## MATH 371 Assignments and Project Spring, 2011

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?

1/20

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
Due:1/26
M&I p5:1-8,11
2 1/25 Euclid. Prop 1-3
The Pythagorean Theorem
.
The Pythagorean Theorem
1.2
1/27 More Euclid
Constructions.
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
Isometries
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.
coordinates/
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
6.
2/22 More Isometries: classification   More on Isometries
2/24  .... Finish Classification of Planar Isometries. M&I: 2.1,2.2

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
M&I:1.6:1-12,17,18
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

10
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
11
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
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
Problem Set 5
13
quiz #2 4/14
4/12Conic Sections.
Pascal and More Duality

Projective transformations. Perspectivities and Projectivities.
4/14Sections
M&I:
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.

4/21Projectivities. Transformations of lines with homogeneous coordinates.
Conics
Pascal's Theorem ?
More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates.
4.10, 5.4, 2.4
4.11
Due :4/27
M&I:4.5:2;
4.6:7,8,9; 4.7:4,7
4.10: 4,5
,7,9,10
[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:

• Proposal Format: The proposal should be typed (neatly hand written proposals are acceptable).
• Contents: The content of your proposal should describe, explain or otherwise demonstrate what your project is as you currently envision it. It should also indicate how you will go about completing the project.

• Below are some specific suggestions on features your proposal description might include:
• Title: Include a name( or list of possible names ) for your project.
• Introduction: (Your topic's core idea.) You should explain the idea of your project. Remember that the Introduction is the first place where the reader hears about your idea. You should also explain how the Proposal is organized in the introduction.
• Form and result: Indicate your vision of the final project's form(s), that is, the appearance of the FINAL PRODUCT. What will your project look like in its final ideal form? Note that all forms must include some written explanatory  component.
• Variations: (optional) Since this is a preliminary proposal, indicate some of the possible variations of both substance and form. It might be useful to distinguish the ideal from what may be a minimal project in both substance and form, and perhaps to see the project in stages from minimal to ideal, just in case you run into practical or time problems.
• References and Tools: List references and tools (books, journals, software, people, etc.) that are relevant to your project and that you might use. If you don't have any specific references yet, then indicate the kind of references you might use and where you will find them.
• Methods- Timeline and Task Delegation (for partnerships): Who will do what? When will they do it? If your project has definite parts or subdivisions, then indicate target dates for the completion of each stage.

• For partnerships:This project is a collective effort and should reflect the work and effort of all. Indicate when and where you will meet outside of class and how often. When possible, estimate the number of hours you are allocating to each task.
• Record keeping: Indicate how you will keep track of the progress of your project and the time spent by each individual participant on the project's work.

Results of Brainstorming and other suggestions from previous courses :)

 Tiling patterns - tesselation 3d tiling MC Escher perspective Curves: conics, etc. optical illusions knots fractals Origami Kaleidescope Symmetry The coloring problem Patterns in dance and other performance arts Flatland sequel (4d) Maps Juggling structural Rigidity dimension Polyhedra bridgemaking (architecture) Models (3d puzzles) paper mache or clay mobiles sculpture A play - movie build three dimensional shapes power point performance website

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}.