Mathematics 271, Fall Semester 2002
Welcome to the Math 271 home page. It is here that you will find a complete listing of homework assignments, test schedules, and other pertinent information about the course.
The class meets Mondays, Wednesdays, and
Fridays from 1:40 until 2:30. The first class meets on Wednesday, September
4, and the last class meets on Wednesday, December 11.
Hot Links
The handout that was distributed at the first class meeting:
The following web sites are useful if you are looking for definitions of mathematical terms, such as "field", "vector space," etc.:
HOMEWORK
Page Exercises Assigned for
24 14
9/06/02
29 8,9,10
9/09/02
29 7,12
9/11/02
41 13
9/13/02
44 9
9/13/02
Projective
Geometry Notes 9/13/02
52 3,6,11,16
9/18/02
special see below
9/23/02
81 6,7
9/23/02
85 1,2,3
9/25/02
85
6 -- 10 9/27/02
Reading: Section 3.4 of the text
92 8,11,12,14
10/02/02
103 1,7,8,11,12,13
10/07/02
106 23,24
10/11/02
327 7
10/11/02
121 1,2,6,7,10,12,22
10/14/02
172 12 -- 16
137 4,8,10,12,16,20
10/25/02 Also: show how to decompose a rotation as a product
of reflections. Same for translation.
160 5,8,9,15,16,18,35,36,38
11/11/02
(answer to 16: 175,220)
Assigned reading: "Euler line": pp173 - 175 (stop at the dingbat);
Exercises 6,7,9,10. Due 11/18/02.
285 4,9,12. Also: Draw the inverses of the equilateral triangle inscribed in the unit circle with respect to the circles with radii 1/2, 1 and 2 and centers at the origin. Hint: remember that the inverse of a line segment is either a circular arc or another line segment. Thus, by finding the inverse of the endpoints and the midpoint of a line segment, you have three points, and can thus draw the circular arc that they determine.
412 6,9,10,11,12
12/9/02
Read "Inaugural
lecture" by Juan Carlos Alvarez
WHAT
HAPPENED IN CLASS
9/4/02 Introduction ot synthetic geometry. Role of undefined terms and how Euclid failed to understand the need for them. Example: Fano geometry (See p. 21 of the text for the axioms)
9/6/02 Axiom systems for geometry. A synthetic geometry is an axiomatic system in which the undefined terms include "point", "line", and "on". Fano geometry models, including (Z2)3. Incidence geometries are those that have among their axioms four basic postulates, which are listed on page 25 of the text. Projective, Euclidean, and hyperbolic (non-Euclidean) geometries are distinguished by parallel lines. In a projective geometry, there are no parallel lines. In Euclidean geometry, there is a unique line parallel to a given line l that passes through a given point p not on l. Dual axiom systems and the solution of Ex. 14. For details, see the Fano notes.
9/9/02 Discussion of homework problems involving affine geometries (those that have the parallel postulate of Euclid or equivalent (often Playfair's version is used).
9/11/02 We talked about finite affine geometries
and projective planes. For a brief synopsis, see the
Projective
Geometry Notes
9/13/02 Hilbert's axioms: Connection and Order
9/16/02 Hilbert's order axioms; equivalence relations; separation of the plane by a line.
9/18/02 Proof that the axioms of connection and order imply that if P, Q, and R are points that do not lie on a straight line and if m is a line that is not on any of these points, then m cannot meet all three sides of the triangle PQR. The proof is as follows: suppose that m meets PQ at X, QR in Y, and PR in Z. Then one of these intersection points is between the other two; we will say X -- Y -- Z. Then the line QR meets only the side XZ of the triangle PXZ. This cannot be, by the axiom of Pasch. We also covered the axioms of congruence, noting that one axiom, III-4, is misstated in the text (but correctly stated in the Appendix). It was noted that the SAS theorem of Euclid (his proposition 4) is taken as an axiom by Hilbert. Euclid's proof assumes that one can "superpose" one triangle on top of another, which is an assumption not covered by his axioms. See Euclid-prop 4. The geometry of Felix Klein (his so-called Erlanger Programm) is based on this idea of rigid motion of the plane, and is a valid alternative to Hilbert. Finally I introduced the hyperbolic plane: this is the part of the x-y plane above the x-axis. Thus points have coordinates (x,y) with y > 0. Lines are vertical lines x = constant and circles (x-a)2 + y2 = r2 with centers on the x-axis.
Exercises (due 9/23/02) 1. Show how to construct the line in the hyperbolic plane determined by two given points, using ruler and compass.
2. Try to construct a rectangle in the hyperbolic plane.
Reference for today: Hilbert, D., Grundlagen der Geometrie. English translation by Anderson: QA 681.H6
9/20/02 Discussion of SMSG axioms, which accomplish what Hilbert set out to do with the two continuity axioms by a set of postulates that assert each line can be placed in 1-1 correspondence with the real numbers, and extablish measures of length and angle. The axioms of order are replaced by one axiom, that a the complement of a line in a plane can be partitioned into two equivalneces classes (the sides of the line) that are convex; that is, any line segment whose ends are on one side of a line is wholly on that side. We can prove the axiom of Pasch holds in the SMSG setting, and also that the Crossbar Theorem holds.
