MATH 371 Assignments and Project Spring, 2011

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TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE at 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 assignments and topics for classes.4/3/10 Blue cells Subject to Revisions
Week Tuesday ?? Thursday Reading/Videos for the week. Problems 
Due on Wednesday 
of the next week
1 1/18 1.1 Beginnings
What is Geometry? 



Starting to look at Euclid. Prop 1-3.
Convexity defined

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 1/25 Euclid. Prop 1-3
The Pythagorean Theorem 
The Pythagorean Theorem
1/27 More Euclid
M&I 1.2, 1.3 
E: I Prop. 16, 27-32, 35-45
Watch online: Here’s looking at Euclid.
Due: 2/2
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.]
3 Reading Report due: 2/1 2/1 Equidecomposable Polygons Constructions 
1.1 Def'ns- Objects 
1.2 Constructions 1.3 Geometry: Constructions and numbers
1.4 Continuity 2/3   Breather:
Start: Transformations - Isometries
M&I 1.3,1.4 
E: III Prop. 1-3, 14-18, 20, 21, 10 
F. Sect. 11, 25, 31
Watch Equidecomposable  Polygons
Due: 2/9
M&I: p17:5, 8-11 
p11: 16-19, 24, *27  Problem  Set 1
4Preliminary Project Proposals  first review 5 p.m. Wednesday, February 9th. 2/8 More on continuity and rational points. Similar triangles
2/10 More on similar triangles and rational points.
Start Inversion.
Orthogonal Circles
M&I:1.5, 1.6, 2.1 
OPTIONAL[E: V def'ns 1-7;VI: prop 1&2 ]
F. [Sect. 32] pp 55-61on Axioms of Continuity

5 Reading Report due: 2/15 2/15 Coordinates and Proofs
Inversions and orthogonal circles. Odds and ends on continuity axiom
More on Cantor
Isometries / 2/17 Transformations - Isometries.
M&I: 2.1, 2.2
OPTIONAL [E: V def'ns 1-7;VI: prop 1&2 ]
E:IV Prop. 3-5
Isometries (Video # 2576 in Library) .
Due : 2/23 [Previously due 2/16]
M&I: p23: 9,10 (analytic proofs)
Problem Set 2
2/22 More Isometries: classification   More on Isometries
2/24  .... Finish Classification of Planar Isometries. M&I: 2.1,2.2
7 Reading Report due: 3/1

3/1 Isometries and symmetries

More on Similarity
3/Begin Affine Geometry
Proportion and Similarity
M&I: 2.2 again;3.1,3.2, 3.5
8Quiz #1 on Thursday 3/10 in class 3/8 3 Inversion and Affine Geometry (planar coordinates)
The Affine Line.
 Affine geometry- Homogeneous coordinates and visualizing the affine plane. 3/10 Visualizing the affine plane. Seeing the infinite.
More on Homogeneous coordinates for the plane.
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)
Due  3/23
Problem Set 3 (Isos Tri)   [3 Points for every distinct correct proof of any of these problems.]
9 Spring break 3/15 No Class  No Class 3/17 No Class

Reading Report due: 3/24
Project progress report
Thursday, March 25th
3/22 More on The affine Plane - A first look at a "Projective plane."
Axioms, consistency, completeness and models.
A non-euclidean universe.
 Begin Synthetic Geometry [Finite] Algebraic-projective geometry: Points and lines.
Spatial and  Planar

3/24Axioms for 7 point geometry.
Begin Synthetic Projective Geometry
M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1
Due : 3/30
 M&I: 3.5: 1,3,4,5,10,11
3.6: 3,7-15 
3.7: 1,4,7,10,13
Problem Set 4
3/29 Homogeneous Coordinates with Z2 and Z3
More on Finite Synthetic Geometry and models.
Proof of Desargues' Theorem
-Projective Geometry -Visual/algebraic and Synthetic..
3/31 No Class CC Day
M&I:4.1, 4.2, 4.3, 2.4
12 Reading Report due: 4/7 (changed 4-3) 4/5 Axioms 1-6 Projective Planes.
More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry.  Proofs of some basic projective geometric facts.
Applications of Projective  Geometry Postulates.1-6

4/7 Triangle Coincidences (Perpendicular Bisectors- the circumcenter)
Desargue's Theorem and Duality
M&I:4.1, 4.2, 4.3, 2.4 Due: 4/13
Prove P6 for RP(2); 
4.2: 2,3, Supp:1 
4.3: 1-6, Supp:1,5,6
Problem Set 5
quiz #2 4/14
4/12Conic Sections.
Pascal and More Duality
Projective transformations. Perspectivities and Projectivities. 
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm)

14 Reading Report due: 4/19 4/19 Perspective
More duality.
Graphs and duality.
Complete quadrangles Postulate 9. 

4/21Projectivities. Transformations of lines with homogeneous coordinates.
Pascal's Theorem ?
More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates.
4.10, 5.4, 2.4 
Due :4/27
4.6:7,8,9; 4.7:4,7 
4.10: 4,5
[Prove P9 for RP(2),optional]
15 Projects are due 5pm Thursday, 4/29
4/26More on coordinates and transformations. Projectivities in 3 space: Harmonic sets: uniqueness and construction of coordinates for a Projective Line, Plane, Space.
Projective generation of conics 
 More on Transformations, Coordinates and Harmonic sets. . 4/28Matrices for familiar Planar Projective Transformations.
5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2 Due: OPTIONAL!
M&I: 4.10:1,3,6; 
5.1:5; 5.4:1-8,10;  5.5: 2,3,7

16 Reading Report due: 5/3 5/3Conics revisited.
Inversion and the final exam.

Quiz #3
Inversion angles, circles and lines. 5/5 Pascal's and Brianchon's Theorem.
The Big picture in Summary. .
17 Final Exam Week.
Office Hours for Exam Week
MTWRF 8:15-10:00 and by appointment or chance

Student Presentations will be made Tuesday 5-10 3:00-4:50 Final Exam
DUE Friday, May 13, before 5 P.M.

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. Tuesday , February 9th.
A progress report on the project is due March 25th.
Final projects are due for review Friday, April 29th. (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.

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.

Problem Set  5

1. 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.

2. 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}.