MATH 371 Assignments and Project Spring, 2009

Back to Martin Flashman's Home Page :)   Last updated: 11/24/08

TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE through Moodle.
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Flatland By E. Abbott, Dover.

Tentative assignmentsand topics for classes. 11/24/08 Blue cells Subject to Revisions
Week Monday Wednesday Friday Reading/Videos for the week. Problems 
Due on Wednesday 
of the next week
1  No Class
1.1 Beginnings 


What is Geometry? 
Starting to look at Euclid. Prop 1.

M&I:1.1, 1.2
E:I Def'ns, etc. p153-5; 
Prop. 1-12,22,23,47 
A:.Complete in three weeks
M&I p5:1-8,11
2 1-26 The Pythagorean Theorem 

More on Pythagoras and Euclid.
Prop 2
Begin  disssection theory,
M&I 1.2, 1.3 
E: I Prop. 16, 27-32, 35-45.
Due: 2-4 
M&I: p10:1,2,5,10,11-13
Prove:The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
[ HELP! Proof outline for the midpoint proposition.]
32/2Convexity defined.
Equidecomposable Polygons
1.1 Def'ns- Objects 

 2/4 1.2 Constructions 1.4 Continuity2/6  1.3 Geometry: Constructions and numbers
M&I 1.3,1.4 
E: III Prop. 1-3, 14-18, 20, 21, 10 
Here’s looking at Euclid
F. Sect. 11, 25, 31
Watch Equidecomposable  Polygons
Due: 2-13 (changed 2-13!)
M&Ip11: 16-19, 24, *27  Problem  Set 1

4 2/9 More on continuity and rational points.
Start Inversion
Orthogonal Circles
2/13More on Cantor
Similar triangles
Odds and ends.
M&I:1.5, 1.6, 2.1 
E: V def'ns 1-7;VI: prop 1&2 
F. Sect. 32
Due 2-18
M&I: p17: 5, 8-11
p23: 9,10 (analytic proofs)
5 2/16 Odds and ends    2/18 Odds and ends
2/20Start: Transformations - Isometries
Transformations - Isometries.
M&I: 2.1, 2.2
E: V def'ns 1-7;VI: prop 1&2 
E:IV Prop. 3-5
Isometries (Video # 2576 in Library) .
Due: 2-25

Problem Set 2
6 2/23More Isometries: Coordinates and Transformations
2/25  More on Isometries
2/27  ....
M&I: 2.1,2.2 Due 3-4
Problem Set 3
(Isos Tri)   [3 Points for every distinct correct proof of any of these problems.]
3/2Finish Classification of Planar Isometries.

Begin Affine Geometry

3/4 Isometries and symmetries
3/6 Proportion and Similarity
M&I: 2.2 again;3.1,3.2, 3.5 Quiz #1 on 3/11
See Sample quiz on Moodle.

3/9 More on Similarity
3/11 Inversion and Affine Geometry (planar coordinates) the Affine Line. 3/13 Constuctions with Straight edge and compass. M&I:3.6, 3.4,3.7
View video (in Library #4376 ) on "Central similarities" from the Geometry Film Series. (10 minutes)
View video (in Library #209 cass.2) on similarity (How big is too big? "scale and form")  "On Size and Shape"  from the For All Practical Purposes Series. (about 30 minutes)

9 Spring break 3/16
No Class
3/18 No Class 3/20 No Class

10 3/23 Visualizing the affine plane. Seeing the infinite.Affine geometry- Homogeneous coordinates and visualizing the affine plane.
More on Homogeneous coordinates for the plane.A first look at a "Projective plane."
Axioms, consistency, completeness and models.
3/27Begin Synthetic Geometry [Finite] Algebraic-projective geometry: Points and lines.
Spatial and  Planar
Axioms for 7 point geometry.
Begin Synthetic Projective Geometry
A non-euclidean universe.
M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1
Due :  4/1
 M&I: 3.5: 1,3,4,5,10,11
3.6: 3,7-15 
3.7: 1,4,7,10,13
11 3/30 Homogeneous Coordinates with Z2 and Z3
More on Finite Synthetic Geometry and models.
Proof of Desargues' Theorem
4/1 -Projective Geometry -Visual/algebraic and Synthetic..Axioms 1-6 Projective Planes. . 4/3. More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry.
M&I:4.1, 4.2, 4.3, 2.4
12 4/6 Proofs of some basic projective geometric facts. 4/8 Triangle Coincidences (Perpendicular Bisectors- the circumcenter)
Applications of Projective  Geometry Postulates.1-6
4/10 Quiz #2
Desargue's Theorem and Duality

Due 4/15
M&I: 4.1:7,15,16; 
Prove P6 for RP(2); 
4.2: 2,3, Supp:1 
4.3: 1-6, Supp:1,5,6
13 4/13 Conic Sections.
 More Duality
4/15  More Postulates-
Sections (more duality)
Complete quadrangles
Postulate 9.  
4/17 Perspectivities and Projectivities.
Projective transformations.
Pascal's Theorem ?
More on coordinates and transformations.
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm)
Watch video: "Conics" [VIDEO #628].
Watch video: For All Practical Purposes on conicsFAPP on conics[HSU Video #209- 3. Stand-up conic (Conic sections) (ca. 30 min.)].
WatchVideo: Orthogonal Projection [Video #4223]