9/23/02 The isosceles triangle theorem, the exterior angle theorem, and its corollary, the alternate interior angle theorem, which establishes that, in neutral geometry, parallel to a given line and on a given point there is at least one line. We established the ASA and AAS criteria for congruence of triangles. There's a test, covering up to section 3.3 of the text, on Monday, 9/30/02. Click here to see a sample.
9/25/02 Inverse of the Isosceles triangle theorem (in a triangle, the longer edge is opposite the greater angle); the triangle inequality, and the hinge theorem. The Saccheri-Legendre theorem states that in neutral geometry, the degree measures of the interior angles of a triangle must have a sum of no more than 180. An equivalent statement is that the measure of an exterior angle is greater than or equal to the sum of the measures of the two remote interior angles. Contrast this with the exterior angle theorem (can you explain why the exterior angle theorem can be restated as "the sum of the degree measures of any two angles of a triangle must be less than 180?")
9/27/02 The sum of the radian measures of the angles
of a triangle on the surface of a sphere (the edges must be arcs of great
circles) is equal to A/r2 + p,
where
A
is the area of the triangle and r is the radius of the sphere. This
was demostrated by peeling an orange. A lune is a region on a sphere
that has as its boundary two great circles. A triangle on a sphere can
be regarded as the intersection of 3 lunes. The area of a lune with vertex
angle q is equal to 2qr2.
If
the angles of the triangle are a, b, and
g, then the total area covered by the 3 lunes is half the area of
the sphere, or 2p. Adding the areas of these
lunes, and remembering that the triangle is covered 3 times (there are
no other points on the sphere covered more than once), we have
2(a + b + g)r2
- 2A = 2pr2
and the formula for the angle sum quoted above follows by simple
algebra.
10/2/02-10/4/02. If there is a rectangle, then all triangles have angle sums of 180 degrees. We discussed Sacceri and Lambert quadrilaterals, midlines of Saccheri quadrilaterals.
10/7/02. Equivalence and triangulation of polygons. A triangulation of a polygon P is a collection of non-overlapping triangles that exactly covers P. Two polygons P and Q are equivalent if there are triangulations TP and TQ of P and Q, respectively, such that the triangles in TP can be placed in 1-1 correspondence with the triangles in TQ in such a way that corresponding triangles are congruent to each other. For example, a triangle is equivalent to its associated Saccheri quadrilateral (see Exercise 12, page 105, of the text). True to its name, "equivalence" is an equivalence relation. The only difficult thing to verify is transitivity. We need to say that a triangulation TQ" is a refinement of a triangulation TQ of a polygon Q if every triangle in TQ" is contained in or equal to some triangle in TQ. One can show that given two triangulations TQ and TQ ' of a given polygon, there exists a triangulation TQ" that is a refinement of both. Thus if P ~ Q and Q ~ R it is possible to find traingulations TP and TQ showing the first equivalence, and TQ' and TR showing the second. We find a common refinement TQ" of TQ and TQ'; that refinement induces refinements TP" of TP and TR" of TR so that the triangles of TP" and TQ" can be put into congruent pairs. Similarly TQ" and TR" can be put into congruent pairs. Composing these 1-1 correspondences, we see that P ~ R.
10/9/02 Area of polygons in hyperbolic geometry. Consider a triangle DABC, with angles a,b, and g. The defect of DABC is defined to be p - a - b - g. Given a triangulation TP of a polygon P, define the defect of TP to be the sum of the defects of the triangles that comprise TP. It is possible to show that the defect of any refinement of TP will be the same as that of TP itself, and it follows from this --- because any two triangulations must have a common refinement --- that we can define the defect of a polygon P to be the defect of any triangulation of P, since all triangulations will yield the same result. This defect satisfies postulates 17 - 19 of SMSG, and thus is a candidate for a way to describe the area of a polygon in hyperbolic geometry.
In Euclidean geometry, the defect isn't interesting, because the defect of every polygon is 0. Thus, we use SMSG 20 to determine area. In hyperbolic geometry, SMSG 20 is meaningless, for it postulates the area of a rectangle, and rectangles don't exist in this context. Thus, in hyperbolic geometry, we define the area of a polygon to be its defect. Thus, all triangles have area < p, all quadrilaterals have area < 2p, and so on.
10/11/02 We proved that the existence of a rectangle implies that Euclid's Postulate 5 is true. Then we moved on to Euclidean geometry, and derived a consequence of the congruence of opposite sides of a parallelogram (theorem 4.2.5 on p. 117 of the text).
