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.
ehud AT math.huji.ac.il
The language of instruction will be Hebrew, but the textbooks and other instructional material
will be in English.
The principal textbook will be:
In the first part of the class, it will be used in conjunction with:
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.
Euclid's Geometry; intrinsic notions of measure of length, angle, area; construction of a regular pentagon.
Connections to the algebra of ordered fields Descarte's theorem on constructibility.
Hilberts' axioms; connections to logic. Rigid motions, geometry and group theory.
Hyperbolic geometry via quadratic forms, and connections to linear algebra.
- 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
- 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
The homework will count for up to 15% of the grade, on a "magen" basis.
Due in class on the date indicated.
- 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
- Explain what is wrong with Example 3.1 in Hartshorne.
- Read the Propositions of Euclid, Book II. Translate the statements
of Book II into modern algebraic language, and explain the
- Explain the proofs of Euclid III 37. (You will need a number of earlier propositions.)
- (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.)
April 16 assignment (pdf file, updated March 26,
11:40. This HW will be counted double.)
- Read Euclid IV 1, 2,5,10,11. Explain a proof of IV 11, using IV 10. (Either Euclid's or another.)
- 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.
- 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.
Graded HW's for the March 19, March 22 sets can be found outside of Einstein 104,
starting tomorrow (March 27).
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.
The April 16 assignment will be due at the end of the strike.
Note that Excercises 0.4 and 0.5 are precisely the content of Theorem 5 of Hilbert's book
(link at top of page), or of Hartshorne's 7.1. You may answer the excercises either
by reading one of these sources, or on your own.
Here is some reading material, and the
June 4 homework set.
and (partial) solutions. (Updated, July 7).
Thanks to Lior, Rima and Tamar for corrections.)
There will be no class on Thursday, June 7. Instead there will be an additional
class on Thursday, May 31, 12-13, in 24 Szprinzak. (The 8-9 class will take place
as usual in 27 Szprinzak.)
synopsis of May 31 classes + homework due June 11 .
Final version will be posted on June 4.
Thanks (+extra credit) for corrections sent before then.
Homeword due June 21.
Notes and corrections: The hint for 0.4 should read: f(DE)=AB (thanks to Lior.)
Homework due July 5.
(Updated with some answers on July 10 and July 14. Thanks to Ayala and Margalit for
This homework will be discussed rather
thoroughly in class.
We agreed to meet on Monday, July 9 at 12:00. Szprinzak 29.
The final exam will take place on July 15. You are strongly encouraged to attend moed aleph!
The oldest extant copy of Euclid is in the Bodleian library, from the year 888. Here is
Book I, Proposition 1