Due : 4/22
Problem Set 4
14 4/20Projectivities. Perspective 4/22Transformations of lines with homogeneous coordinates 4/24 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates. 4.10, 5.4, 2.4 
Due: 4/29
M&I: 4.5:2  ;4.6: 7,8,9;   4.7: 4,7 
4.10: 4,5,9,10
[Prove P9 for RP(2),optional]
Final exam available  4/30.
4/27Harmonic sets: uniqueness and construction of coordinates for a Projective Line, Plane, Space.
4/29More on Transformations, Coordinates and Harmonic sets. 5/1 5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2
Final projects are due for review Tuesday, May 5th.
5/4 Projective generation of conics
 Pascal's and Brianchon's Theorem.
5/6 Matrices for familiar Planar Projective Transformations.
Inversion angles, circles and lines?
5/8 The Big picture in Summary.


M&I: 4.10:1,3,6,7;
 5.4:1-8,10;  5.5: 2,3,7

M&I: 4.10:1,3,6,7; 
5.1:5; 5.4:1-8,10;  5.5: 2,3,7

Conics revisited.
Inversion and the final exam.

Quiz #3

 . Student Presentations

Project Proposal Guidelines and Suggestions
The Project. Each student will participate in a course project either  as an individual as a part of a team. Each team will have at most three. These projects will be designed with assistance from myself . The quality of the project will be used for determining letter grades above the C level. Ideas for projects will be discussed during the third week.
Preliminary Project Proposals should be submitted for first review by 5 p.m., February 10th.
A progress report on the project is due March 25th.
Final projects are due for review Tuesday, May 5th. (These will be graded Honors/Cr/NCr.)

A Project Fair will be organized for displays and presentations during the last day of class. Details will be discussed later.

Guidelines for Preliminary Proposals:

Results of Brainstorming and other suggestions from previous courses :)

Tiling patterns - tesselation
3d tiling
MC Escher
Curves: conics, etc.
optical illusions

The coloring problem
Patterns in dance and other performance arts
Flatland sequel (4d)

structural Rigidity

bridgemaking (architecture)

Models (3d puzzles) paper mache or clay

A play - movie
build three dimensional shapes
power point

Problem Set 1

DEFINITIONS: A figure C is called convex if for any two points in the figure, the line segment determined by those two points is also contained in the figure.
That is, if A is a point of C and B is a point of C then the line segment AB is a subset of C.

If F and G are figures then F int G  is { X : X in F and X  in G }.
F int G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then  int A = { X : for every figure F in the family A, X  is in F }.
int A is called the intersection of the family A.

1. Prove: If F and G are convex figures , then F int G is a convex figure.

2. Give a counterexample for the converse of problem 1.

3. Prove: If A is a family of convex figures, then int A is a convex figure.

4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.] 

Problem Set 2

1. Suppose n is a natural number. Given P0 and P1 , prove by induction that you can construct with straight edge and compass (SEC) a point P sqrt(n)   which will correspond to the number  sqrt(n) on a Euclidean line.

2. Suppose we are given P0, P1, and P a where P a corresponds to the real number a>0. Give a construction with SEC of a point Psqrt(a) which will correspond to the number  sqrt(a) on a Euclidean line.

3. Given points P0, P1, Px, and Py on a Euclidean line corresponding to the real numbers x>0 and y>0, give constructions with SEC for the following points.

a) P x + y b) P x - y c) P x *y d) P 1/x
4. Construct with SEC on a Euclidean line:   sqrt(5)/sqrt(3)  + sqrt(sqrt(6)) .

5. Suppose that d(A,B) = d(A',B') and that l is the perpendicular bisector of the line segment  AA'. Let B'' be the reflection of B across l, i.e., B''= Tl(B). Prove that if  B' is not equal to B''  then A' lies on the perpendicular bisector of the line segment B''B'.

Problem Set 3

1.  Prove: Two of the medians of an isosceles triangle are congruent.

2.  Prove: If two of the medians of a triangle are congruent then the triangle is isosceles.

3.  Prove: The angle bisectors of congruent angles of an isosceles triangle are congruent.

4.  Prove:  If two of the angle bisectors of a triangle are congruent then the triangle is isosceles. 

Problem Set  4

1. Use an affine line with P0 , P1 , and Pinf given. Show a construction for P1/2 and P2/3.

2. Use an affine line with P0 , P1 , and Pinf   given. Suppose x > 1.
Show a construction for  Px2 and Px3 when Px is known.

3. D is a circle with center N tangent to a line l at the point O and C is a circle that passes through the N and is tangent to l at O as well.
Suppose P is on l and PN intersect C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ'  intersect l.

Prove: a) P and Q are inverses with respect to the circle D.
b) P' and Q' are inverses with respect to the circle D.
c) P and P' are inverses with respect to the circle with center at O and radius ON.

4. Suppose C is a circle with center O and D is a circle with O  an element of D.
Let I be the inversion transformation with respect to C.

Prove: There is a line l, where I(P) is an element of  l for all P  that are elements of  D -{O}.