10/14/02 Centers of a triangle: The concurrence of various lines give points that can be considered as "centers" of a triangle. The centroid is the center of gravity of a triangle. Any line through the centroid divides the triangle into two polygons (usually a quadrilateral and a triangle) of equal area, although we will discuss area later. The centroid is the point of concurrence of the medians. The perpendicular bisectors of the sides are also concurrent, they meet at the circumcenter --- the point that is equidistant from the three vertices. The angle bisectors are concurrent at the incenter, which is the center of the circle that is tangent to each of the sides. All proofs depend heavily on the parallel postulate, leaving open a question: what happens in the hyperbolic case?
10/16/02 Another center is the orthocenter, the point where the altitudes of the triangle are concurrent. The three vertices and the orthocenter form a quadrupe of points with the following property: Any one of the points is the orthocenter of the triangle whose vertices are the other three.
Euclid did not address area explicitly. Figures that we would say are equivalent (meaning that they have triangulations consisting of triangles that are congruent to one another), Euclid called "equal." We will discuss area briefly on Friday.
10/18/02 We followed Euclid's proof of the equivalence of triangles on the same base, with third vertices on a line parallel to that base, and the famous Theorem of Pythagoras, which you should read in the Heath version of Eucild and/or on the web at Euclid-I.47.
10/21/02 Transformation Geometry in a nutshell.
An isometry is a mapping T taking points to points, and lines to
lines, preserving the incidence relation, such that any segment AB
is congruent to T(AB). There are four kinds of isometries
of the plane: reflections, rotations, translations, and glide reflections.
Dilations are a type of transformation that are not isometries, and
with the isometries generate a group of transformations of the plane called
similarities.
Two
polygons are similar if the one can be mapped onto the other by a similarity.
We covered the results in the text (sect. 4.4) on similar triangles as
well, and I recommended the following article aas interesting reading:
Review
of Robin Hartshorne's book: Geometry: Euclid and Beyond.
10/23/02 We reviewed what you need to know for the test, and postponed the test until Monday in response to a request from a student. We then moved on to circles, and proved a theorem: if A, B, and C are distinct points on a circle with center O, then the measure of the angle ABC is equal to half the measure of the arc with endpoints A and C, and not containing B. As a corollary: DABC is a right triangle with hypotenuse AC if and only if the circumcenter of DABC is the midpoint of AC.
10/30/02 - 11/4/02 Theorems about circles, etc. (book, sect. 4.5)
11/6/02 Isometries of the plane. There are four kinds: reflections, translations, rotations, and glide reflections. Every isometry can be decomposed as a product (under composition of functions) of at most three reflections. We also discussed similarities, which can be formed by composing dilations (homotheties in the book) with isometries.
11/8/02 Inversion in a circle. The inverse of a point P with respect to a circle C with center O and radius r is the point P' on the ray OP such that OP OP' = r2. This topic is covered in section 5.5 of the text. Inversion is a transformation of the extended plane (a point at infinity is introduced to serve as the inverse of the center of a circle). Although inversion is not an isometry of the Euclidean plane, it is relevant because some isometries of the hyperbolic plane are inversions.
11/11/02 - 11/18/02 Transformations and metrics in hyperbolic geometry. Distance is measured by the cross ratio. Generally, if P,Q,R, and S are points in the Euclidean plane and XY denotes the distance between points X and Y, the cross ratio of the points is [P,Q;R,S] = (PR)(QS)(QR)-1(PS)-1. Now suppose that X and Y are points in the Poincare upper half plane, and let R and L be the ideal points where the semicircle that represents the line XY meets the x-axis. If the X and Y are on a vertical line, then L is the point at infinity. The distance from X to Y in the hyperbolic plane is d(X,Y) = |ln([X,Y;R,L])|. Notice that if L is the point at infinity then YL cancels with XL in the cross ratio; then d(X,Y) = |ln(XR)-ln(YR)|.
Linear fractional transformations of the complex plane are given
by a formula f(z) = (az + b)/(cz+d),
where (a b) and (c d) are the rows of a nonsingular matrix.
Notice that the composition of two such funcrtions is a linear fractional
transformation corresponding to the product of the matrices. Linear fractional
transformations take the complex line onto itself. The image of a circle
under a linear fractional transformation is always a circle ( we take a
line to be a circle of infinite radius, of course). A linear fractional
transformation corresponding to a matrix with real entries preserves the
real line and sends the upper half plane either onto itself or onto the
lower half plane, depending on the sign of its determinant. An isometry
of the upper half plane consists of either (1) a linear fractional transformation
of positive determinant, or (2) the complex conjugation of a linear fractional
transformation of negative determinant. Inversion in a circle with center
on the real axis is an isometry of type (2).
11//25 - 12/6/02 Introduction to projective geometry. We have covered everything up to the book's proof of Desargues' theorem, and will cover projective transformations next. You should read the material in the website mentioned in the homework assignment above.
TEST
SCHEDULE
Date
Tentative syllabus
9/30/02
Chs. 1,2 and 3 through 3.3 (Note change of day}
10/28/02
Sections 3.4 - 4.4; 4.6 (Note change of day!)
Test from Fall
2000
11/20/02
12/18/02, 11:00 - 1:00 Includes a comprehensive
final.