euclid
1. A straight line may be drawn from any one point to any other point.
2. A finite straight line may be produced to any length in a straight line.
3. A circle may be described with any center
at any distance from that center.
4. All right angles are equal.
5. If a straight line meets two other lines, so as to make the two interior angles on one side of it
together less than two right angles, the other straight lines will meet if produced on that side on
which the angles are less than two right angles.
★ In a typical geometry argument, (e.g. drawing a perpendicular line) more postulates
and definitions are necessary.
Hilbert's
Axioms
The undefined terms are: point, line, incidence, betweenness, congruence (≅).
Then, line segments, angles, and triangles
straight
may each be defined in terms of points and
lines.
I.
Incidence
II.
Order
III. Congruence
IV. Continuity
V. Parallels
(You can find the statements in Chapter 24 of the textbook.)
For 2000 years, many attempts were made to prove the parallel postulate using Euclid's
first four postulates.
The fifth postulate isn't self-evident unlike the other postulates.
Equivalent statements to the parallel postulate
Playfair’s postulate ; there exist a unique parallel line which pass through a point not on a
given line.
-
Alternate angle theorem below is needed to prove that the 5th imply the existence of a
parallel line in the Playfair's, while the uniqueness comes from the Separation axiom (II.5).
The sum of
3 angles of a triangle is 180 degrees.
There exist two similar triangles which are not congruent.
Some definitions
A segment
AB is a set consisting of A, B and all the points between
A and B.
An angle consists of a point A and two rays AB and AC.
Two points p and q are in the opposite side of a line l if the line pq and l meets at a
point, say r, and r is between p and q.
Alternate angle theorem:
Two lines whose alternate angles are congruent
never meet.
Proof) Consider two lines m and n which meet a given l at points A and B respectively.
Suppose that m and n meet at a point C. There is a point D such that it lies in the
opposite side of C with respect to l and that BC and AD are congruent
Triangles ABC and ABD are congruent, so angle ABD and angle BAC are congruent. D is
on n. As m and n are distinct two lines, this is a contradiction.
Theorem ;
For a point P not on a line l, there exist a perpendicular line to l through P.
proof)
1. There exist two points A, B on l.
I-3
2. There exists a point X such that it lies in the opposite side of P with respect to l
and that angle PAB and angle XAB are congruent. CA-1
3. There exist a point Q on the half-line AX so that AP and AQ are congruent. CS-1
4.
The line PQ meets l at a point R.
5. By SAS congruence,
6. Angle PRA
and
angle PRA and angle QRA are congruent.
angle QRA are right angles.
*Definition of ‘side’ needs separation postulates.
Importance of proving
Parallel postulate
1. Newton based his calculus on Euclid geometry.
2. Kant in “critique of pure reason” argued that
truth by reason and
a man can perceive(인식) an absolute
an example is the Euclid geometry.
So, if Euclid geometry were not self-evident, then it could mean a big trouble.
Establishment of non-Euclidean geometry
The 19th century finally saw mathematicians exploring those alternatives and discovering
the logically consistent geometries which result.
In 1829, Lobachevsky published an account of acute geometry.
In 1831, János Bolyai included, in a book by his father, an appendix describing acute
geometry, which, doubtlessly, he had developed independently of Lobachevsky.
Carl Friedrich Gauss had actually studied the problem before that, but he did not publish
any of his results.
The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré
into hyperbolic geometry and spherical geometry .
The independence of the parallel postulate from Euclid's other axioms was finally proved
by Beltrami in 1868.
Erlanger Program
In 1872, Klein tried to classify and characterize geometries on the basis of projective
geometry and group theory.
The problems of nineteenth century geometry
Was there one 'geometry' or many? Since Euclid, geometry had meant the geometry of
Euclidean space of two dimensions or of three dimensions .
In the first half of the 19th century there had been several developments complicating the
picture. Mathematical applications required geometry of four or more dimensions;
Non-Euclidean geometry had been born.
Klein proposed an idea that all these new geometries are just special cases of the
projective geometry, as already developed by Poncelet, Möbius, Cayley and others
With every geometry, Klein associated an underlying group of symmetries. The hierarchy
of geometries is thus mathematically represented as a hierarchy of these groups, and
hierarchy of their invariants.
For example, lengths, angles and areas are preserved with respect to the Euclidean group
of symmetries.