Geometry - 80637. Spring 2006

Mondays 29 Szprinzak, 12:00 - 13:45; Thursdays 27 Szprinzak 8-8:45.

Office hours: Thursday 9:00-10:00, Mathematics 104.
e-mail: ehud AT math.huji.ac.il

• Hartshorne, Robin Geometry: Euclid and beyond. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2000. xii+526 pp. ISBN: 0-387-98650-2

• Euclid, The Elements; translated from the text of Heiberg by T.L. Heath Available in Math. library, in Harman, and in the National Library (two copies, not for loan.) Also,
• A hypertext version by David Joyce
• Chapter 2 of Hartshorne's book contains a version of Hilbert's rigorous re-axiomatization of Euclid. An English translation of Hilbert's book is available from Project Gutenberg: The Foundations of Geometry .
  We will only use the axioms and theorems relating to plane geometry.
  
• Closely related material can also be found in the book: Geometry, open university course draft (in Hebrew), mathematics library number 30Un. Notes edited by Shmuel Berger, of lectures by Shlomo Sternberg.

Synopsis of class material.

February 26

Introduction to Euclid.
Book I: Axioms. Basic notions. Definitions.

March 8 (Shahar Mozes)

Euclid I 1,3,4. Continuity axioms (I 1). Modern approach to Euclid I 4.

March 12

Euclid I 16, 22,23. Theory of parallels (I 27); theory of area. (I 35,37.) Review.
Hartshorne, I 1-3, and Euclid, I 1-37 covered to this point.

March 15

Euclid I 47. Euclid's proof; comparison to possible Pythagorean proof by similar triangles (cf. Heath.) Discussion of difference in meaning of statement. Example 3.1 from Hartshorne.

March 19

Euclid II: geometric algebra. II.11 and quadratic equations. Euclid III: circles. III 32,36,37. Angle from a circle point outside an arc, to a segment of the arc, does not depend on point. Criteria for tangency in terms of area.

March 22

Euclid IV 10: construction of angle of 180/5 degrees.

March 26

Descarte's equivalence of algebra and geometry (Hartshorne's Theorem 13.2)

May 24

Hilbert's Axioms: Incidence planes, betweenness.

May 28

Hilbert's Axioms: Betweenness. The crossbar theorem (taken as an axiom.) Congruence axioms. Affine transformations, Pythagorean and Euclidean fields. Theorem: In a Hilbert plane, Complements to equal angles are equal.

May 31 (8-9,12-13).

See notes in homework for June 11.

June 4.

Defined rigid motions. Defined directed rays (= a ray, together with a choice of side of the line containing the ray.) Theorem: in a Hilbert plane, given any two directed rays, there exists a unique rigid motion taking one to the other. Given the two directed rays d,d', we defined a function f and showed any rigid motion taking d to d' must coincide with f. We also showed that f preserves distances, and sends an angle ABC to a congruent angle A'B'C', provided one of the points ABC is the initial point of they ray. Using the converse to the supplementary angles lemma, we showed that lines and betweenness are preserved by f.

June 11,June 14.

End or proof of the theorem on rigid motions. (Sharp transitivity on directed rays.) Discussion of converse: deducing the congruence axioms from the sharp trasitivity (see June 21 HW set, and this supplementary remark. )

June 18,21,25,28.

We used rigid motions to prove side-angle-side, side-side-side, angle-angle-side, justifying Euclid's terminology of "placing one triangle on another." Partial ordering on angles. Corollaries; in particular all right angles are congruent. Addition of angles when the sum is less than two right angles. Angles-with-rotation-numbers (zaviot sibuv) were defined as pairs consisting of an integer and an angle less than two right angles (or zero); addition defined. Saccheri quadrangles ABCD defined. (AC,BD congruent segments, perpendicular to AB and on the same side.) The angles at C,D shown to be congruent. Given a point P with C*P*D, and Q on AB with PQ perpendicular to AB, we showed that the angle at C is acute iff PQ < AC. Lemmas lead up to classification of Hilbert planes as Euclidean (parallel axiom holds), hyperbolic type (sum of angles in any triangle is < two right angles) or elliptic type (sum of angles in any triangle is > two right angles.) (Hartshorne, section 34 up to 34.7).

July 2 , July 5

Hyperbollic geometry via a symmetric form on $\Rr^3$ of type (--+); see HW.

Homework

March 19

Hartshorne 1.2,1.6-1.10; 2.1, 2.2-2.7
1.2 is a reading excercise: read the statements of the Propositions of Euclid, Book I (e.g. in above link); and understand the proofs of Propositions 4,5,8,15,26,27,29,30,32 in Book I.

March 22

1. Explain what is wrong with Example 3.1 in Hartshorne.
2. Read the Propositions of Euclid, Book II. Translate the statements of Book II into modern algebraic language, and explain the differences.
3. Explain the proofs of Euclid III 37. (You will need a number of earlier propositions.)
4. (3.3) Given a triangle ABC and a segment DE, construct a rectangle with side DE and with equal content to ABC.

March 26 (Should be done by March 26, but can be handed in together with the next assignment.)

1. Read Euclid IV 1, 2,5,10,11. Explain a proof of IV 11, using IV 10. (Either Euclid's or another.)
2. Let a be a positive real number. Let C be a circle on the real plane, passing through the points (-1,0) and (a,0), and with center on the x-axis. Find the points of intersection of the circle with the y-axis.
3. Let F be a field. Define a point to be an ordered pair (a,b) with a,b in F. Define a line to be the solution set to an equation ax+by+c=0, where a,b,c are in F, and a,b are not both zero. Show that the three incidence axioms below are valid.

April 16 assignment (pdf file, updated March 26, 11:40. This HW will be counted double.)
Graded HW's for the March 19, March 22 sets can be found outside of Einstein 104, starting tomorrow (March 27).

Hilbert's axioms

I. Incidence Axioms.

I.1: Given two distinct points there exists a unique line containing both.

I.2: Any line contains at least two points.

I.3: There exist three non-colinear points.